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Dear Colleague,
It is our pleasure to welcome you to the (4th) 1997 International Meeting on Fully ThreeDimensional Image Reconstruction in Radiology and Nuclear Medicine, or 3D97.
The aim of this meeting is to bring together people actively researching problems related to
fully three-dimensional tomography in Radiology and Nuclear Medicine. To encourage
discussions, we have chosen the Nemacolin Woodlands Resort, in the Laurel Highlands
region of the Appalachian Mountains near Pittsburgh, Pennsylvania. Nemacolin has state
of the art meeting facilities and a gracious and relaxed setting with many amenities.
Many people have worked to make this meeting a success, but in particular we would like
to thank the tremendous effort of the local organizing committee: Ms. Kathie Antonetti, Dr.
Claude Comtat, and Ms. Ruth Hall. We also received valuable assistance from Forbes
Travel (5835 Forbes Avenue, Pittsburgh, PA 15217, Tel: 412-521-7037) and the
Nemacolin Woodlands Resort (PO. Box 188 Farmington, PA 15437, Tel: 800-422-2736)
:1
In addition, we would like to acknowledge the excellent reviewing provided by the
Scientific Committee:
LJ
Dale Bailey, Ph.D.
Harrison Barrett, Ph.D.
Michel Defrise, Ph.D.'
Lars Eriksson, Ph.D.
Pierre Grangeat, Ph.D.
Grant Gullberg, Ph.D.
Ronald Huesman, Ph.D.
Ronald J aszczak, Ph.D.
Hiroyuki Kudo, Ph.D.
Tom Lewellen, Ph.D.
Gerd Muehllehner, Ph.D.
Christopher Thompson D .Sc.
Hammersmith Hospital, London, England
University of Arizona, USA
Free University of Brussels, Brussels, Belgium
Karolinska Institute, Stockholm, Sweden
LETIICEA, Grenoble, France
University of Utah, USA
Lawrence Berkeley Laboratory, USA
Duke University, USA
University of Tsukuba, Tsukuba, Japan
University of Washington, USA
UGM Medical Systems, Philadelphia, USA
McGill University, Montreal, Canada.
We are pleased to acknowledge the generous support of this meeting from:
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ADAC International
CTI PET Systems
eV Products
GE Medical Systems
Hamamatsu Photonics
National Cancer Institute
Oxford Positron Systems
Picker
Siemens Medical Systems
UGM Medical Systems
U. S. DOE Office of Energy Research
"".---
Paul Kinahan
David Townsend
Conference Co-Chairs, 3D97
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11997 International Meeting on Fully 3D Image Reconstruction
Table of Contents
Letter from the Organizers
Table of contents
2
Scientific program
6
Papers
System Modeling and Spatial Sampling Techniques for Simplification of Transition
Matrix In 3D Electronically Collimated SPECr
12
A. C. Sauve, A. O. Hero, W. L. Rogers and N. H. Clinthorne
Application of Spherical Harmonics to Image Reconstruction for Compton Camera
16
R. Basko, G. L. Zeng and G. T. Gullberg
A Fourier Reblnnlng Algorithm Incorporating Spectral Transfer Efficiency
E. Tanaka and Y. Amo
Performance of the Fourier Aeblnning Algorithm for PET with Large Acceptance
Angles
S. Matej, J. S. Karp and R. M. Lewitt
High Resolution 3D Bayesian Image Reconstruction for microPET
J. Oi, R M. Leahy, E U. Mumcuoglu, S. R Cherry, A. Chatziioannou and T. H.
20
24
28
Farquhar
Reconstruction of Truncated Cone-Beam Projections using the Frequency-Distance
Relation
M. Defrise and F. Noo
Fast and Stable Cone-Beam Filtered Backprojection Method for Non-Planar Orbits
32
36
H. Kudo and T. Saito
Iterative and Analytical Reconstruction Algorithms for Varying Focal-Length ConeBeam Projections
40
L. G. Zeng and G. T. Gullberg
Practical Limits to High Helical Pitch, Cone-Beam Tomography
44
M. D. Silver
Exact Cone Beam CT with A Spiral Scan
48
K. C. Tam, S. Samarasekera and F. Sauer
Energy-Based Scatter Correction for 3-D PET: A Monte Carlo Study of "Best Possible"
Results
52
D. R. Haynor, R. L. Harrison and T. K. Lewellen
Algorithms for Calculating Detector Normalisation Coefficients in 3D PET
55
R. D. Badawi and P. K. Marsden
Normalization for 3D PET with a Low-Scatter Plane Source: Technique,
Implementation and Validation
T. R. Oakes, V. Sossi and T. J. Ruth
Axial Slice Width in 3D PET: Potential Improvement with Axial Interleaving
M. E. Daube-Witherspoon, S. L. Green and R. E. Carson
11997 International Meeting on Fully 3D Image Reconstruction.,
59
63
[]
[]
Implementation and Evaluation of Iterative Three-Dimensional Detector Response
Compensation in Converging Beam SPECT
E. C. Frey, S. Karimi, B. M. W. Tsui and G. T. Gullberg
Minimal Residual Cone-Beam Reconstruction with Attenuation Correction in SPECT
67
71
V. La and P. Grangeat
Simulation of Dual-Headed Coincidence Imaging using the SimSET Software Package
R. L. Harrison, S. D. Vannoy, W. L. Swan, M. S. Kaplan and T. K. Lewellen
Quantitative Chest SPECT in Three Dimensions: Validation by Experimental Phantom
Studies
75
77
Z. Liang, J. Li, J. Ye, J. Cheng and D. Harrington
[]
3D Reconstruction from Cone-Beam Data using Efficient Fourier Technique Combined
with a Special Interpolation Filter
M. Magnusson Seger
Iterative Reconstruction for Helical CT: a Simulation Study
J. Nuyts, P. Dupont, M. De frise, P. Suetens and L. Mortelmans
Iterative Reconstruction of Three-Dimensional Magnetic Resonance Images from
Boron Data
81
85
89
F. Rannou and J. Gregor
Adaptive Inverse Radon Transformer
9.3
A. F. Rodriguez, W. E. Blass, J. Missimer and F. Emert
l]
[1
The Effect of Activity Outside the Direct FOV on Countrate Performance and Scatter
Fraction in the ECAT EXACT3D
T. J. Spinks, M. Miller, D. Bailey, P. M. Blommfield and T. Jones
Binning List Mode Dual Head Coincidence Data into Parallel Projections
97
101
W. L. Swan, R. S. Miyaoka, S. D. Vannoy, R. L. Harrison, T. K. Lewellen and F. Jansen
Characteristics of an Iterative Reconstruction Based Method for Compensation of
Spatial Variant Collimator-Detector Response in SPECT
B. M. W. Tsui and E. C. Frey
An Exact 3D Reconstruction Algorithm for Brain SPECT Using a Parallel-Plus
Collimator
105
109
C.Wu
On Combination of Cone-Beam and Fan-Beam Projections in Solving a Linear System
of Equations
G. T. Gullberg and L. G. Zeng
Circular and Circle-and-Line Orbits for Conebeam X-ray Microtomography of Vascular
Networks
113
117
R. H. Johnson, H. Hu, S. T. Haworth, C. A. Dawson and J. H. Linehan
Kinetic Parameter Estimation from SPECT Cone-Beam Projection Measurements
121
R. H. Huesman, B. W. Reutter, L. G. Zeng and G. T. Gullberg
:-.1
t_
An Analytic Model of Pinhole Aperture Penetration for 3-0 Pinhole SPECT Image
Reconstruction
M. F. Smith and R. J. Jaszczak
Comparison of frequency-Distance-Relationship and Gaussian-Diffusion Based
Methods of Compensation for Nonstationary Spatial Resolution in SPECT Imaging
V. Kohli, M. King, S. Glick and T.-S. Pan
126
130
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11997 International Meeting on Fully 3D Image Reconstruction
31
Comparison of Scatter Compensation Methods In Fully 3D Iterative SPEer
Reconstruction: A Simulation Study
F. J. Beekman, C. Kamphuis and E. C. Frey
Inversion of the Radon Transform In Two and Three Dimensions using Orthogonal
Wavelet Channels
1 34
1 38
E. Clarkson
Towards Exact 3D-Reconstruction for Helical Cone-Beam Scanning of Long Objects.
A New Detector Arrangement and a New Compieteness Condition
P.-E. Dan/elsson, P. Edholm, J. Eriksson and M. Magnusson Seger
141
3D Efficient Parallel Sampling Perturbation In Tomography
145
L. Desbat
Estimation of Geometric Parameters for Cone Beam Geometry
Y.-L. Hsieh, L. G. Zeng and G. T. Gullberg
Simulation Studies of 3D Whole-Body PET Imaging
1 50
1 54
C. Comtat, P. E. K/nahan, T. Beyer, D. W. Townsend, M. Oefrise and C. Michel
Advantage of Algebraic Regularized Algorithms over Feldkamp Method in 3D Cone8eam Reconstruction
I. Laurette, j. Darcourt, L. Blanc-Feraud, P.-M. Koullbaly and M. Barlaud
A New Symmetrical Vertex Path for Exact Reconstruction in Cone-Beam CT
F. Noo, R. Clack, T. J. Roney and T. A. White
Fast Accurate iterative Reconstruction for Low-Statistics Positron Volume Imaging
A. J. Reader, K. Erlandsson, M. A. Flower and R. J. Ott
Design and Implementation Aspects of a 3D Reconstruction Algorithm for the JOlich
TierPET System
1 58
162
166
170
A. Terstegge, S. Weber, H. Herzog, H. W. MOiler-Gartner and H. Halling
3D-Reconstruction during Interventional Neurological Procedures
K. Wiesent, R Graumann, R Fahrig, D. W. Holdsworth, A. J. Fox, N. Nava.b and A.
1 74
Bani-Hashemi
Development of an Object-Oriented Monte Carlo Simulator for 3D
Tomography
Positron
1 76
H. Zaidi, A. Hermann Scheurer and C. Morel
Block-Iterative Techniques for Fast 40 Reconstruction Using A Priori Motion Models in
Gated Cardiac SPECr
1 80
D. S. Lalush and B. M. W. Tsui
SPECr Reconstruction - The
Jl'tO'
Model
184
D. L. Gunter
Strategies for Fast Implementation of Model-Based Scatter Compensation in Fully 3D
SPECr Image Reconstruction
D. J. Kadrmas, E. C. Frey and B. M. W. Tsui
3D Tomographic Reconstruction Using Geometrical Models
X. L. Batt/e, G. S. Cunningham and K. M. Hanson
Symmetry Properties of an Imaging System and Consistency Conditions in Image
E. Clarkson and H. Barrett
11997 International Meeting on Fully 3D Image Reconstruction
187
191
195
i~l
, J
Experience with Fully 3D PET and Implication for Future High Resolution 3D
Tomographs
199
D. L. Bailey, M. Miller, T. J. Spinks, P. M. Bloomfield, L. Livieratos, H. Young and T.
Jones
[]
Inter-Comparison of Four Reconstruction Techniques for Positron Volume Imaging
with Rotating Planar Detectors
A. J. Reader, D. Visvikis, R. J. Ott and M. A. Flower
A Fully-3D, Low Cost, PET Camera Using Hidac Detectors with Sub-Millimeter
Resolution for Imaging Small Animals
A. P. Jeavons, R. A. Chandler and C. A. Dettmar
201
205
Author Index
210
List of Participants
212
[]
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[]
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[]
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11997 International Meeting on Fully 3D Image Reconstruction
51
SCIENTIFIC PROGRAM
Wednesday, June 25
9:00-9:30
Welcome
Paul Kinahan and David Townsend, University of Pittsburgh
SESSION 1: GENERAL RECONSTRUCTION I,
CHAIR: DAVID TOWNSEND, PHD
9:30-10:00
System Modeling and Spatial Sampling Techniques for Simplification of
Transition Matrix in 3D Electronically Collimated SPECT
A. C. Sauve, A. O. Hero, W. L. Rogers and N. H. Clinthorne; University of
Michigan, U.S.A.
10:00-10:30
Application of Spherical Harmonics to Image Reconstruction for Compton
Camera
R. Basko, O. L. Zeng and G. T. Gullberg; University of Utah, U.S.A.
SESSION
2: PET RECONSTRUCTION~
CHAIR: MICHEL DEFRISE, PHD
11 :00-11 :30
A Fourier Rebinning Algorithm Incorporating Spectral Transfer Efficiency
E. Tanaka and Y. Amo; Hamamatsu Photonics K.K., Japan
11:30-12:00
Performance of the Fourier Rebinning Algorithm for PET with Large
Acceptance Angles
S. Matej, 1. S. Karp and R. M. Lewitt; University of Pennsylvania, U.S.A
12:00-12:30
High Resolution 3D Bayesian Image Reconstruction for microPET
1. Qi, R. M. Leahy, E. U. Mumcuoglu, S. R. Cherry, A. Chatziioannou and T. H.
Farquhar; University of Southern California, U.S.A.
SESSION 3: CONE BEAM RECON I,
CHAIR: HARRISON BARRETT, PHD
14:30-15:00
Reconstruction of Truncated Cone-Beam Projections using the FrequencyDistance Relation
M. Defrise and F. Noo; Free University of Brussels, Belgium
15:00-15:30
Fast and Stable Cone-Beam Filtered Backprojection Method for Non-Planar
Orbits
H. Kudo and T. Saito; University of Tsukuba, Japan
15:30-16:00
Iterative and Analytical Reconstruction Algorithms for Varying Focal-Length
Cone-Beam Projections
L. G. Zeng and G. T. Gullberg; University of Utah, U.S.A.
11997 International Meeting on Fully 3D Image Reconstruction
61
f~-1
SESSION 4: X-RAY CT,
t 1
\-1
CHAIR: RONALD HUESMAN, PHD
16:30-17:00
Practical Limits to High Helical Pitch, Cone-Beam Tomography
M. D. Silver; Bio-Imaging Research, Inc., U.S.A.
17:00-17:30
Exact Cone Beam CTwith A Spiral Scan
K. C. Tam, S. Samarasekera and F. Sauer; Siemens Corporate Research, Inc.,
U.S.A.
BREAKOUT SESSION,
CHAIR: DALE BAILEY, PHD
, I
17:30-18:30
An open discussion organized around a topic of interest
Thursday, June 26
SESSION 5: PET QUANTIFICATION,
r~
09:00-09:30
Energy-Based Scatter Correction for 3-D PET: A Monte Carlo Study of UBest
Possible" Results
D. R. Haynor, R. L. Harrison and T. K. Lewellen; University of Washington,
U.S.A.
dr';";"
09:30-10:00
Algorithms for Calculating Detector Normalisation Coefficients in 3D PET
R. D. Badawi and P. K. Marsden; Guy's and St. Thomas' Clinical PET Centre,
U.K.
10:00-10:30
Normalization for 3D PET with a Low-Scatter Plane Source: Technique,
Implementation and Validation
T. R. Oakes, V. Sossi and T. J. Ruth; University of British Columbia / TRIUMF
PET Centre, Canada
1
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o
CHAIR: PAUL KINAHAN, PHD
SESSION 6: POSTER SESSION I
11:00-12:30
[]
i
1
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o
Axial Slice Width in 3D PET: Potential Improvement with Axial Interleaving
M. E. Daube-Witherspoon, S. L. Green and R. E. Carson; National Institutes of
Health, U.S.A.
Implementation and Evaluation of Iterative Three-Dimensional Detector
Response Compensation in Converging Beam SPECT
E. C. Frey, S. Karimi, B. M. W. Tsui and G. T. Gullberg; The University of
North Carolina at Chapel Hill, U.S .A.
Minimal Residual Cone-Beam Reconstruction with Attenuation Correction in
SPECT
V. La and P. Grangeat; Labo~atoire d'Electronique de Technologie et
d'Instrumentation (Comissariat al'Energie Atomique - Technologies Avancees),
France
11997 International Meeting on Fully 3D Image Reconstruction
Simulation of DualNHeaded Coincidence Imaging using the SimSET Software
Package
R. L. Harrison, S. D. Vannoy, W. L. Swan, M. S. Kaplan and T. K. Lewellen;
University of Washington, U.S.A.
Quantitative Chest SPECT in Three Dimensions: Validation by Experimental
Phantom Studies
Z. Liang, 1. Li, J. Ye, 1. Cheng and D. Harrington; State University of New York
at Stony Brook, U.S.A.
3D Reconstruction from ConeMBeam Data using Efficient Fourier Technique
Combined with a Special Interpolation Filter
M. Magnusson Segel'; Linkoping University, Sweden
Iterative Reconstruction for Helical CT: a Simulation Study
J. Nuyts, P. Dupont, M. Defrise, P. Suetens and L. Mortelmans; Katholieke
Universiteit Leuven, Belgium
Iterative Reconstruction of ThreeMDimensional Magnetic Resonance Images
from Boron Data
F. RannOll and J. Gregor; The University of Tennessee, U.S.A.
Adaptive Inverse Radon Transformer
A. F. Rodriguez, W. E. Blass, J. Missimer and F. Emert, The University of
Tennessee, U.S.A.
The Effect of Activity Outside" the Direct FOV on Countrate Performance and
Scatter Fraction in the EeAT EXACT3D
T. J. Spinks, M. Miller, D. Bailey, P. M. Blommfield and T. Jones; Medical
Research Council, U.K.
Binning List Mode Dual Head Coincidence Data into Parallel Projections
W. L. Swan, R. S. Miyaoka, S. D. Vannoy, R. L. Harrison, T. K. Lewellen and
F. Jansen; University of Washington, U.S.A.
Characteristics of an Iterative Reconstruction Based Method for Compensation
of Spatial Variant CollimatorMDetector Response in SPECr
B. M. W. Tsui and E. C. Frey; The University of North Carolina at Chapel Hill,
U.S.A.
An Exact 3D Reconstruction Algorithm for Brain SPECT Using a ParallelMPlus
Collimator
C. Wu; Positron Corporation, U.S.A.
11997 International Meeting on Fully 3D Image Reconstruction
81
SESSION
r-1
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7: CONE BEAM RECONSTRUCTION II,
CHAIR: HIROYUKI KUDO, PHD
14:30-15:00
On Combination of Cone-Beam and Fan-Beam Projections in Solving a Linear
System of Equations
G. T. Gullberg and L. G. Zeng; University of Utah, U.S.A.
15:00-15:30
Circular and Circle-and-Line Orbits for Conebeam X-ray Microtomography of
Vascular Networks
R. H. Johnson, H. Hu, S. T. Haworth, C. A. Dawson and J. H. Linehan;
Marquette University, U.S.A.
15:30-16:00
Kinetic Parameter Estimation from SPECT Cone-Beam Projection
Measurements
R. H. Huesman, B. W. Reutter, L. G. Zeng and G. T. Gullberg; Lawrence
Berkeley National Laboratory, U.S.A.
SESSION 8: SPECT COLLIMATORS,
CHAIR: PIERRE GRANGEAT, PHD
16:30-17:00
An Analytic Model of Pinhole Aperture Penetration for 3-D Pinhole SPECT
Image Reconstruction
M. F. Smith and R. J. Jaszczak; Duke University Medical Center, U.S.A.
17:00-17:30
Comparison of frequency-Distance-Relationship and Gaussian-Diffusion
Based Methods of Compensation for Nonstationary Spatial Resolution in
SPECr Imaging
V. Kohli, M. King, S. Glick and T.-S. Pan; University of Massachusetts Medical
School, U.S.A.
--1
[j
BREAKOUT SESSION,
17:30-18:30
CHAIR: DAVID TOWNSEND, PHD
An open discussion organized around a topic of interest
Friday, June 27
(I
Cl
SESSION 9: INVITED ~PEAKER
9:00-10:30
Functional Neuroimaging applications in cognitive research
Jonathan Cohen, MD, PhD, Department of Psychiatry, Carnegie Mellon
University, Pittsburgh, U.S.A.
[I
SESSION 10: POSTER SESSION II
r- ,
11:00-12:30
l_1
Comparison of Scatter Compensation Methods in Fully 3D Iterative SPECT
Reconstruction: A Simulation Study
F. J. Beekman, C. Kamphuis and E. C. Frey; Utrecht University Hospital, The
Netherlands
U
11997 International Meeting on Fully 3D Image Reconstruction
Inversion of the Radon Transform in Two and Three Dinzensions using
Orthogonal Wavelet Channels
E. Clarkson; University of Arizona, U.S.A.
Towards Exact 3D-Reconstruction for I-Ielical Cone-Beam Scanning of Long
Objects. A New Detector Arrangement and a New Completeness Condition
P.-E. Danielsson, P. Edhohn, 1. Eriksson and M. Magnusson Seger; Linkoping
University, Sweden
3D Efficient Parallel Sampling Perturbation in Tomography
L. Desbat; TIMC-IMAG, France
Estimation of Geometric Parameters for Cone Beam Geometry
Y.-L. Hsieh, L. G. Zeng and G. T. Gullberg; University of Utah, U.S.A.
Simulation Studies of 3D Whole-Body PET Imaging
C. Comtat, P. E. Kinahan, T. Beyer, D. W. Townsend, M. Defrise and C. Michel;
University of Pittsburgh Medical Center, U.S.A.
Advantage of Algebraic Regularized Algorithms over Feldkamp Method in 3D
Cone-Beam Reconstruction
I. Laurette, J. Darcourt, L. Blanc-Feraud, P.-M. Koulibaly and M. Barlaud;
University of Nice-Sophia Antipolis, France
A New Symmetrical Vertex Path for Exact Reconstruction in ConeNBeam CT
F. Noo, R. Clack, T. J. Roney and T. A, White; University of Liege, Belgium
Fast Accurate Iterative Reconstruction for Low-Statistics Positron Volume
Imaging
A. J. Reader, K. Erlandsson, M. A. Flower and R. J. Ott; Institute of Cancer
Research, U.K.
Design and Implementation Aspects of a 3D Reconstruction Algorithm for the
Jillich TierPET System
A. Terstegge, S. Weber, H. Herzog, H. W. Muller-Gartner and H. Halling;
Forschungszentrum Julich GmbH, Germany
3D-Reconstruction during Interventional Neurological Procedures
K. Wiesent, R. Graumann, R. Fahrig, D. W. Holdsworth, A. J. Fox, N. Navab
and A. Bani-Hashemi; Siemens AG, Germany
Development of an Object-Oriented Monte Carlo Simulator for 3D Positron
Tomography
H. Zaidi, A. Hermann Scheurer and C. Morel; Geneva University Hospital,
Switzerland
11997 International Meeting on Fully 3D Image Reconstruction
101
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SESSION 11: SPECT QUANTIFICATION,
CHAIR: GRANT GULLBERG, PHD
14:30-15:00
Block-Iterative Techniques for Fast 4D Reconstruction Using A Priori Motion
Models in Gated Cardiac SPECT
D. S. Lalush and B. M. W. Tsui; The University of North Carolina at Chapel Hill,
U.S.A.
15:00-15:30
SPECT Reconstruction - The
Ji/UJ
Model
D. L. Gunter; Rush-Presbyterian St. Luke's Medical Center, U.S.A.
15:30-16:00
SESSION
Strategies for Fast Implementation of Model-Based Scatter Compensation in
Fully 3D SPECT Image Reconstruction
D. J. Kadrmas, E. C. Frey and B. M. W. Tsui; The University of North Carolina
at Chapel Hill, U.S.A.
12: GENERAL RECON,
CHAIR: CHRISTOPHER THOMPSON, DSc
16:30-17:00
3D Tomographic Reconstruction Using Geometrical Models
X. L. Battle, G. S. Cunningham and K. M. Hanson; Los Alamos National
Laboratory, U.S.A.
17:00-17:30
Symmetry Properties of an Imaging System and Consistency Conditions in
Image Space
E. Clarkson and H. Barrett; University of Arizona, U.S.A.
OPTIONAL BREAKOUT SESSION
17:30-18:30
An open discussion organized around a topic of interest
Saturday, June 28
SESSION 13: 3D PET INSTRUMENTATION,
CHAIR: GERD MUEHLLEHNER, PHD
09:00-09:30
Experience with Fully 3D PET and Implication for Future High Resolution 3D
Tomographs
D. L. Bailey, M. Miller, T. J. Spinks, P. M. Bloomfield, L. Livieratos, H. Young
and T. Jones; Medical Research Council, U.K.
09:30-10:00
Inter-Comparison of Four Reconstruction Techniques for Positron Volume
Imaging with Rotating Planar Detectors
A. J. Reader, D. Visvikis, R. J. Ott and M. A. Flower; Institute of Cancer
Research, U.K.
10:00-10:30
A Fully-3D, Low Cost, PET Camera Using Hidac Detectors with SubMillimeter Resolution for Imaging Small Animals
.
A. P. Jeavons, R. A. Chandler and C. A. Dettmar; Oxford Positron Systems Ltd,
U.K.
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11997 International Meeting on Fully 3D Image Reconstruction "
System ril0deling and spatial sampling tecllniques for simplification of
transition matrix in 3D Electronically Collimated SPECT
Anne C. Sauvel , Alfred O. !lerol, W. Leslie Rogers2 and Neal H. Clinthorne 2
January 15, 1997
Abstract
In this paper we will present numerical studies of the per"
formance of a 3D Compton camera being developed at the
University of Michigan. We present a physical model of
the camera which exploits symmetries and an adapted spatial sampling pattern in the object domain. This model
increases the sparsity of the transition matrix to reduce
the very high storage and computation requirements. This
model allows the decomposition of the transition matrix
into several small blocks that are easy to store. Finally we
discuss a real time algorithm which calculates entries of
the transition matrix based on a Von Mises model for the
conditional scatter angle distribution given the Compton
energy measurement as well as a vector reformulation or'
the computation of the probabilities.
I
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Jet,
.
Figure 1: Illustration of the Compton scatter collimator
The I ·rays from the point source, X, that reach det 1
are Compton scattered by the solid state detector, detl
1 3D Compton scatter SPEC'.f (Fig. 2). Those scattered photons are then detected by
the second detector in coincidence with the events in deti'
camera
The energy deposited in detl increases as a function of
Application of the Compton scatter aperture to imaging the scattering angle () according to Compton scattering
in nuclear medicine was first proposed in 1974 by Todd and statistics.
Everett. This camera uses an innovative electronical cone~
recoil electron
beam collimator based on the Compton scattering effect.
Its requires a 3D image reconstruction. Singh [1] proposed
.•.••••
in 1983 a l~near image reconstructi~ll for the Compton
with energy Eo
"',
0
camera. This reconstruction is computationnally good but
'.
"
uses an inaccurate model of the system since it neglects
", scattered photon
with energy E
the Poisson nature of the measurements. Leahy [2] implemented an MLE reconstruction from the transition matrix
Figure 2: The Compton sca.tter effect
that takes into account the Poisson noise for a prototype
system. This algorithm is computationnally demanding
Since the vector describing the scattered photon is
since a 3D image of moderate size (128 3 pixels) requires
known
from the two position signals, the direction of the
already a very big matrix T (resp 1286 ).
original photon can be computed within a conical ambiBackground
guity.
The aperture consists of a position sensitive solid state
Although mechanical collimation is simple and reladetector (detI) with a high energy resolution. This detively inexpensive to implement, it has the fundamental
tector is paired with a second position sensitive detector,
drawback that it has a poor sensitivity. Even in the case
det2, which can be a scintillation camera with lower energy
of the pinhole collimator and without considering attenuresolution.
ation and scatter effects, only about 10- 4 of the emitted
1 A. Sauve (corresponding author: [email protected]) and
photons are detected based on geometric factors alone.
A. Hero are with the Dept. of Electrical Engineering and Computer
Efforts have been made to develop electronical collimaScience, The University of Michigan, Ann Arbor, MI 48109-2122.
tors.
Those collimators, since they utilize as manyemit2L. Rogers and N. Clinthorne are with the Dept. of Nuclear
ted photons as possible from all directions, improve the
Medicine, The University of Michi~an, Ann Arbor, MI 48109-0552.
InCI~~~~?~!~? L).~. . . . . . . . . . . . .
11997 International Meeting on Fully 3D Image Reconstruction
121
!l
)1
,lJI
[]
jl
\LJI
[J
u
solid angle of detection and therefore provide an improved
detection efficiency and sensitivity over mechanical collimators.
.
For the Compton scatter collimator, each resolution element of detl can be thought of as a "pinhole" whose
response functi'on at each energy interval is an ellipse on
the surface of det2. Sensitivity gains derive from the fact
that, unlike the case for a real pinhole, a resolution element
in det2 is sensitive to primary 'Y rays incident from many
angles. Further, since the position of the scatter event in
deh is known, the number and density of these resolution elements may be increased without introducing any
ambiguity involving the particular "pinhole" the photon
passed through. This means that the sensitivity increases
in approximate proportion to the solid angle subtended by
deti'
It has been shqwn that we can detect 60 times more
photons with such a collimator than with a comparable
mechanical cone-beam collimator, [3]. Moreover, electronical collimation provides multiple views simultaneously.
One of the open questions is whether it will be possible
to attain equivalent or better resolution than with pinhole
collimators using fast algorithms based on sparse matrix
computations and sparse systems modeling.
2
Analytical model
Mises density ,
using
1_- _______________________ _
Figure 3: definition of the angles
The conditional distribution of f given emitter position
x and detector cell d l is given by the Klein-Nishina dis-
tribution [4] :
.
P(fldl, x) = -.572 ['1 + ( 2 - -fa)2
fa
f
+ (1 -
1
€
-fa)2 .
f
fa
In the classical approach, the conditional distribution
of the detected energy E (E
f + n) given 0, d l and x
is assumed to be Gaussian [1]. However, this is a nonphysical model even for variance of moderate magnitude
since it assigns non-zero probability to negative values of
E. Under the Gaussian assumption, the joint probability
of an incident event received in detector cell d 2 and energy
bin E given d l and x can be computed by:
=
Von
As with many statistical imaging systems, the camera is
entirely defined by its transition matrix T. We developed
a program' that analytically computes T. This eliminates
the need for Monte Carlo simulations for determining the
transition matrix. Then, we developed a simplified model
for the camera so that T will be easy to manipulate, i.e.
sparse and well conditioned. It is important to simplify
the structure of T as much as possible t'o implement computationally tractable 3D reconstructions and to optimize
the system through CR bound computations.
Analytical computation of T '
The elements of the transition matrix T of the camera
are the joint probabilities P(d 2 , E, dIlx), often called trarisition probabilities, where:
d 2 : det2 cell where scattered photon is recorded,
dl : deh cell where incident photon is scattered,
E : detected Compton energy,
x : a source point. '
A cell d2 is completely defined by the two scattering
angles 0 and cfJ (Fig. 3).
We assume the azimuthal angle cfJ to be uniformly distributed
1
P(cfJI0,d1,x) = 27f"
1
From the Compton relation the energy f of the scattered
photon and the scatter angle 0 are related by [4]:
fa
f=
.
1 + a(1, - cos 0)
Here PIC is the probability of a single Compton scatter
in detl (here we assume negligible photo-electric absorption) and n is the set of angles (0, cfJ) which define rays
passing through position x, d l to the surface of the cell
d2 •
The Von Mises model described below is an alternative
which ensures that negative energy measurements are assigned probability zero. First, since it is more convenient
to measure the energy E instead of the angle 0 we express
P(Old l , x) as a function of P(Eld b x). Using the Compton
and the Klein-Nishina relations, we can make the following
change of variable :
The Von Mises model specifies the conditional distribution
of the scattering angle 0 given E, d l and x as a Von Mises
distribution [5] with centrality parameter arccos (2 and width parameter {3:
t)
LlI
P ( E, db x)
(7
=
exp [{3 cos (0 - arccos (2 27f'I (f3)
a
t ))] ,
where Ia({3) is the modified Bessel function of the first
kind and {3 is a shape parameter which we estimate from
Monte Carlo simulations. The Von Mises model is physically appropriate for the angle distribution because it is
27f' periodic.
11997 International Meeting on Fully 3D Image Reconstruction
Using the Von Mises conditional density for () we get:
P(d 2 , E/(h, x) ,=
~~
II
rotation of the system
around 0
"
<I>
. ..... ~~~~~~:::::: .....
t
y'
-.
(.'\ :. -.-f.:'\ -:::-~~~~:f=!::}..:;..'~.~:;.
~. ~l::\1
.... ~........ '"
" , ~
P(()/E, db x)P(E/d 1 , x)d¢dO.
n
'.
Under the assumption that d 2 cell surface area is small
with respect to the spread! of the Von Mises density, this
integral relation can be approximated.
"
"
:,
'-.
I :
...
r'-f\'.;' . .
.
fiIeId 0 f' view
1 _ IIPfiX(cr.a;)II
lliIiCl;II~
.. l.·'\
..'y!o
/~
/
E 2
2)
I
'1,c;.~1 detector 2
""
I :
' , ' ,.1" , t'"
• - - - ••• -',:: • _ ••• 1 ........ ol. _. _ ,,".J.,.. ,.
-~II'
\
center of the
=
;or",,"
•• ,'
'" II ",
•••I
..1.......
" " ''"
.
1- (2
,,'
'--.:'
\\
~~ >"
t,
.. _" ....
x
: :tetector 1
I,
"'",,~.1 ,
\
\
\
soun;e.....
........',
hemispHeres
I
I
I
- ... "
Figure 5: Hemispherical source sampling and Rotation of
the detectors 1 and 2 around the field of view
]
E
detector 1
where the projection into plane with normal Cl'iX is
denoted by the operator:
:= I - (~~f , and
tlS is the det2 pixel area.
The above expression is rich in vector operations and
is therefore -:--:+
suitable for fast on line computation. Note
that when d 1d2 has constant length and the pixel areas
AS are constant, the p.d.f. P( d2 , E/d 1 , x) has symmetries
which can be exploited to reduce computation. This oc~
curs when the detector det2 is an hemisphere centered at
dl . Let P( dl/x) be the probability that a i-ray emitted at
x, intercepts the detector surface dl (computed from the
solid angle subtended by the cell d l from the source point
x). This probability when combined with the above rela- Figure 6: Source emissions leading to event trajectories
tion gives the required elements of the transition matrix (LL') and (M M'), respectively, have identical transition
from:
probabili ties.
PP
to be reconstructed (ie the source intensity) in cartesian
coordinates, and N accounts for system mismodeling erSymmetry exploitation
rors and counting statistics of the Poisson events.
We first assume that det2 is an hemi~phere and that detl
T can be arranged as a the concatenation of the tran ..
is a unique cell at the center of det2. As we will show, this sition matrices obtained for the p different Compton energies Ei, and Q is the concatenation of the interpolation
matrices for the n different rotations of the system:
indexed over d2, E and dl.
T =
detector I
[T~I],
Tsp
Q
= [Q~I]
.
Qif?n
Figure 4: Camera Model
can
write
TSj
=
[HEjR]
where:
HEj = ((P(d 2 ,Ej /d l ,x)))d2.X 1
{ R [Pllk .. 'PLlkJ.
• k is the number of source pixels on one hemisphere
and I is the number of hemispheres intersecting the field
y=TQ..\+N,
of view.
where y is a vector containing the measured coincidence
• HEj co'ntains the transition probabilities for the enevents, T is the system. matrix, .Q is a cartesian to hemi. . ergy Ej when d 2 varies over det2 and x varies over a single
spherical bilinear interpolation matrix [6J, ..\ is the image hemisphere of the source.
We
greatly reduces the storage and computation requirements
due to resulting symmetries in the transition matrix.
The measurement equation can be written as:
11997 International Meeting on Fully 3D Image Reconstruction
=
141
(
-
i
I
n
I (
l J
f)
I!
fl
[]
• Pi is the solid angle subtended by d l from the hemi- Here ~ = diag(TQA) is a diagonal matrix constructed
sphere i.
from the vector of mean system responses to a source of
Finally, the transition matrix can be written in the com- intensity A. The matrix F).. is symmetric of dimension
pact form:
m 3 x m3 where m 3 is the number of pixels in the (presumed
cubic) imaging volume. Even for relatively small problems
it is not practical to attempt to invert F).. directly. By
exploiting the sparseness and symmetry of the transition
matrix T we develop fast recursive equation solvers to calBecause we uniformly sample over a hemispherical grid culate the CR bound for 3D reconstruction tasks such as
in object space, H Ej does not depend on the particular uptake and contrast estimation in a region of interest. The
hemisphere. Moreover, HEj depends on relative angles bound is used obtain estimator-independent comparisons
only and is therefore approximately circulant and diago- between diff'erent camera configurations, e.g. spatial samnalizable via 2D FFT methods. R also is very sparse. This pling and interpolation schemes, Compton scatter energy
structure for T will significantly simplify the computation- resolution, and number of camera rotation angles. These
ally demanding 3D reconstruction algorithms. Moreover, results 'Yill be presented in the full paper.
we do not have to store the very large T matrix but only
the HEj and the L scalars Pi leading to reduction of storReferences
age requirements by several orders of magnitude.
Annulus sinogram obtained for this camera [1]
M. Singh and D. Doria, "An electronically collimated
model
gamma camera for single photon emission computed
tomography. part II: Image reconstruction and preliminary experimental measurements," Transactions on
Medical Physics, vol. 10, no. 4,pp. 428...:;43,5.,.1983.
[2] T. Hebert, R. Leahy, and M. Singh, "Maximum likelihood reconstruction for a prototype electronically collimated single photon emission system," in Proc. SPIE
Medi.cal Imaging, vol. 767, pp. 77-83, 1987.
[I
rl
l
J
[)
L
Figure 7: Planar projection of the sinogram obtained on
det2 from a 511kev photon coming initially from a point
on the camera symmetry axis and that Compton scattered
400I( ev (for f3
700 and 6400 cells
from detl with E
in det2)'
=
3
=
CR Bound for 3D' Reconstruction Tasks
l
.J
[I
The uniform CR bound [7] provides a lower b9und on
the variance of any estimator with bias gradient length
bounded by the user specified parameter {) > O. It is a
useful tool for establishing fundamental performance limitations of tomographic systems [8]. With g an estimator
of a smooth function 9
g( A) of the 3D intensity A the
bound is of the form
=
[J
[]
where el, dl are vectors related to g, {) and F).. are described in [7], A + denotes pseudoinverse of a matrix A,
and F).. is the Fisher information matrix
[3] M. Singh, "An electronically collimated gamma camera for single photon emission computed tomography.
part I: Theoretical considerations and desig.n criteria,"
Transactions on Medical Physics, vol. 10, pp. 421-427,
July/August 1983.
[4] G. F. Knoll, Radiation Detection and Measurement.
Wiley, 1979.
[5] N. 1. Fisher, T. Lewis, and B. J. J. Embleton, Statistical Analysis of Spherical Data. Cambridge University
Press, 1987.
[6] R. N. Bracewell, Two-Dimensional Imaging. Prentice
Hall, 1995 .
[7] A. O. Hero, J. A. Fessler, and M. Usman, ((Exploring estimator bias-variance tradeoff's using the uniform
CR bound," IEEE Transactions on Signal Processing,
vol. 44, no. 8, pp. 2026-2041, 1996.
[8] N. H. Clinthorne, C. yi Ng, C. ho Hua, J. E. Gormley,
J. W. Leblanc, S. J. Wilderman, and W. L. Rogers,
('Theoretical performance comparison of a comptonscatter aperture and parallel-hole collimator." To appear in the Conference Record of the 1996 IEEE Nuclear Science Symposium, 1996.
11997 International Meeting on Fully 3D Image Reconstruction
151
Application of spherical harmonics to image reconstruction
for Compton Camera
Roman Basko, G. Larry Zeng and Grant T. Gullberg
Departlnent of Radiology, University of Utah, Salt Lake City, UT 84132, USA
I. Introduction
Conventional gamma cameras used in SPECT localize gamma emitters by a mechanical collimator.
This technique leads to very low efficiency because only a fraction of the radiation is recorded through the
collimator. Also at any given time only one view of the object is obtained, so the camera needs to move
relative to the patient in order to collect all the data necessary for image reconstruction.
A new type of gamma camera for SPECr, originally proposed by Everett et ai. [1] and by Singh [2]
and further investigated in [3-5], utilizes Compton scattering for gamma source localization. Using
electronic collimation as an alternative to mechanical collimation provides both high efficiency and mUltiple
views of the object. The camera collects data that are projections along cone surfaces. Several approaches to
image reconstruction from cone projections are described in [6-8].
This paper presents a new approach to reconstruction for the Compton camera, based on estimating
Radon projections for the gamma source from its cone projections. Once Radon projections are known, the
filtered backprojection algorithm can be used to reconstruct the gamma source.
II. Compton camera design
The camera consists of two plane gamma detectors positioned one behind the other. An incident
photon undergoes Compton scattering in detector 1 and is absorbed by detector 2 (Fig. l(a)).
Detector 2
(a)
(b)
Figure 1.
Corresponding positions 0 and 0' as well as energy /:ill deposited in the first detector are measured.
Angle ~ can be found by using
A
mc2~E
cos p = 1 - (E _ ~E)E .
(1)
Assuming that 0, 0' and ~ are known, one can conclude that the gamma source is located somewhere on
the cone surface (Fig. ·1 (b»).
The relationship between 3D gamma source distribution f(x) and the rate of photon counting q( 0, 0', ~)
for specific 0, 0' and ~ is given by
q(O, O',~)
oc
f
fdA.
cone
Application of spherical harmonics to image reconstruction for Compton CameraJanuary 14, 1997
11997 International Meeting on Fully 3D Image Reconstruction
(2)
Therefore data acquired by the camera can be considered as samples of q( 0, 0', ~) and are usually called
cone projections.
ITI. Reconstruction
r-'
,j
In what follows we establis? the relationship between cone projections associated with a fixed point 0
on the front detector and Radon projections for planes intersecting O. This relationship allows efficient
estimation of Radon projection for a given set of planes intersecting 0 from a corresponding set of cone
projections. Having done this for a sufficient number of points 0, the filtered backprojection algorithm Can
then be used to reconstruct the image.
For a fixed point 0 on the front detector let us define two functions, qk(~) and p(n), where both k
and n are unit vectors (Fig.2), as follows:
pen)
= f f( 0 + nr)rdr,
qk(~)
If vector
k is in the direction
f
(3)
p(n)ds.
s(k. p)
o
0' 0 , then
f
qk(~) =
[J
fdA
oc
q( 0, 0', ~).
(4)
cone
Therefore qk(~) , as a function of both
and ~, describes all cone projections associated with point O. It
k
also follows from the definition of qk(~) that qk(rc/2) is equal to the Radon projection along a plane perpendicular to
k
and intersecting point O.
Il
I I
L J
n
[1
(a)
Figure 2.
(b)
Using spherical coordinates (Fig.2(a» pen) can be expressed in terms of harmonic expansion as
follows:
I
p(S, $)= ~
£..i
~ Plm' PI m( cos S) e im<jl
£.oJ
(5)
.
1= Om =-1
[J
{-'I
l_
[J
The following fundamental relationship between qk(~) and
qk(~)
= 21tSin~/~
00
(
m
I
k,
established in the appendix,
m·~_/Im· PI (cosSk)e
where Sk and $k are spherical coordinates of a unit vector
tion qk(rc/2) for any direction
Plm'
provided that
Plm
.!.)
1m,!,};
.
0
PI (cos~),
(6)
k (Fig.2(b», allows calculation of Radon projec-
are known. Expansion coefficients
Plm
can be esti-
mated by a least square fitting of cone projections associated with point 0 into (6). The properties of
Legandre polynomials as well as a fast Furier transform can be explored for efficient implementation.
Application of spherical harmonics to image reconstruction for Compton CameraJanuary 14, 1997
11997 International Meeting on Fully 3D Image Reconstruction
IV. Algol'ithm.
A set of cone projections associated with a fixed point on the front detector is described by the function
qk(~) defined on three dimensional manifold S2 x [O,n] . The functionjJ(n) defined on a unit sphere S2
allows compact representation of qk(P) as an integral alon~ circle S(k, P) with the center k and radius
sin~. Moreover, an integral of pen) along a grate circle S(k, n/2) is equal to Radon projection along a
plane perpendicular to k.
This allows to approach the reconstruction problem as follows:
Step 1. For every point 0 on the front detector, the values of pen) are estimated from samples of qk(~) provided by the camera.
Step 2. Estimated values of pen) are used to calculate Radon projections along planes intersecting point O.
Step 3. The filtered backprojection algorithm is used to reconstruct the image from Radon projections.
Expansion of pen) in terms of spherical harmonics allows efficient implementation of the first two
steps of the algorithm.
V. Summm'Y
A new reconstruction approach for Compton camera is proposed based on estimation of Radon
projections followed by application of the filtered backprojection algorithm. Using expansion in spherical
harmonics allows efficient implementation of the algorithm. A complete set of planar projections can be
formed from only one camera position if the detector has infinite extent.
APPENDIX
With any vector k we can associate a spherical coordinates (e, <1» with e measured from k (Fig.3),
and introduce a function Pl(e, cJ»_ that represents pen) in those coordinates. Since both ~ and e are
measured from the same direction k , we have:
21t
f
qk(~) = sin~ Pk(~' cJ»dcJ>.
(7)
o
Detector plane
~=n/2
plane
Figure 3.
Expressing P'k(O, <1» in terms of expansion in spherical harmonics
00
~
I
~
P'k(O, cJ»= £..J £..J P'k,lm' PIm (cos 0) eim$ ,
(8)
1= Om =-1
we obtain from (7)
Application of spherical harmonics to image reconstruction for Compton CameraJanuary 14, 1997
11997 International Meeting on Fully 3D Image Reconstruction
181
L Pk,lO' p/O(cosP).
qk(P) = 2nsinp
(9)
1=0
Let us now fix the coordinate system corresponding to some vector kO (we can specify
being perpendicular to the detector plane) and use the following notation
p(8,~)
Any unit vector
p/ l11
= Pk
(10)
o,/I1I'
k is uniquely represented in this coordinate system by two angles,
8k, ~k ' and
p(k) = Pk(~~) = P(8~$k)'
r1
t
= Pko(8, ~),
kO , for instance, as
(11)
which can be written in terms of Legandre expansion as
J
I
L Pk,lO = L L p/
[J
/ =0
ilmjl,ii
III
I1I '
(12)
PI (cos8k)e
1= 0111 =-/
Since 2t+1 dimensional space of spherical harmonics {PI1II(cos8)/m cJl , m= -I, ... , t} is invariant under
rotations, it follows from (12) that for any 1 = 0, ... ,00
/
Pk,lO
= L
• m
m
llll't',ii
m
. m
Im't',ii
Plm' P / (cos8k)e
- (13)
111=-/
Finally, combining this result with (9), we obtain
00
qk(~) = 2nsinp
I
L L
I=
Plm' p/ (cos8k)e
°
. PI (cosP)·
(f4)
Om =-/
REFERENCES
[1] D. B. Everett, J. S. Fleming, R. W. Todd, and J. M. Nightingale, "Gamma-radiation imaging system based
on the Compton effect", Proc. lEE, Vol. 124, pp. 995-1000, 1977.
[2] M. Singh, "An electronically collimated gamma camera for single photon emission computed
tomography. Part I: Theoretical considerations and design criteria'\ Med. Phys., Vol. 10, pp. 421-427,
1983.
[3] M. Singh and R. R. Brechner, "Experimental test-object study of electronically collimated SPECT.", J.
Nucl. Med., Vol. 31, pp. 178-186, 1990.
[4] N. H. Clinthorne et at., "Theoretical performance limits for electronically-collimated single-photon
imaging systems.", 1. Nucl. Med., Vol. 37, pp. 116-117, 1996.
[5] J. E. Gormley et at., "Effects of shared charge collection on Compton camera performance using
pixellated Ge arrays.", J. Nucl. Med., Vol. 37, pp. 170-171, 1996.
[6] M. Singh and D. Doria, ''An electronically collimated gamma camera for single photon emission
computed tomography. Part II: Image reconstruction and preliminary experimental measurements.", Med.
Phys., Vol. 10, pp. 428-435, 1983.
[7] T. Hebert et at., "Three-dimensional maximum-likelihood reconstruction for an electronically collimated
single-photon-emission imaging system.", J. Opt. Soc. Am., A, Vol. 7, pp. 1305-1313, 1990.
[8] M. J. Cree and P. J. Bones, "Towards direct reconstruction from a gamma camera based on Compton
scattering.", IEEE Trans. Med. Imag., Vol. 13, pp. 398-407, 1994.
[]
fl
~- _,
[
rl
LJ
l"
\
J
Application of spherical harmonics to image reconstruction for Compton CameraJanuary 14, 1997
(
1
J
11997 International Meeting on Fully 3D Image Reconstruction
------------------------------- ---------- ----------
-----------~---
----------------- ----
A Fourier Rebinning AlgorithtTI incorporating Spectral Transfer Efficiency
for 3D PET
Eiichi Tanaka l and Yuko Amo2
1 I-Imllatnatsu Photonics K. K., Tokyo, Japan
2 National Institute of Radiological Sciences, Chiba,
Abstract
This paper presents an improved Fourier rebinning
algorithm for 3"ditnensional image reconstruction in PET.
The algorithm incorporates a concept of spectral transfer
function, which suggests the need of discarding low ~
frequency components fi'om reb inning. It also includes the
correction for rebinning efficiency which was evaluated by
simulations as a function of oblique angle of projections.
The performance was optimized by a high8pass filter and
Its parameters. The algorithm yields satisfactory images
with negligible axial crossMtalk for a maximum oblique
angle up to 26.6°. The statistical noise was evaluated in
terms of "noise equivalent number of oblique angles," and
reasonable results were obtained in view of the theoretical
expectation. Ring artifact due to noise is negligibly small.
Japan
inverse 2D FT. The 3D image is obtained by the conven2D recollstl1lction method from the 2D sinograms
slice by slice.
'
We consider that a Fourier coefficient P( ~ k, zo, 0) of
an oblique sinogram along slice Q is reb inned into the
matrix P(OJ, k, z) of a transaxial slice S, as shown in figure
1(a). The two slices have a common region, C, shown as
the shaded area. Accurate rebinning for the Fourier
coefficient is expected only when the wave-length of the
component is sufficiently shorter than the length of region
C along each slice. For low frequency components having
wave-length comparable or longer than the region, the
information extends beyond the region, and the reb inning
process transfers a reduced fraction of the components of
slice Q together with those of the neighboring slices. This
may cause image distortion and axial cross-talk.
tiona
1. Introduction
The Fourier rebinning algorithm (FRA) developed by
Defrise [1,2] provides a computationally efficient method
for three-dimensional (3D) image reconstruction in
positron emission tomography (PET). The algorithm is
based on the frequencYMdistance relation introduced by
Edholm et al. [3] for 2D radon transform. The FRA is an
approximate method, and its performance depends on the
various parameters involved in the algorithm, but the
details have not been made clear yet. Defrise [2] reported
an analysis of the accuracy of the FRA and he proposed an
exact reb inning equation, but it is more complex than the
original one and it has not yet been implemented. We
considered the accuracy of FRA from a different point of
view, and developed a new 'FRA incorporating spectral
transfer efficiency from oblique projections to transaxial
slices. This paper describes the theory and the results of
simulation studies on the new algorithm.
2. Theory and principle of the new algorithm
The FRA is based on acquisition of an oblique
sinogram, per, l/J, Zo, 0) for each detector ring combination
(Zh Z2), where,zl and Z2 are the axial coordinates of the
detector rings, Zo= (Zl + z2)/2, and 8 = IZI - ~I. After 2D
Fourier transform (FT) of those sinograms, the 2D maps of
the Fourier coefficient P(ro, k, Zo, OJ are reb inned into a
matrix P( cq k, z) of a set of transaxial slices, using the
frequency-distance relation, Z = Zo - (k I OJ) to, where t 0=
tanO and 0 is the oblique angle of the sinogram. After
normalizing for rebinning density on P( ~ k, z), the 2D
sinogram per, l/J, z) on each transaxial slice is obtained by
11997 Ihternational Meetingon Fully 3D Image Reconstruction
(a)
to
(b)
:=
0.125
T(m)
Figure 1.(a) An oblique slice Q and a transaxial slice S
(b) Spectral transfer function, 1{ OJ)
We can suppose that region C plays a role ofa low-pass
filter which traps the highMfrequency components to be
transferred. The frequency response of the low-pass filter
will be given by FT[t(s)], where FT[e] denotes Fourier
transform and t(s), normalized for unit area, represents the
profile of region C along slice Q (see figure 1(a»). The
response of the information transfer will then be given by
T(ro) = 1 - Ff[t(s)].
(1)
The function, t(s), is approximated by an isosceles triangle
having a half-width equal to dlto, where d is the slice
thickness. Equation (1) then becomes
T(ro)=I-{sinaro/(aro)}2.
a=dl(2to )
(2)
We refer to 1{ro) as "spectral transfer function" (see
figure l(b». The function has the first maximum (=1) at a
201
I-I
J
!.
"critical frequency" mo=2 rr te/d. The rebinning is expected
to be accurate for frequencies close to or higher than lib,
and the lower frequency components should be damped or
discarded by a high-pass filter in the FRA. It was also
found that, by simulation studies described later, the
rebinning efficiency is lower than unity even for m> mo ,
depending on O.
The correction for the reb inning
efficiency should also be included in the algorithm.
The new FRA is then expressed by
r-l
I;
r-'
I I
L)
[--]
'LP(m,k,zo'o) H(m,o)
P(m,k,z) = .
W(m,k,z)
z=zo-(klm)te
W(m,k,z)= LH(m,o)E(o),
[-J
l
r--
l
[-I
L
[-I
[]
[]
[:
_J
[I
U
[I
[I
(3a)
3.2 Rebinning efficiency
To confinn the validity of the consideration described
in Section 2, we perfonned the following simulation.
Assum ing a point source on a trans axial slice (z\), we first
generated the rebinned matrix pew, k, z\) of the slice from
the data of a ring difference, 0, assuming H( m, 0)= 1 and
E(O)=1. We then took the inverse ID FT on k, converted it
to the amplitude and averaged on cp to obtain the ID
frequency spectrum, Ir (m, 0). The spectrum was normalized by the similar spectrum obtained with 8=0 to yield a
normalized frequency response (NFR):
In (ro,o) = fi(ro,o)1 fiero, 0).
(3 b)
(4)
Some examples of the NFR are shown in figure 2, together
with
It is seen that the NFR is roughly constant at
m~ mo and decreases with decreasing ro at m<ut. We
where W( m, k, z) is a correction matrix for the reb inning
density, H(m, 8) is the high-pass filter and E(O) is the
assumed the reb inning efficiency, E( 0), is given by the
reb inning efficiency. Equation (3b) requires axial interpovalue of the NFR at the critical frequency, In(mo , 0).
lation to detennine slice number z. The ranges of the
Figure 3 shows the rebinning efficiency thus obtained with
arguments, (m, k), in the summations will be discussed
a point source positioned at various distances from the
later in 3.3.
center of the FOY. The small oscillation observed in the
curve for each source position is due to the discretization
3. Simulation studies
in equation (3b). The data were obtained with the nearest
neighbor interpolation. The linear interpolation.-yielded a
3.1 Detector geometry and digital implementation
slightly lower efficiency.
Although the rebinning
We consider a multi-ring PET scanner. The size of
efficiency depends somewhat on the position of the point
image matrix is 64 x 64, and the number of azimuthal view - source, we assumed that E( 0) is simply expressed by, as
angles is 64 in rr. The slice thickness is equal to pixel size,
the average,
and the detector ring width is equal to two pixels. The
diameter of the detector ring is assumed to be 128 pixels
E(O) = 0.95 - 0.9 tf)
nearest neighbor interpolation (5)
(twice of the matrix size). Then te=8/64. Assuming the
= 0.90 - 0.9 te. linear interpolation
pixel size of 4 mm, the ring diameter is 51.2 cm, ring
width is 8 mm, the matrix size is 25.4 cm and the slice
thickness is 4 mm. We consider two modes of axial
sampling: stationary mode (.1=2) and scan mode (.1=1),
where .1 is the axial sampling pitch. In the stationary
mode, a slice belongs to either of "direct slice" or "cross
slice" as in the conventional PET scanners.
Projection data are produced assuming the axial
response of the coincidence detection is triangular having
the full width equal to the detector ring width (two pixels).
This assumption is only accurate at the mid point between
two detectors, but we neglect the inaccuracy at off-center
positions. Attenuation and scattering of photons are
neglected. Effect of positron range and angular fluctuation
Figure 2. Nonnalized frequency response, !n(ro, 0)
of annihilation photons are also not taken into account.
The maximum ring difference, omax, is assumed to be
constant in the field of view (FOY).
The FRA is implemented as follows.
First, the
~
........ 80
projection data for each detector ring difference, 0, are
~
c:
arranged in a 3D sinogram, per, ¢, zo, 8) sampled over 2rr
.~ 60
angle. The merged sinogram is transfonned to P(ro, k, ~,
~
0) map by 2D FFT with zero-padding for r. The size of the
g> 40
'cc:
map is 64(m)x 128(k) (ro~O). After reb inning according
:0
20
to equation (3), P(ro, k, z) is converted to 2D sinogram, per,
Q)
a:
cp, z), sampled over 2rr by inverse 2D FFT. The sinogram
00
30
10
15
20
25
is reduced to the sinogram over rr using per, ¢, z) = p(-r,
5
¢+n:, z). The final image is reconstructed with the
Ring difference 0
convolution backprojection method using the Shepp-Logan
Figure 3. Rebinning efficiency, E( 0)
filter.
(3c)
rem).
~
\J
0
_I
-I
j
-,
I
11997 International Meeting on Fully 3D Image Reconstruction
211
3.3 High-pass filter and the range of the arguments
For data with o<L1, all Fourier coefficients P( ~ k, ZQ, 0)
are assigned into zOHslice assuming H( ro,O)= 1 and E( 0)= I
as the conventional 2D PET. Fourier coefficients with k=O
deal with rotationally symmetric components of the source
distribution or noise, and special treatment is required. If
all coefficients for o~ tl are assigned, rotationally sym
metric components of the images suffer from low~
frequency distortion. On the other hand; if all coefficients
are discarded, rotationally symmetric components of
statisticai noise are reconstructed only using the data with
8<L\, which results in the appearance of ring artifacts due
to the poor statistics. We then implemented a variable
10wMcut filter having the cutRoff frequency at' given by
M
COc' ==
(10/64) n+
1C to
.
mode. About 74% of the total image density is deposited
to the source slice, and 13% to each of the adjacent slices.
The fractions are similar to those in the conventional 2D
reconstruction (omox==:I). The cross"talk to the next slices
but one is negligibly small. To demonstrate the effect of
the highHpass filter, the similar images obtained without
high-pass filter arc shown in figure 5.
(6)
As the result, a uniform circular disc phantom (20 cm
diameter) was reconstructed with nonHuniformity less than
0.6% in 1'1118 error within the central circular area of 80% in
diameter, for 0.118" up to 32. Ring artifacts were eliminated
effectively (see figure 7).
For the data with o~ L\, we implemented a high-pass
filter given by
H(ro, 0) = [T(ro, 8)]111
= 1
=0
roc = aroo,
if at< ro < roo .
if at, .< 0)
(7)
otherwise
0< (X::; 1
(8)
where at is a o..dependent cut-off frequency, m and a are
constants. Note that when m=O, the filter is reduced to a
simple 10wMcut filter. The perfonnance of the FRA with
the highHpass filter was studied with various omax up to 32.
The test phantom was a 5 cm off-centered disc phantom
having 10 cm diameter and 4 mm (one pixel) thickness.
With m=O and a=0.9~1, quite reasonable images are
obtained with negligible axial cross-talk, but the reduction
of statistical noise is not sufficient at large omax. By
lowering a-value, the noise at large omax decreases, but the
image distortion and the cross-talk increases gradually.
Suitable combinations of m=1-2 and a-:=0.5-0.75 provide
reasonable results, but a slight low-frequency distortion
and crossHtalk are observed. It was found that the image
distortion is sensitive to a-value for small oblique angles
but not for large oblique angles. On the contrary, the
statistical noise at large omax is decreased by lowering a
effectively.
.
Then, we redefined the following cut-off frequency
0< {3
where
(9)
f3 is a constant.
A simple low-cut filter (m =0) with
yields quite satisfactory results. We have also tested
the combination of m=1~2 and f3=1~3, but the improve"
ments were not remarkable (see table 1). The linear axial
interpolation in equation 3(b) yields slightly larger axial
cross-talk and lesser statistical noise than the nearest
neighbor interpolation. The data shown in this paper were
obtained with the nearest neighbor interpolation.
Figure 4 shows images obtained with m=O and (J=l.
The phantom is on a direct slice (slice-O) in stationary
{J=1~2
11997 International Meeting on Fully 3D Image Reconstruction
Figure 4. Images for various omax with m:=::O & f3::;:1
The phantom is on the direct slice (Slice-O).
Figure 5. Similar images with figure 4 obtained without
highHpass filter (omax == 24)
Slice
direct
direct
direct
direct
direct
direct
cross
cross
cross
cross
mfJ
o0
o1
o 1.5
o2
12
2 3
o0
o1
o 1.5
o2
max error
+2.4 / -2.5
+2.7/-2.7
+3.1 / -3.0
+3.0/-3.6
+3.0/-3.5
+3.0/-3.7
+3.2/-3.9
+3.7/-4.5
+3.9/-5.1
+3.9/-5.0
(%)
rms error (%) NENA
UO
22.8
1.20
30.3
1.41
32.4
34.2
1.49
36.1
1.49
37.5
1.62
29.5
1.53
40.6
1.73
46.8
1.71
50.7
1.71
N
32
Table 1. Effects of parameters in high-pass filter
The performance of the FRA with various values of m
and f3 is summarized in table 1, where "max error" and
"rms error" are the maximum deviation of th~ pixel values
and the root mean square error from their mean,
respectively, in a circular region of 80% in diameter
221
centered on the hot area. The listed values are the worst
ones among the data tested with omax=4~32 at step 4.
"NENA -32" is the noise equivalent number of oblique
angles (described later) for omax=32. The performance for
the source on a cross slice is almost similar to that on the
direct slice but image distortion is slightly larger, as shown
in the table.
Acknowledgments
The authors acknowledge to Drs. M. Defrise, H.
Murayama and H. Kudo for their useful discussions. They
also thank Dr. T. Yamashita and other stuffs of the PET
Center of Hamamatsu Photonics KK for their kind support.
3.4 Statistical noise
Statistical noise was tested assuming a uniform disc
phantom having 20 cm diameter and 4 mm thickness, with
various omax. The total count with 8=0 was 500 k. The rms
noise was evaluated from the fluctuation of pixel values in
the central circular area (80% in diameter) of the phantom.
The decrease of noise with increasing omax was evaluated
by calculating "noise equivalent number of oblique angles
(NENA)" defmed by
[]
NENA = (rms noise with 0 = 0)2
rms noise with omax .
[]
10
(10)
[)
[1
20
25
30
35
Figure 6. Noise equivalent number of oblique angles
Figure 6 shows the NENA for different values of f3
(m=O). It is shown that a small value of f3 spoils the
increase of NENA at large 8max. In fact, lQ, reaches the
Nyquist frequency at 8=32, where no data are assigned
when /3=0. The dashed curve in the figure shows the
expected number of oblique angles, NENA exp , in the scan
mode, which takes into account the actual number of
oblique angles (=20max-l) and the reb inning efficiency
given by equation (5)
[I
15
Maximum ring difference 8max
Oma.~
NENA exp
= 1+
L2E(0).
(11)
0=1
NENA obtained with simulation studies is appreciably
larger than NENA exp • The reason for this will probably be
that the noise components spread to the neighboring slices
far away than the signal components. The NENA with the
stationary mode is smaller than that with the scan mode by
a factor of about 2, as expected. Examples of images with
statistical noise are shown in figure 7. The total count with
8=0 was 50 k, and m=O and {J== 1.
[]
4. Discussion and conclusions
[J
[J
o
1
I
_1
The performance of the new FRA has been investigated
by simulation studies. With a suitable high-pass filter and
variable cut-off frequency depending on 0, the new
algorithm provides satisfactory images with omax up to 32.
The statistical noise decreases with increasing qnax as
expected theoretically. Ring artifact due to noise is
eliminated by 8-dependent low-frequency cut-off for k=O
components. The improvement of signal/noise ratio by
increasing 8max tends to saturate, and a practical limit may
be around t(}=0.5 (8==26.6°), because the critical frequency,
coo, reaches the Nyquist frequency at the angle.
The need for relatively high cut-off frequency suggests
a similarity between the FRA and the pseudo-3D algorithm
proposed by Tanaka et al.[4], although the FRA has great
advantages in the computational speed and in the flexibility for non-uniform axial acceptance angle in the FOY.
Slice-O
Slice-1
Slice-2
Figure 7. Images with statistical noise (m=O, f3=I)
References
[1] Defrise M., Kinahan P. and Townsend D.: A new
reb inning algorithm for 3 D PET: Principle, implementation and performance. Proc. 1995 International Meeting on Fully 3D Image Reconstruction in Radiology and
Nuclear. Medicine., pp235-239, 1995.
[2] Defrise M.: A factorization method for the 3D X-ray
transform. Inverse Problems 1l:983-994, 1995.
[3] Edholm P.R., Lewitt R.N., Lindholm B.: Novel
properties of the Fourier decomposition of the sinogram.
Workshop on Physics and Engineering of Computerized
Multidimensional Imaging and Processing, Proc of the
SPIE 671:8-18,1986.
[4] Tanaka E., Mori S., Yamashita T.: Simulation studies
on a pseudo three-dimensional reconstruction algorithm
for volume imaging in multi-ring PET. Phys Med BioI
39:389-400, 1994.
11997 International Meeting on Fully 3D Image Reconstruction
231
Perforlnance of the Fourier Rebinning Algorithm
for PET with Large Acceptance Angles
Samuel Matej, Joel S. Karp and Robert M. Lewitt
Department of Radiology, University of Pennsylvania
423 Guardian Drive, Philadelphia, PA 19104-6021, USA
Abstract
The theory of the FORE method [2] is based on the
frequency-distance relation for the 3D x-ray transform
and on the stationary-phase approximation. It enables
one to approximate the spectrum of the no-tilt (co-polar
angle, 0
0°) projection data from the spectrum of
the oblique (tilted) data. Thus, the measured projection
data at non-zero tilts can be resorted into 2D sinograms
using formula
Modern Positron Emission Tomography (PET) scanners characterized by a large axial Field Of View (FOV)
provide data from a large axial acceptance angle. Direct
reconstruction of the 3D image from these data (e.g. by
using popular 3D-RP technique [1]) is of large computational complexity and typically requires long reconstrucP(w, k, z, 0) ~ P(w, k, z - (tanO (k/w», 0) , (1)
tion times. Rebinning techniques, approximating (with
an error which increases with the acceptance angle) plawhere wand k are the 2D sillogram spectrum variables
nar sinogram data from the oblique projection data, enrelated to the radial distance from the center and proable the use of multiplanar 2D reconstruction techniques
jection angle, respectively; z is the z coordinate (slice
characterized by much lower computational demands.
number) and 0 is the oblique angle of the tilted sinoThe recently proposed Fourier Rebinning (FORE) techgram.
nique [2] was shown [3, 4] to provide a very good approximation for moderate (around ±100) acceptance angles.
In this study we show performance of the FORE techII. EXPERIMENTAL RESULTS
nique for a wide range of acceptance angles and compare reconstruction performance of the FORE (followed In our experimental study we have used simulated and
by multislice 2D reconstruction) to the 3D-RP technique real data from a large acceptance angle PET scanner.
for large acceptance angle data (±26.25°).
The real data were obtained from the HEAD PENNPET scanner [5] having a cylindrical detector of radius
420mm and active axial height 256mm. The data were
I. INTRODUCTION
acquired in list-mode and reb inned into sets of 2-D panels of projections (x-ray transforms), each consisting of
Several reb inning algorithms have been used for PET 128 x 128 pixels (parallel bins), 2 mm on a side. The azscanners having small to moderate acceptance angles imuthal range (around scanner z-axis) was divided into
(angles up to ±100). On the HEAD PENN-PET scanner 96 bins, leading to an azimuthal bin size of 1.875°, and
having acceptance angle ±26.25°, the Single Slice Rebin- the co-polar bin size was set to twice this (3.75°), leading
ning (SSRB) algorithm results in very significant distor- to 15 co-polar angles (the so-called "tilts") within effection of the point spread function, whereas the Multislice tive acceptance angle of the scanner. By the acceptance
Rebinning (MSRB) algorithm results in noise propaga- angle we understand co-polar angle of the bin center of
tion from axial de-blurring. The most promising is the the extreme tilt. Projection data had a series of correcrecently proposed Fourier Rebinning (FORE) algorithm tions applied for sampling pattern normalization, atten[2]. However, so far, the' FORE algorithm has only been uation correction,' 'and scatter subtraction as described
tested [3, 4] for a scahner geometry with a moderate axial elsewhere [5]. The projection data were reconstructed,
acceptance angle (around ±100).
by 3D-RP reconstruction method [1] with Colsher window, and by FORE/Il'BP technique, into images of 128
This work was supported by the National Institutes of Health
under Grants HL-28438 and CA-54356, and by the Department of x 128 x 128 cubic voxels, each 2mmon a side. To study
how the FORE behavior depends on the axial acceptance
Energy under Grant DE-FG02-88ER60642.
=
11997 International Meeting on Fully 3D Image Reconstruction
241
\l
\
J
[]
angle we used subsets of the data - from 15 tilts, for full
acceptance angle (()Max = ±26.25°), gradually down to
1 tilt (()Max = 0°), representing use of only direct nonoblique data (actually, data within central co-polar bin).
The reconstruction time, for the full acceptance angle
data, on the SPARe 10 workstation, was 364 min for
the 3D-RP versus 7 min for the FORE reconstruction
(5.1 min for FORE algorithm and 1.7 min for multislice
image reconstruction using 2D-FBP method) speedingup reconstruction about 50-times.
3.50
~
2D-FBP
3.00
FORE 3
1=1 2.50
FORE 7
0
.~
ro
'>
FORE
2.00
SSRB
4)
[I
Q
1.50
~
"C)
1.00
U5
0.50
a
0.00
0.00
3D-RP
50.00
100.00
150.00
200.00
those slices. The· reconstruction using SSRB has very
similar noise levels (62% for SSRB versus 57% for FORE
in the center axial locations) and general noise behavior.
3D-RP has similar noise level in the center slices (54%),
but the noise increases considerably slower while moving from the center axial location (e.g. the noise standard deviation is 92% for 3D-RP and 169% for FORE, at
z=100mm from the center). This is due to an effective
improvement of statistics in the oblique angles given by
the estimation of missing portions of the projection (xray transform) data in the 3D-RP. However, the improvement in noise comes with a loss of resolution (as shown
later) in the far off-center slices, which are more influenced by the estimated data. Actually, we have found
that if we individually filter (using 2D smoothing kernel) each slice, in the multislice reconstruction, by such
an amount that the axial noise behavior and noise levels
of FORE are matched to those of the 3D-RP, then we
obtain also matched transverse (radial and tangential)
resolution in both of the methods.
250,00
18.00 ...---.,---r----,r----r--r----r----,r-----, r - - - - ,
Slice number [mm]
S
u
Axial
16.00
'Figure 1: Noise standard deviation values at different ax- §
ial locations inside the image of cylindrical phantom recon1=1
o
structed using: 2D-FBP - multislice reconstruction using only
nonoblique data (acceptance angle ()Ma:c = 0°); FORE 3, 7 - . ~
FORE reconstruction from limited acceptance angle data us~
ing center 3, 7 tilts (8Ma:c = ±3.75°, ±11.25°, respectively);
and complete data (8Ma:c = ±26.25°) reconstruction using
FORE, SSRB and 3D-RP.
Radial
14.00
12.00 _______•_____.__ .. _. __
!..~~:~~~
FWTM
10.00
:::
===~=.~::~:::=.:::::::;:"--,,..
~=--r;;,;"n";;;;; =.=:::::::__-::-_==;:::.~;:,._==="="'~nn""
....==·::..,...;;
..
___
MUU
FWHM
4.00
2.00
0.000.00
3.75
7.50
11.25
15.00
18.75
22.50
. 26.25
Axial Acceptance Angle' [deg]
[1
[J
u
r
The noise behavior was studied using data containing
90M counts collected from a cylindrical phantom of di- Figure 2: FWHM and FWTM of FORE reconstruction, in
ameter 20cm and height 24cm. Noise standard deviation the center slice at radius lOOmm from the center, as function
was calculated inside a circular ROI of diameter 15cm of the axial acceptance angle.
centered in the cylinder slices. The presented results are
for images reconstructed without any axial or transverse
filtering in both 3D and multislice 2D reconstructions.
The reconstructions using filtering exhibited a similar
relative behavior. Fig.l shows the noise standard deviation (in % of the ROI mean value) as a function of the
slice axial location. The noise levels, reflecting sensitivity of the scanner for different axial locations, decrease
from the value similar to the 2D case, at the extreme
slices, to approximately one fifth of that value, in the
center slices. This axial nonuniformity of the noise level
can be undesirable, especially for whole body scans, and Figure 3: FORE reconstruction (Transverse slice - left imcould require normalization or overlapping of the PET age, Sagittal slice - right image) of a small sphere from
measurements of the consequent body axial sections. It simulated data containing only two extreme tilts of angle
can be clearly seen from the graphs, that by reducing 8 = +26.25° and 8 = -26.25°.
the acceptance angle we get uniform noise behavior in
For resolution (point spread function) measurements
the center slices but for the cost of noise increase within
L
')
I
11997 International Meeting on Fully 3D Image Reconstruction
251
18.00
18.00
-
16.00
,......,
E
14.00
E!
~
12.00
CU
~
4.00
8
'-I
-
0
~
].....
FWHM
-
2.00
0.00
S 14.00
oS 10.00
8.00
6.00
:a
-
~
::I
CI)
L,.,..,/
14.00
16.00
J::: 12.00
0
'.z:l 10.00
~
~
S
S
-
.9
s:::
'0
,......, 16.00
18.00
--
0
40
80
120
Radial Position [mm]
~I)
fa
[-l
.2
~
-
10.00
~
8.00
6.00
~
';j
6.00
~
4.00
2.00
0.00
A 3D-RP
z= 100
.........................
::l
(/)
-
D,3DMRP z=Q
.....................
12.00
'0
FWHM
OPORE z=Q
• FORE z=lOO
I=l
8.00
4.00
-
FWHM
2.00
a
40
120
80
Radial Position [nun]
0.00
0
40
80
120
Radial Position [mm]
Figure 4: Resolution of the FORE and 3D-RP reconstructions at different radial locations in the center slice (z=Omm; open
symbols) and near the extreme z location (z=100mm; closed symbols) of the scanner axial field of view.
we moved a small Imm point source to different radial
and axial positions. Data were not corrected for scatter or attenuation since the point was suspended in air.
The resolution was characterized by measuring the fullwidth at half-rnuximum (FWHM) and the full width at
tenth-maximum (FWTM) in radial, tangential and axial
directions, using linear interpolation between sampling
points. This was done rather than fitting a Gaussian
profile, as the point spread function profiles are rarely
Gaussian in practice. Fig.2 shows dependence of the
resolution measures on the acceptance angle size, for the
point source located in the center slice and close to the
extrelne radial position (at lOOmm) - the most challenging location for the FORE algorithm (other z locations
are less challenging because missing portions of the data
reduce actual acceptance angle size). We can see that
by increasing the acceptance angle the axial resolution
deteriorates (with radius as shown later) as predicted by
Defrise [2]. However, the radial and tangential resolution is not affected by the Fourier reb inning at all, with
the exception of tangential FWTM where it actually improves (this is caused by poor noise statistics for low acceptance angle case, causing low amplitude streaky artifacts influencing tangential FWTM): Fig.3 shows a small
sphere phantolIl reconstructed from the idealized (complete, line integral) projection data. It was reconstructed
using only the two extreme tilts of angle B ::: +26.25 0 and
B = -26.25 0 • We can see that ev'en in this extreme case
the resolution within the transverse slice (left image) is
not affected. The sagittal slice (right image) shows the
typical axial smoothing and axial artefacts caused by the
FORE approximation (it is fair to note that by using all
of the tilts these effects have lower overall contribution
to the final image). When using SSRB for the same data,
we observed very strong tangential and axial artefacts,
as expected.
Fig.4 shows comparison of the resolution of the FORE
(solid line) and 3D-RP (dotted line) at different radial
locations in the center slice (open symbols) and slice near
the axial FOV boundary (z::::lOOmmj closed symbols). In
the center slice, the methods have comparable resolution
in both radial and tangential direction and the resolution slightly deteriorates with the radius. The exception
is tangential FWTM which is deteriorated in the 3D-RP
by the low contrast streaky artefacts. In the "boundary"
slice the 3D-RP (solid black triangles) has slightly worse
radial and tangential resolution comparing to the FORE,
as discussed in the paragraph describing the noise behavior. The major difference between the methods can be
seen in the axial resolution, where in the center slice (reconstructed from the full acceptance angle data) FORE
resolution deteriorates with the radial. distance from the
center. However in the extreme z locations, where the
axial range of actually detected data is limited, i.e. effective acceptance angle is limited for FORE (to about 3
tilts, or BMax = ±3.75°, at z=lOOmm) whHe substantial
amount of data is estimated for the 3D-RP, the axial resolution is consistently better for FORE for all radii (the
transverse filtering,. to match the noise levels of both
methods, does not affect the axial resolution).
119!;}7 International Meeting on Fully 3D Image Reconstruction
261
FORE 10 deg
FORE 16 deg
FORE 26 deg
FORE 36 deg
SSRB 26 deg
3D-FBPM 26 deg
11
II
[J
[]
Figure 5: Comparison of the FORE reconstruction, using several sizes of axial acceptance angle, with the SSRB and 3DFBPM (3D-RP) using large acceptance angle data. First row shows Transaxial slices and second row shows Sagittal slices
of the simulated "box" phantom.
Fig.5 is an illustration of the practical performance
of the FORE algorithm for axial acceptance angles
() a corresponding to different PET scanners: common·
whole body PET scanner of moderate acceptance angle
- (}a
±10° (used for previously published FORE studies [3, 4]), large FOV whole body scanner - (}a = ±16°)
large FOV brain scanner (for example HEAD PENNPET scanner) - (}a = ±26°) and limiting case of coincidence imaging scanner using a pair of gamma cameras - (}a = ±36°. For comparative purposes we are
showing also SSRB and 3D-RP reconstructions, for the
large acceptance angle case. We used simulated idealized
(line integral) projection data of the "box" mathematical phantom filling practically the whole axial and radial FOV. The geometrical characteristics (sizes) of the
data were same as those described earlier. To obtain the
same amount of nonmissing data we have changed the acceptance angle by changing the diameter of the scanner
(keeping the same axial FOV). Visual appearance confirms that FORE is providing very good results for the
moderate acceptance angles (as published before), but
deteriorates with radius for the high acceptance angles.
It is clear that SSRB does not provide acceptable results
for the large acceptance angle. It can be seen that the
3D-RP is not perfect, as well, for the large acceptance
angle data (as discussed earlier), since large amounts of
data must be estimated.
=
\1L J
[]
[]
o
[]
III.
CONCLUSIONS
With increasing acceptance angle the axial resorllition deteriorates, while the transverse resolution is affe~t'~cl only
very little. For the acceptance angle of ±26°, the noise
levels and transverse resolution of FORE can be matched
with those of 3D-RP. In the axial direction, FORE has
worse performance for the full acceptance angle data. On
the other hand 3D-RP has worse axial resolution in those
regions which were reconstructed from large amounts of
estimated (reprojected) data. More detailed studies and
clinical examples will be presented in the conference paper.
REFERENCES
[1] P. E. Kinahan and J. G. Rogers, "Analytic 3D image
reconstruction using all detected events," IEEE Trans.
Nucl. Sci., vol. 36, pp. 964-968, 1989.
[2] M. Defrise, "A factorization method for the 3d x-ray
transform," Inverse Problems, vol. 11, pp. 983-994, 1995:
[3] M. Defrise, P. E. Kinahan, and D. Townsend, "A new
rebinning algorithm for 3D PET: Principle, implementation and performance," in Proceedings of the 1995 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, pp. 235239, Aix-Ies-Bains, France, 1995.
[4] M. Defrise, M. Sibomana, C. Michel, and D. Newport,
"3D PET reconstruction with the ECAT EXACT HR using Fourier rebinning," in Proceedings of the 1995 IEEE
Nuclear Science Symposium and Medical Imaging Conference, vol..~, pp. 1316-1320, San Francisco, CA, 1995.
[5] J. S. Karp, R. Freifelder, P. E. Kinahan, M. Geagan,
G. Muehllehner, L. Shao, and R. M. Lewitt, "3D imaging characteristics of the HEAD PENN-PET scanner,"
Journ. Nucl. Med., 1997. In press.
As confirmed by our study, FORE reconstruction is providing good results for low axial acceptance angle data.
11997 International Meeting on Fully 3D Image Reconstruction
27\
High Resolution 3D Bayesian Image
Reconstruction for microPET
tJinyi Qi, tRichal'cl M. Leahy, tErkan U. Mluucuoglu, *SiInon R. Cherry
tArion Chatziioannou, *Thomas H. Farquhar
tSignal and llnage Processing Institute, University of Southern California, Los Angeles, CA 90089
*Crtunp Institute for Biological Imaging, UCLA School of IVleclicine, CA 90024.
Abstract- A Bayesian method is desc1'ibed fOl' high l'eSO
lutioll reconstruction of images fl'om the UCLA micl'oPET
system. High l'esolutioll images are achieved by model..
illg spatially vadant geometl'ic efficiency, intrinsic detoctor
efficiency, photon pail' llollHColinol.U'lty, llon .. uuiforll1 sin0
gram sampling, Cl'ystnl penoh'ation and 11ltOl'"cl'ystai scat..
tel'. To reduce stol'age and computatiollal l'oquiroments,
nonacoHnoarity, intet'~cl'ystal scat tel' and penetl'ation are factored from the geometric matrix and appl'oximated using 20
sinogl'am bhu'rillg kernels. Fm,thel' savings in storage were
realized by exploiting sinogl'am symmetries. We demonstl'ate that inclusion of the siuogl'am blurring kernels can
produce almost uniform l'adial l'esolution of approximately
lmm FWHM out to a l'adiuB of 4cm. Further improvements
in l'esolutioll were obtained using wobbling. By choosing the
relative wobble positions to form the 4 opposing corners of·a·
single voxel, we avoid any inCl'ease in the storage required to
l'epl'eSent the projection matrix. Preliminary results using a
O.5mm point source indicate that we can achieve resolution
better than lmm FWHM in all directions with radial and
tangential resolutions approaching O.5mm at the center of
the field of view.
TABLE I
M
3D PnODr,E:-'i DI:-.m:-lSION Fon MlCnoPET
8
I.
INTRODUCTION
We have developed a 3D maximum a posteriori (MAP)
reconstruction method for the microPET system with the
goal of prod ueing high resol u tion images through accurate
system modeling and system wobble, while attempting to
minimize the storage and computational costs involved in
computing these images. MicroPET is a high resolution
PET scanner designed for imaging small laboratory animals [1). It consists of a ring of 30 position-sensitive scintilw
lation detectors, each with an 8 x 8 array of small lutetium
oxyorthosilicate (LSO) crystals coupled via optical fibers
to a multi-channel photomultiplier tube. The detector ring
diameter of microPET is 172 mm with an imaging field of
view of 112 mm transaxially by I8mm axially. The scanner
has no septa and operates exclusively in 3D mode.
The 3D MAP reconstruction method was developed to
achieve as high resolution as possible. We use the standard
Poisson model for PET data in which the measured projection data are independent Poisson random variables with
means equal to a linear transform of the mean positron
intensity in each voxeldefined by the system projection
matrix P. Since we are looking at small animals, scatter
in the data can be ignored. Similarly, the low count rates in
small animals result in very small randoms fractions, and
these are also ignored in our model. The blurring effects
due to non-colinearity, inter-crystal scatter and penetration, are included as a local, spatially variant blur in the
ring diameter, mm
detectors per ring
number of rings
angles per sino gram
rays per angle
sinograms
projections per sinogram
total projection rays
object size, mm
object size, voxels
voxel size, mm
full size of Pij
size of P Geom with symmetry reduction
size of P Blur
size of P Attna11dPEll
actual storage size of P
_.
172
240
8
120
100
64
12,000
768,000
100 x 100 x 18
128 x 128 x 24
0.75 3
3 x 1011
16 Mbytes
25 Kbytes
1.4 Mbytes-18 Mbytes
sinogram space as described in [2]. These factors were
determined using Monte Carlo calculations. Further savings in storage requirements were realized by exploiting the
symmetries in P in a similar manner to that described in
[3] and [4]. The method was applied to FOG data from
a baby vervet monkey study. Resolution was investigated
using a 0.5mm point source. We also investigated the effect
of using a 3D wobble to further enhance spatial resolution.
II.
IMPLEMENTATION OF
3D MAP
RECONSTRUCTION
A. Factored System Model
The theory of 3D MAP reconstruction is essentially the
same as for the 2D problem, differing only in the specifics
of the projection matrix [5). Even for microPET, which is
small compared to the latest generation of clinical 3D PET
systems, the original size of the P matrix is huge (Table I).
Clearly it is necessary to store this matrix in a sparse form
and to exploit symmetries as previously described in [3) [4].
In developing our 3D approach, we have also extended the
use of the factored system matrix that we presented previously for 2D PET [6], (2). Here we outline the development
of this model for the microPET system.
In order to reduce the storage size of the P matrix, we
11997 International Meeting on Fully 3D Image Reconstruction
281
factor the P matrix as follows:
P
[J
r"-]
l
J
ilL'' '
',-lJ'
[
-:,
[_
"]"
[
11
LJ
r-
1
l"l
__.1
[
-J)
l
". ·J1
[.
,
r-,',
1,1
t_, 1
r_--']
\_
r"]
l__
(-1
lJ
(1)
PGeom is the geometric projection matrix with each element (i, j) equal to the probability that a photon pair
produced in voxel j reaches the front faces of the detector
pair i in the absence of attenuation and assuming perfect
photon-pair colinearity. It is calculated based on the solid
angle spanned by the voxel j to the faces of the detector
pair i.
Although the full size of P Geom is extremely large,
P Geom is very sparse and has redundancies of which we
can take advantage to reduce the storage size. By choosing
the voxel size in the z direction to be an integer fraction of
the ring distance, there are the following symmetries in the
PGeom matrix [3] [4]. In-plane rotation symmetries, resulting from rotating the projection rays by () = 90°,180°,270°,
and a () = 45° reflection symmetry provides a tot.al factor
of 8 reduction (with the exception of angle () = 0° and
the ray passing through the image center where there is
no reflection symmetry). Axial reflection symmetry provides an additional factor of 2 reduction for ring difference
other than zero. The symmetry of sinograms with a common ring difference Rd provides a reduction by a factor
of (Nr - Rd) where N r is the number of rings in the system. Combining these, the total reduction factor from the
symmetry operations is approximately 64 for microPET.
Therefore, we need only store the nonzero components of
the base-symmetry lines of response (LORs) , which amount
to 12440 rays for microPET. Efficient storage of these components is achieved by storing the elements in an ordered
fashion so that only the base pixel index need be stored
with the remainder generated using automated indexing.
P Blur is the sinogram blurring matrix used to model
photon pair non-colinearity, inter-crystal scatter and penetration. In principal, uncertainties in the angular separation of the photon pair should be included in P Geom. However, this will reduce the sparseness and and result in a loss
of some of the symmetry characteristics in P Geom, leading
to a dramatic increase in storage size. To a reasonable approximation, the non-colinearity effect can be assumed to
be depth-independent and hence can be lumped into the
P Blur matrix.
These effects can result in radial, angular and intersinogram blurring so that a 3D sinogram blurring model
should be used. In our current implementation, we have
assumed that these blurring effects can be confined to a
single sinogram and use' a 2D blurring model. Furthermore, we assume that the blurring kernels are identical for
sinograms for all ring differences. By treating each crystal
as a separate detector and ignoring the crystal block effect,
we can assume a rotational symmetry of the blurring kernels due to rotational invariance of the detector geometry
[2]. Therefore, we only need compute and store the blurring kernels for the projection rays for a single projection
angle, which saves both computational time and storage.
The blurring factors were computed using the Monte
Carlo code described in [2]. Statistical modeling of non-
-]
U
,"
= PEJjPBlur P AttnPGeom
ray-1
o
0
0
0.0062
0.0063
0
0
0.0103
0.1169
ray-3
ray-2
0
0.0014
0
0
0.0018
0
0.0092
0.0055
0.0100
0
0.0099
0
ray+2
0
0.0432
0.2459
0.0449
0
ray+1
0
0.0028
0.1686
0.1711
0.0025
ray+3
0
0.0056
0.0600
0.0604
0.0052
ray+4
0.0016
0.0077
0.0140
0.0079
0.0014
ray+5
0
0.0019
0.0019
0.0018
0.0022
ray+1
0
0
0.0618
0.0630
0
ray+2
0
0.0081
0.0060
0.0093
0
ray+3
0
0.0017
0
0
0.0022
(a)
ray-1
0
0
0.0651
0.0658
0
I
0
0
0.0058
0.6778
0.0054
0
(b)
Fig. 1. Sinogram blurring kernels for the sinogram component shown
by bold face corresponding to (a) the 20th and (b) the 50th ray
out of the 100 projection rays.
colinearity, crystal penetration and inter-crystal scatter in
the LSO detectors was used to prod~ce the blurring of the
sinogram element under consideration into the n~ig~boring
elements. Figure 1 shows two examples of these 'blurring
kernels. Note the significant crystal penetration for offcenter detector pairs due to the small crystal size.
The attenuation matrix P Attn is a diagonal matrix that
contains the attenuation correction factors for each detector pair and can be obtained from the ratio of a blank to
transmission scan. Alternatively, it can also be computed
by forward projecting a reconstructed attenuation image.
In the monkey images presented below, we assumed uniform attenuation throughout the head and computed the
factors using an estimated head volume determined from a
preliminary emission reconstruction.
The detector efficiency matrix P E JJ is again a diagonal
matrix. Each element is computed using daily calibrations
from a uniform cylindrical source.
B. Reconstruction
Using the factored matrix approach we gain substantial
savings in both storage and computational requirements.
We note however I that to realize this saving, it is necessary
to consider all data and all pixels at each iteration. Therefore, we cannot use coordinate-wise methods nor ordered
subset methods. Here we use the preconditioned conjugate gradient approach applied previously to 2D PET data
in [6].
C. 3D wobble
To further improve the spatial resolution, a 3D wobble
was investigated. For each wobble position we potentially
need to compute and store a different P Geom matrix. To
overcome this problem, we choose each wobble position to
be sited at the corner of an image voxel. In this case, rather
than generate a new PGeoml we simply perform a corresponding shift of the pixel indices for each new wobble po-
11997 International Meeting on Fully 3D Image Reconstruction
ra,Ulruolullon
400
I
-measured
'1'
- - prcdicled wllh full modol
350
",
I
....• prediclcd wllhoul blurrklg kernels
,.
t'5 . ,....... :..... ;. __
300
.j
1 2 - - - - - _... . _ •. -
-'
"~I
',::,-,>'/
•.
..
II
"
I! 1.5
.
---.".. ..- ..
0.5
00
0.8
1.6
2.8
4.8
3.6
oils.' (cml
(a)
Itngtn~61li1l1oMon
-3
-2
-1
offsol(cm)
3.6
1:-
2
.6
12
sition and use the same PO eom • This approach also avoids
the need for any interpolation when moving between the
different wobble data sets. The wobble motion we used in
this study has four positions with {x, y, z} indices (0, 0, 0),
(I, 1,0), (1,0, I), and (0, I, 1) where the voxel size was chosen as {.5mm x .5mm x .75mm}.
III. EXPERIMENTAL
A. Point Source Measurements
... -30W,P
• - 3D MAP wlthou' blUrMg ktlMll
3 .. , .'." 30FBP
Fig. 2. Pl'ofilcs of thc measurcd and calculatcd sinograms for the
point SOUl'CC at diffCl'cnt positions in the field of view.
....
....
.~ ~.,":'.:~.::~;-
.. ...... ~. - - ••.•. _._.- •. - - -
-.... _
1.6
0.6\- .. ·· .. · ....... ; .......... ; ......... , ................:................... ,
2.6
oII;oI(cm)
1.5
- 4.5
3.6
(b)
RESULTS
A 0.5 mm diameter, 0.5 mCi Na-22 point source was
scanned at different positions in the field of view of microPET. Each data set contained approximately 3,000
events per sinogram. To test our factored system model,
we compared the sinogram profile measured wit.h that. predicted using the factored system model, The results are
shown, with and without the inclusion of PBlur in Figure 2. There is clearly good agreement between the measurements and the full factored system model. By using
the blurring kernels, the asymmetry and peak shift due to
crystal penetration are successfully followed.
The data were reconstructed using the proposed 3D
MAP method. Because the Poisson likelihood with positivity constraint can create artificially high resolution for
a point source in zero background, we also used a quadratically weighted least squares (WLS) method [8], without
a positivity constraint, to measure the resolution. Profiles
were taken through the point source images and the resolution determined by measuring the FWHM. Figure 3 shows
the resolution plots for the radial and tangential components of the transaxial resolution and the axial resolution
compared to the results obtained using the 3D PROMIS filtered backprojection method. These clearly show improved
resolution of the MAP method in comparison to FBP, with
and without the use of the blur kernels. As one would expect, the most dramatic improvement in resolution from
the use of the blurring kernels occurs in the radial direction where we see a resolution of Imm FWHM' out to a
4cm radius. The axial resolution is not improved by the
...
.ldlllillOlut!on
:::J:
-: :. ~l
3.5
3
•f . E
:
'
- :
::- : ::: wliloul blunlno ktrnill
........... ·_ .. 30FBP
:
. " ...
............................................................ ..
~2.6
t ~:::,c>":.~"";" :.. -...: • • .
&
...... "': - - _.• - -: ....... _....... . .
1
.... ,,'
....................\....
:
:
...............
.
:
..( ...... ..
:
0.6 ............... , ........ , ........., ........ ~ .........:' .... "'... .
1.5
2.6
oIIH'(cml
3.6
4.5
(c)
Fig. 3. Resolution of the point source image using FBP and MAP,
with and without modeling of the sinogram blur: (a) radial res~
olution, (b) tangential resolution, (c) axial resolution.
2D blurring kernels because we do not currently model the
axial blur.
To examine the resolution improvements resulting from
the 3D wobble, we collected data for the point source at
three positions: 0 em, 1 cm, and 2 cm for each of the
four wobble positions. Figure 4 shows the resolution of
the reconstructed point ~ource, again using WLS. We see
in this case a resolution of better than Imm FWHM in
all directions and approaching 0.5mm transaxially at the
center of the field of view.
11997 International Meeting on Fully 3D Image Reconstruction
301
1.8 ....... ~ ....... (
...... , ..
1.6 ....... ; .. ······~·········i··
1.4
...
.
..
:
:
~
:
:
~
:
:
:
:
:
:
:
:
:
:
:
:
'-----'
········i·· ......;.........;........ .j .........;......... .: ........ -!-······ .. ·,.········i········
j~: __ I_:J:-:.j::I:I;~EI,~~:--;~.~;·
0.4 ..
0.2
[1
°0
[]
······1····· .. ··:·········~·········:····· .. ···;.. ········(···....!· .. · .. ·;· .... ·. ·l·· .. ·· ..
········f·········;·········f· .. ·.. ···f········t······ ..
0.2
0.4
0.6
0.8
t. . . . +........;......... ]........
1
1.2
offset (em)
1.4
1.6
3D MAP method with full system model
1.8
Fig. 4. Resolution of the 0.5 mm point source image obtained using
data from four wobble positions.
B. Monkey Images
Data were collected from a 3 month old baby vervet monkey scanned using the microPET scanner after injection of
~.. l.l 2.2mCi of FDG. The total counts were about 1 million per
sinogram for a collection time of 40 mins. Figure 5 shows
the image reconstructed by the 3D MAP method (with and
without use of the sinogram blur kernel) in comparison to
the 3D FBP method. The reconstructed field of view in
[".1J these
figures is a circle of diameter Bcm with the maximum
diameter of the brain approximately 6cm. These images
1.... appear to confirm the resolution enhancement observed in
l the point source studies. Note also that. modeling of penetration effects compensates for the spatial distortion that.
is seen towards the edge of the field of view in the FBP and
MAP images without blurring kernels.
[
3D MAP method without blurring kernels
11:
· ... ·.1).
3D FBP method
[
IV.
CONCLUSIONS
Fig. 5. Baby monkey brain images.
[l
We have described a fully 3D MAP reconstruction
j method for the high resolution microPET animal scanner. We have shown that we can model and deconvolve
\~l the system response within this framework to achieve unil J forlll transaxial resolution of 1mm FWHM for objects up
to a 4cm diameter, and a resolution of about 1.2mm up
to an Bcm diameter. Further improvements were achieved
[ 1 using wobble data. The computation cost on a Sparcsta... tion60 is on the order of 10mins per iteration or 1.5-3 hours
for a single study that requires 10-20 iterations. The use
of the blurring kernels increases the computation time by
_ approximately 10%. In comparison, the FBP algorithm
takes on the order of 30 minutes. These results are prelimfi inary - additional studies of the resolution and quantitative
I J properties of the MAP method applied to the microPET
. scanner will be presented at the meeting.
·..... 11
[2]
[3]
[4]
[5]
[
[I
V.
ACKNOWLEDGMENTS
[6]
[7]
[8]
This work was supported by the National Cancer Institute under Grant No. R01 CA579794
Imaging Conference, 1996.
E. Mumcuoglu, R. Leahy, S. Cherry, and E. Hoffman, "Accurate
geometric and physical response modeling for statistical image
reconstruction in high resolution PET," IEEE Medical Imaging
Conference, 1996.
C. Johnson, Y. Van, R. Carson et ai, "A system for the 3D reconstruction of retracted-septa PET data using the EM algorithm,"
IEEE Trans on Nuclear Science, Vol. 42, pp. 1223-1227, 1995.
C. Chen, S. Lee, and Z. Cho, "Parallelization of the EM algorithm for 3-D PET image reconstruction," IEEE Trans on
Medical Imaging, Vol. 10, pp. 513-522, 1991.
P. Kinahan, C. Michel, M. Defrise, "Fast iterative image reconstruction of 3D PET data," IEEE Medical Imaging Conference,
1996.
E. Mumcuoglu, R. Leahy, and S. Cherry, "Bayesian reconstruction of PET images: Methodology and performance analysis,"
Physics in Medicine and Biology, pp. 1777-1807, 1996.
E. Mumcuoglu, R. Leahy, S. Cherry, and Z. Zhou, "Fast
gradient-based methods for bayesian reconstruction of transmission and emission PET images," IEEE Trans on Medical Imaging, Vol. 13, pp. 687-701, 1994.
J .A. Fessler and W.L. Rogers, "Spatial Resolution Properties of
penalized-likelihood Image reconstruction: Space-invariant Tomographs," IEEE Trans. Image Processing, Vol. 5, pp 13461358, 1996.
REFERENCES
[1]
S. Cherry, Y. Shao, R. Silverman et aI, "MicroPET: a high resolution PET scanner for imaging small animals," IEEE Medical
11997 International Meeting on Fully 3D Image Reconstruction
311
RECONSTRUCTION OF TRUNCATED CONE-BEAM PROJECTIONS
USING THE FREQUENCY-DISTANCE RELATION.
M. Defrise and F. Noo
Free University of Brussels (VUB) and University of Liege (Belgium)
Recently tho frcquency,.distnncc relation [1] for the 2D Radon transfonn has been applied to various problems in
tomography, such as ihe dcconvoiution of the distance-dependant blurring in SPECT [2,3] and the
reconstruction of 3D PET data by Fourier rebinning [4]. The aim of this paper is threefold: to propose a new
geometrical interpretation of the frequcncYMdistance relation, to generalize this relation to the case of fan-beam
sampling and of linogram sampling, and finally to apply these results to the approximate reconstruction of
truncated conc-bemn (CB) data acquired when the vertex of the CB projections moves along a helicoidal path. This
last problem has applications for future medical CT scanners based on detectors consisting of several rows
(lypically 32) of detector elements. The data measured with such devices are truncated axially (Le. along the
direction of Ule axis of the helicoidal path) and the only exact method known for truncated CB data [5] requires a
vertex path containing a circle, and is not applicable to the helix. Approximate algorithms have been developed
for helicoidal CB data [6,7] and the link between these methods and the frequency-distance relation will be
clarified.
1. A geometrical interpretation of the frequency-distance relation.
The frequency-distance relation for 2D paraIIelMbeam data
-too
I dt
Ppa"(S,</»::::
(1)
f(s cos</> - t sin</>, s sin</> + t cos</»
000
is derived [1] by applying the stationary-phase approximation to the 2D Fourier transform of the sinogram data
+00
P(ro,k)::::
2rc
I ds f dcp
000
0
exp(-i ro s - i k <1» Ppar(s,<I»
k E Z, ro
E
R
(2)
Specifically this approximate relation states that Lhe main contributions to P(ro,k) arise from sources located at a
fixed distance t:::: &k/ro along their respective line-of-response, where t is the integration variable in eqn (1). This
result can be seen as providing a kind of virtual time-or-flight information.
In view of the 2D central section theorem, it is not surprising that the frequency-distance relation can also be
interpreted in image space, as was already proposed in [1]. Calculate a line integral through the sinogram, as
2n
q(a,~) =
f d<l>
o
where the line
Ppar(a </> + ~,<I»
L(a,~)
slope of the line) and
={(s,<I» Is:::: a
(3)
<I> +~)
in the sinogram is parametrized using linogram parameters a. (the
p. Each point along L(a,~) corresponds to a straight line ,,-(s,<I»
in image space (Le. x y
space) of eq uation
x cos</> + y sin</> :::: s
11997 International Meeting on Fully 3D Image Reconstruction
(4)
321
Thus, L(a,p) corresponds to a family of lines in image space, and the value of q(a.,P) is the sum of the integral
of f(x,y) along all these lines. As can easily be seen (and as will be shown in the paper), the set of lines L(a,p)
il
has an envelope, which is a curve in image space defined by a certain equation
r a,p(x,y) = 0
l1
(5)
Two further results will be needed:
a) Intuitively, it is evident that the value of q(a.,P) receives contribution essentially from the points (x,y) lying
on the envelope (cfr figure 1), since it is in these points that the "density" of lines is the highest. Rigorously,
one can show that the contribution of a point (x,y) to q(a,p) is inversely proportional to the tangential distance
[I,J
between the point and the envelope.
<P
['j
sinogram
o
Figure 1 : each point along the line L(a,p) in the sinogram (left) corresponds to one line in image space (right)
[]
The second result is a remarkable geometrical property of the envelope:
b) Consider any point (x,y)
[]
E
r a,p, and
the corresponding line A(S,Q» which is tangent to
r a,p at
(x,y).Then
the distance t = -x sin</> + y cos</> between the point (x,y) and the projection of the origin (x=O,y=O) onto A(S,</»,
is equal to t = a, and is the same for all points along the envelope.
Combining a) and b), we now see that the value of q(a,p), the 2D Radon transform of the sinogram, receives
contributions mainly from points located at a distance t=a along their line of response. This approximate
property is the image space expression of the frequency-distance relation.
r- I
I
I
lJ
2. Generalization to fan-beam sampling.
Consider fan-beam data
[J
Pfan(O', </» = Ppar(R sinO', </> + n/2 - 0')
where the angle </> is the position of the X-ray source along a circle of radius Rand
(6)
0'
denotes the position on the
detector located along an arc of circle (figure 2, left).
[1
o
Following the same idea as with parallel-beam data, calculate line integrals of the fan-beam sinogram:
2n
q(a,p) = d</> Pjan(a </> + p,</»
(7)
f
o
The line L(a,p) = (O',</» 10'= a</>+ P} in the fan-beam sinogram corresponds to a family of lines in image
space, and the value of q(a,p) is the sum of the integral of f(x,y) along all these lines. The envelope of the set of
lines L(a,p) is a curve r a,p in image space, given by the parametric equation:
11997 International Meeting on Fully 3D Image Reconstruction
331
R
x == 2M2a. (lH2a) cos</> + cos[</> H2(a(~+B)])
y:::
2~a
(1-2a.) sin</> + sin[</> - 2(ac~+B)]}
(8)
This curve is an epicycloid and can be shown to have the following remarkable property (corresponding to
property b in section 1) :
Consider a point PEr a,~ and the line "-(<P;O') tangent to
r a,~ in P. Denote by A and n the intersections of
"-(</>,0') with the circle of radius R. Then the ratio
IAPI
1
t =lAB I = 2(1-a)
(9)
is the same for all points P along the envelope. Combining this property with the approximation a) in section 1
(which holds for any piece-wise smooth curve), leads to the following result: q( a,/3) receives contributions
essentially from points which divide the line joining the X -ray source to the detector in two segments of relative
length given by 1-2 a.
~
____________________~__________-+________________~x
x
B
Figure 2: definition of the fan-beam coordinates (left) and illustration of the envelope (right)
One consequence of eqn (9) is that, for consistent data, q(a,~) is negligible when t = 1/2(1-a) exceeds the range
defined by the support of the object. The property derived in this section can also, alternatively, be obtained as a
frequency-distance relation by applying the stationary-phase approximation to the 2D Fourier transform of the
data.
3. Application to helicoidal cone-beam CT scanning.
The result obtained in section 2 has been applied to reconstruct CB data acquired with a helicoidal orbit. The basic
idea is the same as in the Fourier rebinning algorithm for 3D PET data, and the algorithm is briefly sketched
below:
The (weighted) cone-beam data are described by
11997 International Meeting on Fully 3D Image Reconstruction
341
\.1
1
II
Pcb(cr, </>, v) =
f dt
o
feR cos</> - t R (cos</> + cos(</> + 20"),
(10)
R sin</> - t R (sin</> + sine</> + 20"», h </> + t v)
[I
where v is the axial coordinate of the detector relative to the X ray source, h </> is the axial coordinate of the
source, and cr and </> have the same meaning as for 2D fan-beam data, eqn (6). Note also that the integration
variable t is equal to the ratio IAPI/IABI in eqn (9) and in figure 2.
For simplicity we only consider the reconstruction of one slice, say zO = h re, from the helicoidal data measured
for 0 ~ </> < 2re (the proposed algorithm can be generalized for short-scan reconstruction).
Our aim is to estimate the 2D fan-beam data for a slice, zo = h re, i.e. Pjan(cr, </>, ZO). To achieve this, we first
calculate the 2D Radon transform of Pjan(cr,
<1>,
zO), for each a and
~,
as :
2re
o
o
n
o
D
lJ
q(a,~, ZO)
= f d<l>
o
pcb(a
<I>
+ ~,
<1>,
v(</>, a, ZO))
(11)
where for each ray the axial coordinate on the detector is taken as :
(12)
v(</>, a, zo) = 2 (1-a) ( zo - h<l»
This choice is such that each ray contributing to the integral in eqn (11) intersects the slice at the point that. gives
the main contribution according to eqn (9).
For the values of a such that the value t = l/2(1-a) given by equation (9) is outside the known support of the
image, the frequency distance relation cannot be applied, and we use instead the value t = 1/2, thereby selecting
for each ray the value of v such that the ray intersects the slice in its mid-point. For consistent data the
corresponding values of q(a,~, zO) are small (this problem is closely related to the handling of low frequencies in
the Fourier rebinning algorithm [4]).
When
q(a,~,
zo) has been calculated for all values of a and
beam data Pjan(cr,
<1>,
~
required to obtain a sufficient sampling, the 2D fan-
zO) are recovered using a 2D inverse Radon transform. The image is finally recovered using
any 2D reconstruction method.
This algorithm has been implemented, and a first series of simulations demonstrates improved image quality
compared to that obtained by using instead of eqn (9) the value t = 1/2 for all
(a,~).
This latter approach is
equivalent to the single-slice rebinning approximation used in 3D PET. A theoretical and practical comparison
with the more accurate algorithms proposed by Wang et al [6] and by Schaller et al [7] will be presented at the
[I
[]
[]
[]
conference.
[1] Edholm P R, Lewitt R M and Lindholm B, Int Workshop on Physics and Engineering of Computerised
Multidimensional imaging and processing, Proc of the SPIE 671 8-18, 1986
[2] Glick S J, Penney B C, King M A and Byrne C L, IEEE Trans Med Imag MI·13 363-74, 1994
[3] Xia W, Lewitt R M and Edholm P, IEEE Trans Med Imag MI·14 100-115, 1995
[4] Defrise M, Inverse Problems 11 983-994, 1995 .
[5] Kudo H, SaitoT, Proc 1994 Nuclear Science Symposium (Norfolk VA)
[6] Wang G , Lin TH, Cheng PC, Shinozaki DM, IEEE Trans Med Imag MI·12 486-496, 1993
[7] Schaller S and Flohr T, to appear in the Proceedings of the 1997 SPIE Symposium on Medical Imaging
(Newport Beach)
rl
J
11997 International Meeting on Fully 3D Image Reconstruction
351
Fast and Stable Cone-Beam Filtered Backprojection Method for Non-Planar Orbits
Hiroyuld Kudo
and
Tsuneo Saito
Institute of Information Sciences and Electronics, University of Tsukuba, Tsukuba, 305 Japan
[Extended Abstract]
Cone-beam tomography aims at recovering a 3-D object from a set of line integrals crossing
a specified orbit. This problem possesses various medical imaging applications such as volume xray CT and cone-beam SPECT. The standard approach to cone-beam tomography has been to use
the single circular orbit and implement the Feldkamp approximate filtered backprojection (FBP)
reconstruction method. However, this approach suffers from severe axial image blurring and low
quantitativity unless the orbit is a sufficiently large circle. This drawback can be overcome by
using non-planar orbits such as two-orthogonal-circles orbit and circle-and-line orbit. There exist
two classes of exact reconstruction methods for the non-planar orbits. The first class converts
a set of projections into the 3-D Radon data and then inverts it after the reb inning step. This
class was developed by Tuy, Smith, and Grangeat and called RADON method. The second class
was developed by Kudo and Saito [1] and Defrise and Clack [2]. This class also converts a set of
projections into the 3-D Radon data but each projection is backprojected independently into 3-D
space as in the standard Feldkamp method. This class can be considered a kind of FBP where each
projection undergoes space-variant filtering (not space-invariant filtering). The space-variant FBP
method is advantageous over the RADON method from a few practical perspectives such as memory
space but the original space-variant FBP method has the following drawbacks compared with the
standard Feldkamp method. First, computational time of the space-variant filtering step is huge
due to explicit computation of the 3-D Radon data. Second, the space-variant filtering introduces
considerable discretization errors due to resampling and interpolation. Some researchers overcome
these drawbacks by using the mathematical equivalence between the space-invariant filtering and the
space-variant filtering [3-5]. However, these works are limited to specific orbits, do not lnaximally
utilize the data redundancy, and lack generality; The contribution of this paper is to develop a more
general FBP method which maximally utilizes the data redundancy, enables fast implementation,
and significantly reduces discretization errors of the space-variant FBP method. The proposed
method is based on the hybrid FBP framework developed by Kudo and Saito [3]. The hybrid FBP
framework utilizes the mathematical equivalence between the ramp filtering and the space-variant
filtering where each projection undergoes both the ramp filtering and the space-variant filtering before
the backprojection into 3-D space. Kudo and Saito [3] demonstrated that the hybrid FBP method
produces various benefits such as improvement of data sufficiency condition for exact reconstruction
and possibility of region of interest reconstruction for rod.:like objects. This paper demonstrates that
the hybrid FBP method with adequately designed filtering weights enables fast implementation and
significantly reduces discretization errors.
We explain the theory. We represent an object supported in a finite region n by F(x) where
x = (Xl,X2,Xa)T. Let us consider the situation where cone vertices lie along the orbit expressed
as ¢>(A) :;:;;: (¢>1(A),¢>2(A),¢>a(A))T j A E A. We represent cone-beam projections by G,,\(Y,Z) where
(Y, Z)T denotes a position on the detector plane defined such that the Z-axis coincides with the
tangential direction of the orbit. Let D,,\ denote the source-to-detector distance. The general hybrid
FBP method consists of the following three steps. The first step is to multiply each projection by
the weighting factor as in the Feldkamp method. The second step is to modify each projection by
adding the ramp filtering result with the weight will and the space-variant filtering result with the
weight Wi 2 )(r,O). The third step is to compute the cone-beam backprojection as in the Feldkamp
method to obtain a reconstructed image. Mathematically; this procedure can be written as follows.
F(x) :;:;;: C[B 1 (W1 ») + B 2 (W1 2 )(r, O))]AG,,\(Y, Z)
l
11997 International Meeting on Fully 3D Image Reconstruction
(1)
II
!
!
<Weighting Operator A >
C(2)(y Z) - G (Y Z)
). , -).. , J D~ +D)..y2 + Z2
fj
(2)
<Ramp Filtering Operator B 1 (W2»)>
[-1
[]
(3)
<Space-Variant Filtering Operator B2 (W~2) (r, B))
>
(4)
[]
p(4)
(r , B) -- ~
p(3) (r , B)
)..
dr)"
[]
p~5)(r,B)
= '--pl4)(r, 0)Wl 2 )(r, 0)
G~3)(y, Z) = j7r/2
, [1
.J
(5)
" ¢'(A)
II
pl5)(y sin B+ Z cos 0, O)dO
-7r/2
(7)
~; ."oi-'i."
G(4)(y
)..
, Z) =. ~G(3)(y
dZ)..
, Z)
[J
(6)
(8)
<Cone-Beam Backprojection Operator C >
[]
[J
[j
1
F(x) = 41["2
r
D)..
(9)
wl
(10)
l.J
1Wi~)6((q,(>,)
= 1wi~)(e)6((q,(>,)
Ni1)(o
[]
2
Ni )(e)
- q,(>")) • e) I q,'(>.') . eI d>.'
=
[]
LJ
r:
(Y)..(x), Z)..(X))dA
where el denotes the unit vector which towards the detector center from the orbit point ¢(A). The
1
) and W~2) (r, B) plays a role to compensate for the data redundancy of
pair of filtering weights
acquired cone-beam projections. These functions must satisfy the following normalization condition
to achieve exact reconstruction [3].
fl
I_'
(4)
JA [(x _ ¢(A)) . el]2 G)..
w1
2
) (')
- q,(>")) . e) I q,'(>.') . eI d>.'
== w1 2 ) (r, B)
(11)
(12)
(13)
e
where denotes the unit normal of plane specified by (>', T, 0). A most natural class of the filtering
weights which maximally utilizes the data redundancy is characterized by
(14)
[J
M",(e) =
u
1
6((q,(>.) - q,(>")) . e) I q,'(>.') . eI d>.'
11997 International Meeting on Fully 3D Image Reconstruction
(15)
where M -\ (e) denotes the number of points on which the plane specified by (A, r, 0) intersects the
orbit. Geometrically, the choice of (ILl) implies that multiplly measured redundant 3-D Radon data
are averaged with equal weights. For example, this choice is desirable when the data contains noise
to maximally reduce effects of noise.
The following discussion considers only the filtering weights W~l) and VV~2) (e) satisfying (14)
but there still exist a number of such filtering weights. Therefore, it is desirable to choose the
filtering weights which enable fast hnplementation and reduce discretization errors bacause they
have significant effects on computaional time and discretization errors. The strategy to design the
filtering weights are explained as follows. The space-variant filtering causes more discretization errors
and requires more than ten times of computational time compared with the ramp filtering due to
explicit computation of the Radon transform (ll) and the backprojection (7). Thus, we wish to design
the filtering weights such that W~2) (r, 0) = 0 at a large number of points (A, r, 0) as many as possible
and the value of I W~l) 1/11 W~2)(r, 0) II for each A is sufficiently large where
(16)
r('\)
= {(r, 0) I
plane (A, r, 0) meets O}.
(17)
The condition W~2) (r, 0) = 0 at a large number of points ('\, r, 0) ensures that the space-variant
filtering can be omitted or implemented with little computational time. The sufficiently large I
wp) I / II W~2) (r, 0) II ensures that the space-variant filtering has little contribution to the filtered
projections, which produces the advantage that reconstructed iInages suffer from less discretization
errors. Let H-\(a) denote the histogram of the function l/M)"(e) with fixed ,\ which is computed
within the region f('\). The histogram H-\(a) has non-zero frequencies only at a finite number of
points a (a few in most cases) because 1/1\11-\(0 is a piecewise constant function fronl its definition.
Then, two simple choices of the filtering weights satisfying the above conditions are given by
<Ramp Filtering with Averaged Weight>
J
(1) _
Jr(-\) I/M)"(e)drdO
----.:...,.~---­
Jr().,) drdO
W-\
W~2)(r,0)
(18)
J
W~l) for (r,O)
= {oI I M-\(e) -
E
otherwise
r('\)
(19)
<Ramp Filterinr; with Most Frequent Weight>
W~l)
W~2)(r,0)
= argmax H-\(a)
a
(20)
W~l) for (r,O)
E r('\).
(21)
otherwise
The first choice performs the ramp filtering with the weight obtained by averaging the function
I/M)"(e) with respect to
The second choice performs the ramp filtering with the most frequent
value of the function I/M)"(e). Clearly, these choices greatly reduce the number of non-zero values
of the function W~2)(r, 0) compared with the simple case W~1) = 0 for all'\. The reconstruction
procedure of hybrid FBP method with the above filtering weights are summarized as follows. Process
each projection by the following five steps.
[STEP 1] Perform the weighting (2).
•
[STEP 2] Apply the ramp filtering (3) with the weight Wp).
•
= {I / M-\(e) -
o
e.
11997 International Meeting on Fully 3D Image Reconstruction
381
w1
n
II
I
L
[]
[;
[J
2
\ r, fJ) efficiently by using
[STEP 3] Apply the space-variant filtering (4)-(8) with the weight
the following two facts. First, it is not necessary to compute the Radon transform (4) and the
2
backprojection (7) for (r, fJ) such that
)(r, fJ) = O. Second, this step can be omitted when
2
•
) (r, fJ) = 0 for all (r, fJ) E r(A).
[STEP 4] Add the ramp filtering result with the space-variant filtering result.
•
[STEP 5] Perform the cone-beam backprojection (9).
•
We performed simulation studies to demonstrate the validity of the proposed method. The
3-D Shepp phantom is used and the two-orthogonal-circles orbit is assumed. The object support
is a unit sphere and the radius of orbit D is 3. Each projection consists of 256x256 pixels and
the number of projection is 240. The reconstructed image size is 256x256x256. We summarize
computational time and normalized mean squared errors (MSE) in Table 1 where four methods
(original space-variant FBP (FBP-ORG), linogram implementaion of FBP (FBP-LIN) by AxelssonJacobson et al. [6], hybrid FBP with averaged weight (BYB-AVE), hybrid FBP with most frequent
weight (BYB-FRE)) are compared. The results clearly show that the proposed method significantly
reduces both computational time and discretization errors of the original space-variant FBP method.
In particular, computational time of the BYB-FRE method is comparable to that of the Feldkamp
method.
[1]
[2]
[J
[3]
[!
[4]
J'
[5]
lJ
[6]
[1
Wl
wl
[References]
B.Kudo and T.Saito," Derivation and implementation of a cone-beam reconstruction algorithm
for non-planar orbits," IEEE Trans.Med.Imaging, 13, pp.196-211, 1994.
M.Defrise and R.Clack," A cone-beam reconstruction algorithm using shift-variant filterill'grand
cone-beam backprojection," IEEE Trans.Med.Imaging, 13, pp.186-195, 1994.
B.Kudo and T.Saito," Exact cone-beam reconstruction with a new completeness condition,"
Proc. of 1995 International Meeting on Fully 3-D Image Reconstruction in Radiology and
Nuclear Medicine, Aix-les-Bains (France), pp.255-259.
B.Kudo, T.Saito, and T.Takeda," 3-D computed tomography using cone-beam monochromatic
x-rays," Conference Record of 1996 IEEE Medical Imaging Conference, in printing.
F.Noo, M.Defrise, and R.Clack," FBP reconstruction of cone-beam data acquired with a vertex
path containing a circle," Conference Record of 1996 IEEE Medical Imaging Conference, in
printing.
C.Axelsson-Jacobson et aZ.," Comparison of three 3-D reconstruction methods from cone-beam
data," in 3-D Image Reconstruction in Radiology and Nuclear Med~cinej P.Grangeat and J-L.
Amans Eds., Kluwer Academic, pp.3-18, 1996.
FBP-ORG FBP-LIN HYB-AVE HYB-FRE
Method
[J
i]
[J\
Computational
Time (s)
circle
two-orthogonal-circles
Orbit
filtering
8845
1754
backprojection
Reconstruction eITOrs (MSE)
3289
518
Feldkamp
138
2880
1921.0
406.6
391.8
400.8
--
HP 735/125 workstation is used to evaluate computational time
Table 1 Computational time and normalized mean squared errors.
..
,1
I I
L.J
11997 International Meeting on Fully 3D Image Reconstruction
391
Iterative and Analytical Reconstruction Algorithms for Varying
Focal.. Lellgth Cone. . Beam Projections
G. Larry Zeng and Grant T. Gullberg
Departlnent of RadiologYt University of Utah, Salt Lake City, UT 84132, USA
Background:
The idea of using varying focal length collimation was first proposed by Hsieh in 1989 [1]. The
same idea was also suggested by laszczak et al, at the first Fully 3D Image Reconstruction Meeting
held in 1991 [2]. In a varying focal-length collimator, as shown in Figure 1, the focal lengths increase
from a nrlnimum at the center to a maximum at the edge of the collimator. In such a way the central
region of interest is imaged with short focal.Iengths and high sensitivity. Tissues close to the edge of the
patient body are imaged with nearly parallel rays, hence projection truncation, an inherent problem for
convergent beatn imaging, can be avoided.
Recently, various reconstruction algorithms have been developed for the varying focal-length
convergent bemn projections. laszczak et al, used an iterative algorithm to reconstruct the inmge [3].
Conjecture held that no convolution-backprojection exists for the varying focalMlength fan-beam
imaging geometries. In 1993 Zeng et al. developed a summed convolution-backprojection algorithln
that convolved the varying focal-length fan:..beam projection data with a series of kernels and
backprojected the sum of the convolved projections [4]. This algorithm was based upon the finite
approximation of the infinite series of orthogonal Chebyshev polynomials. Cao and Tsui in 1994
published a filtered backprojection algorithm with a spatially varying filter that could not be
implemented as a convolution [5]. Later an exact backprojection-filtering algorithm was proposed by
Zeng and Gullberg [6] for the varying focal-length fan-beam projections. Their algorithm first
backprojected the projection, then performed a two-dimensional shift-invariant filtering.
Goals:
This paper extends the 2D varying focal-length fan-beam algorithms to obtain the 3D varying focallength cone-beam algorithms. We must point out that in the 2D varying focal-length fan-beam
geometries, the projection data set is complete if the camera rotates in a circle. However, for the 3D
varying focal-length cone-beam geometries, the projection data set is incomplete if the camera only
rotates in a circle; Therefore, one can obtain only approximated algorithms, such as. the Feldkamp
algorithm for the fixed focal-length cone-beam algorithm [7], if the projection data are not sufficient. In
this paper the gamma camera rotates around the patient in a circular orbit or in a circular sinewave orbit.
The iterative algorithm used is the popular ML-EM algorithm, while the analytical algorithm is the
backprojection-filtering algorithm. [Note: A convolution-backprojection 'algorithm does not exist for the
varying focal-length convergent beam geometries.]
11997 International Meeting on Fully 3D Image Reconstruction
401
i
"
J
Focal-Length Function:
i1
i j
Each hole on the detector has its own focal-length: D(s), where s is the distance from the hole to
the center of the detector. This paper uses D(s) = a + ks 2 with a = 63 cm and k = 0.219 1/cm, and
the detector size is 64 x 64. The detector pixel size is 0.7 cm, and the detector is 22.4 cm (32 pixels)
from the axis of rotation. Let's consider a spherical object with a radius of 17.5 cm and a spherical
region of interest at the center with a radius of 7 cm, as shown in Figure 2.
The parallel, fixed focal-length cone-beam, and varying focal-length cone-beam geometries are
compared in Figure 2 and Table 1. It is observed that the varying focal-length geometry has almost the
same sensitivity for the central region of interest as the fixed focal-length geometry. The varying focallength geometry does not truncate the object, while the fixed focal-length geometry does.
r-;
IIu
Reconstruction Algorithms:
Table 1 indicates that the tilt-angles for the projection rays in the varying focal-length cone-beam
are small and close .to zero. Therefore the backprojection blurring is dominant in the planes vertical to
the detector. In [6] Zeng and Gullberg learned that a varying focal-length fan-beam image can be
exactly reconstructed by first backprojecting the projection data then filtering the backprojected image
with a 2D ramp filter. We propose to do the same for the 3D varying focal-length cone-beam geometry.
First a 3D voxel-driven backprojector is used to backproject the projection data. Then a 2D ramp filter
is applied to each slice vertical to the detector. The 2D filtering is performed in the frequency~()main,
and the image array is zero-padded before the Fourier transformation. The iterative ML-EM algorithm
for the varying focal-length cone-beam geometries is almost the same as the one for the fixed focallength cone-beam geometries [8]. A 3D ray-driven, line-length-weighted projector/backprojector pair is
used in the algorithm. The central slice is exactly reconstructed with either analytical or iterative
algorithms if a circular orbit is used.
[I
Computer Simulations:
A 3D Shepp-Logan head phantom was used in computer sim~lations; the focal-length function is
given in Table 1. The projection data were exact line-integrals calculated via analytical formulae. There
were 120 views over 360°. The reconstruction array was 64 x 64 X 64 and the image voxel size was 0.7
I
,
I
\
l_J'
fl
lJ
axis of rotation
beam
axis of rotation
varying focal-length beam
r~
1
1 \
LJ
I1
D(s)
LJ
r- 1[
Figure 1. Varying focal-length cone-beam geometry.
L~~
Figure 2. Comparison of three imaging geometries.
Truncation occurs in the cone-beam geometry.
[-]
'IJ
11997 International Meeting on Fully 3D Image Reconstruction
411
cm, equal to the detector pixel size. Figures 3 and 4 show the analytical reconstructions (central cuts)
with a circular orbit ancl a circular sinewave orbit, respectively. Figures 5 and 6 show the iterative MLEM reconstructions (central cuts) with a circular orbit and a circular sinewave orbit, respectively; 20
iterations were used in both cases. The orbits were defined as:
Orbit 1 ::::( cose, sine, O)T and Orbit 2:::( cose, sine,
Sin(~e)y
for 0 S; e < 21C.
Both circular orbit and circular sinew ave orbit were tested via con1puter siIllulations. The central
slice was exactly reconstructed if a circular orbit was used. The iterative ML-EM algoritlull was also
used to reconstruct the in1ages.
Conclusions:
The varying focal-length cone-beat11 iInaging geometry offers high sensitivity at the region of
interest while keeping the whole object within the field of view. An efficient analytical backprojection
filtering algorithm is proposed for this imaging geolnetry in order to take advantage of the small tiltangle of the projection ray. It is required that the detector has to parallel the axis of rotation.
References
[1] Hsieh J 1989 Scintillation camera and multifocal fan-beam collimator used therein United States Patent
. 4,823,017
[2] Jaszczak R J, Li J, Wang H and Coleman R E 1992 Three-dimensional SPECT reconstruction of combined
cone beam and parallel beam data Phys. Med. Bioi. 37 535-548
Table 1: Comparison of Imaging Geometries
collimator geometry
parallel
varying focal-length
cone-beam
fixed focal-length
cone~beam
focal-length function
(cm)
D(s) == 00
D(s) = 63 + 0.21s2
D(s) 563
normalized total count
of central region of
interest
1
1.2944
1.2948
radius of image in
central region of interest
7cm
(lOpixels)
9cm
(13 pixels)
10.86 cm
(15.5 pixels)
any projection
truncation for given
object?
no
no
yes
maximal tilt-angle of
projection ray
0°
5.5
at s=11.9 cm (i.e. at
17 pixels
~-
[max(tan- I _ S _ ) 1
s
D(s)
0
11997 Internatiohal Meeting on Fully 3D Image Reconstruction
19.57°
at s=22.4 cm (i.e. at
32 pixels)
r
I
j
r-'I
[3] laszczak R 1, Li 1, Wang H and Coleman R E 1992 SPECT collimation having spatially variant focusing
(SVF) 1. Nucl. Med. 33 891 [Abstract]
[4] Zeng G L, Gullberg G T, laszczak R 1 and Li 1 1993 Fan-beam convolution reconstruction algorithm for a
spatially varying focal length collimator IEEE Trans. Med. Imag. 12575-582
[5] Cao Z and Tsui BMW 1994 An analytical reconstruction algorithm for multifocal converging-beam SPECT
Phys. Med. Bioi. 39281-191
[6] Zeng GLand Gnllberg G T 1994 A backprojection filtering algorithm for a spatially varying focal length
collimator IEEE Trans. Med. Imag. 13549-556
[7] Feldkamp L A, Davis L C and Kress J W 1984 Practical cone-beam algorithm J. Opt. Soc. Am. A 1612-619
[8] Gullberg G T, Zeng G L, Tsui BMW and Hagins J T 1989 An iterative reconstruction algorithm for single
photon emission computed tomography with cone beam geometry Int. J. Imag. Sys. Tech. 1169-186
\
,
I
I
I
\.
J
Figure 3. Central cuts (left to
right: transverse, coronal,
sagittal) of backprojection
filtering reconstruction with
circular orbit.
[J
[I
Figure 4. Central cuts (left to
right: transverse, coronal,
sagittal) of backprojection
filtering reconstruction with
circular sinewave orbit.
fl
L)
r'"
'; \
L,.,.1
rl '1,
LJ
[]
Figure 5. Central cuts (left to
right: transverse, coronal,
sagittal) of iterative EM
reconstruction with circular
orbit. Twenty iterations were
used.
Figure 6. Central cuts (left to
right: transverse, coronal,
sagittal) of iterative EM
reconstruction with circular
sinewave
orbit.
Twenty
iterations were used.
r"l
I
I
I
, I\
L...J
11997 International Meeting on Fully 3D Image Reconstruction
431
Practical Limits to High Helical Pitch, Cone-Beam Computed Tomography
Michael D. Silver
Bio-bnaging Research, Inc., 425 Barclay Blvd, Lincolnshire, IL 60069, USA
We Lnvestigate the practical limits to high helical pitch, cone..ooam computed tomography. Helical, £One42eam
£Omputed lomogfaphy, HCBCT, merges one recent trend, helical scanning,l,2 with one potential breakthrough in
CT, cone..ooam scanning. 3.' While HCBCT is not yet a commercial product,6 theoretical and simulation studies,7-12
and experimental tabletop platfonns l3.14 have been reported. A practical HeBCf system has constant, axial
translational velocity of the patient table while the vertex of the cone of x rays-the x..ray tube-follows a circular
orbit with an opposing two-dimensional detector array that rotates with the tube, as indicated in Fig. 1. We define
helical pitch as the axial translation
velocity of the patient table, v, times the
rotation period, T. If w is the nominal
DETECTOR ARRAY
X·RAYTUBE
~
slice width (the axial aperture of a
single element of the x..ray detector
projected at the rotation center of the
scanner), then the helical pitch ratio,
r n , is given by
'h:_~. ._ _ . \
COLLIMATO~
fH
= vTlw,
-""'"::.."..
~
.........,.
c(~:~~
\\
-7Y;,-X-·
'--V~Ji
the ratio of the helical translation per
\ \
......
-<!:Ii:!'--gantry revolution to the projected axial CENTER OF I'
\\
aperture of a detector element. Current ROTATION Y
\\
\ \
commercial helical scanners, other than
\, \
the Elscint cr-Twin, have a single row
\
of detectors, require many rotations to
cover the patient volume of interest,
and generally keep rn S 2, although it
is a matter of some controversy as to the Fig. 1. Helical, cone-beam CT-scanner with x..ray lube and two-d/men ..
sional detector array rotating continuously around a patient underlargest rH and still maintain image
going
constanl translallon through the cone beam.
quality.l,2 HCBCT has the potential to
cover the same patient volume with just
a few gantry rotations, depending on the axial coverage of the two-dimensional array of detectors, for very fast scan
times. Applications include screening procedures when patient throughput is of prime importance and
CT-angiography, crA, where the short scan tiines will improve CTA-imaging because of less susceptibility to
patient motion artifacts and expand the scope for dynamic studies. U
We ignore the question of mathematical completeness of the Radon space for this three-dimensional scanning
geometry. Instead, we rely on the weaker two-dimensional completeness condition as a guide as to when image
quality-the relative absence of artifacts-can be expected to be adequate. That is, the three-dimensional
completeness condition that every plane that intercepts the object must also intercept the source orbit16 is replaced
by the condition that every line through a reconstructed slice must intercept the projection of the source orbit onto
the plane of the slice. We use the weak (or two-dimensional) completeness colldition to estimate a maximum rn
where the image quality might be adequate, at least for screening and CTA where short scan times are more
important than the best possible diagnostic image quality.
A given cross-sectional slice in HCBCT is continuously irradiated as it translates through the rotating cone beam:
at times receiving radiation from large cone angles and at other times from the midplane of the cone. Notice how
11997 International Meeting on Fully 3D Image Reconstruction
441
this differs from non-helical, cone-beam cr. There, only the central reconstructed slice is irradiated by the
midplane of the cone-tbe plane containing the source and perpendicular to the rotation axis-while slices
progressively further from the midplane are irradiated by x rays over a range of increasingly greater cone angles.
Thus, from an image quality standpoint, all slices from an HCBCT-scanner are equivalent, unlike from the
non-helical cone-beam scanner.
Image quality in HeBCT depends on several factors: the cone-angle subtended by the detector array, the helical
pitch, the slice width w, scatter rejection, and choice of reconstruction algorithm. Furthermore, scan objects that
exhibit a large degree of axial invariance are imaged more faithfully than those with high contrast variations along
the axial axis and those of a smaller transaxial diameter are easier to image faithfully than those extending to the
edge of the transaxial field-of-view. This investigation focuses only on the question of the helical pitch ratio.
l J
fl
I
LJ
f1
I
We consider eveI)' pixel in any reconstructed slice, since all HCBCT slices are equivalent. We assume a twodimensional detector array that is a section of a cylinder, focused on the source, the same as the linear detector
array for an ordinary medical Cf-scanner. To make the analysis independent of the number of rows in the detector
array, we introduce the normalized helical pitch ratio, IN' defined by the helical pitch divided by the full axial
length of the detector array (projected at rotation center):
\
lJ
where N is the number of rows in the detector array. The angular range of source positions, or views, designated by
angle 13, that contain a ray path from the focal spot of the source through the pixel at x,y to the detector array is
given by
C]
13
R~
121t L(f3, x,y)
I
~
1
0.40
2"
0.30
where:
13 is defined such that 13 = 0 is when the
source is in the slice plane,
R is the radius of the orbit of the source,
L 2(f3,x,y) = (Rsinf3 +X)2 +(RcosI3 _y)2,
which is the distance squared from the
source at 13 projected onto the slice plane
to the pixel at x,y.
\ I
lJ
If the inequality is satisfied, the pixel is in the
conc beam for view 13; if not, then the ray
r- '1
from the source passing through the pixel
misses the detector array. Fig. 2 shows the
trajectories of selected pixels through the
rows of the detector array as a function of 13.
Each pixel is in the cone beam for a different
range of views. Maps can be made to show
range of coverage by the cone beam for each
pixel.
LJ
11
I
,
I,-.J'
~
0.20
~
~
.!!
0.10
2l
0.00
:fi
-0.10
~
a:
-0.20
)('y=O,O
-0.30
-0.40
·180
·135
-90
-45
0
45
Gantry Angle (degrees)
90
135
180
Fig 2. Trajectories ofthe pixel at rotation center and the north,
east, west, and south-most extreme pixels for a fleld-of-view of
500 nun, R = 600 rom, and r'H = 1.5.
Because we want the largest plausible IN' we sort the cone-beam ray-sums into parallel-beam ray-sums in the
transverse plane while maintaining the cone angle. Analogous to two-dimensional CT, although 1800 plus the fan
angle worth of source views are required for a complete data set (in the two-dimensional sense), sorting to parallel
requires less data (a smaller portion of the Radon space) than a Parker-like fan-beam reconstruction.17 The sort
11997 International Meeting on Fully 3D Image Reconstruction
451
equations are given below for going from ..--..--------..................--....-..-..-----......------,
divergent projections, p(p,,,(,n), to
y
source
semi..parallel projections, p(9,t,n'); Fig.
At.
3 explains the notation and n(1) is the
....
dctcctor row index:
e~f3+'Y
t = Rsiny
n'
=n ..... ·~rH.
2n
The first two sort equations are
familiar;·8 the third represents the
dctector row shift due to the patient
translation that occurs among the
different t..ray..sums for a given view 9
derived from the first two sort equations.
Now maps of the e ..view covemge for
each pixel in a slice can be made. They
show that each pixel has at least the
minimally required 1800 data coverage
for image reconstruction up to a certain
value of In that depends on the source
radius and reconstruction diameter
(field--of..view). Above that limit,
portions of the field-of-view no longer
obey the weak completeness condition.
L
array
Fig 3. Schematic of Transverse Plane of Cone-Beam Geometry.
We modify the helical, Feldkamp algorithm3,8 to allow each pixel its own backprojection range while insisting on
proper three..<fimensional backprojection. We propose such a reconstruction algoritlun: rnCB, which stands for
inconsistent, helical, ~ne"12eam reconstruction algorithm, given by:
ft(x,y) =
41:: 1:
ro(9,x,y)p(9,t,n')g(t-t',n')dt'de
where /, (x,y) is the reconstructed pixel at location x,y for slice I,
t =xcos e + ysine,
nI = e rHCOS"(
.. ,
~(e
27t U(e,x,y)
'\ = 1 _ xsin9
ycos9
,x,y,
R + R '
g(t- t', n') is the convolution filter with optional row weighting, and
ro(9,x,y) is a weight or interpolation function discussed below.
The backprojection range, 9. to 92, covers all possible views that could contribute to the slice. The view angle is
relabeled for each slice so that the view at 9 = 0 has the focal spot in the 'slice plane. Thus, if the algorithm is
implemented as written each slice is rotated with respect to its neighbor by 21CAz/vT, where Ilz is the axial pitch
between slices. There are two choices for the weight or interpolation function ro. The minimally required data set
uses ro(9,x,y) = 1 if9 is within the 1800 range given by the 9.,9 2 pixel maps; 0 otherwise. ,To use all data,
because most pixels have a G-coverage greater than 1800 , then L ro(9 +Im,x,y) = 1, where k is an integer such
k
that e1 ~ 9 + 1m ~ 92. A distance weighted function that peaks for 9 + 1m = 0 to reduce large cone-angle artifacts
may be best for ro.
11997 International Meeting on Fully 3D Image Reconstruction
461
,
J
fl
I
(j
The llICB algorithm can be contrasted with a £Onsistent, helical £One-heam reconstruction algorithm, CHCB.
After the sort, the simplest CHCB algorithm does not use the weighting function and the backprojection limits are
9 1,2= ±rc/2. However, the maximum helical pitch ratio that doesn't violate the
weak completeness condition throughout the field-of-view is significantly
w/o sort with sort
smaller than with the mCB approach. Table 1 summarizes the investigation so
far: we compare maximum r'H for rnCB and CHCB, with and without the sort
CHCB
0.92
1.17
from fan-beam to parallel-beam in the transverse plane for the same source
IHCB
1.41
1.70
orbit and field-of-view as in Fig. 2. For r'H :s: 1.17 with the sort and for
r'H :s: 0.92 without the sort, the mCB and CRCB algorithms are the same.
Table 1. Maximum normalComputer simulations are underway to judge the performance of the IHCB
ized helical pitch ratio.
algorithm and to compare it with CHCB at high helical pitch.
REFERENCES
1. Willi A KaIender, "Principles and performance of spiral cr," in L.W. Goldman, J.B. Fowlkes, Eds., Medical
CT and Ultrasound: Current Technology and Applications, AAPM, College Park, :MD, 379-410 (1995).
2. M W. Vannier and O. Wang, "Principles of spiral cr," in M. Remy-Jardin and J. Remy, eds., Spiral CT ofthe
Chest, Springer-Verlag, Berlin, 1-32 (1996).
3. L.A. Feldkamp, L.C. Davis, and J. W. Kress, "Practical cone-beam algorithm, "J. Opt. Soc. Am. A, 1, 612-619
(1984).
4. Bruce D. Smith, "Cone-beam tomography: recent advances and a tutorial review," Opt. Eng., 29, 524-534
(1990).
S. Thomas J. Beck, "CT technology overview: state of the art and future directions," in R O. Gould, I.M Boone,
eds., Syllabus: A Categorical Course in Physics: Technology Update and Quality Improvement ofDiagnostic
X-ray Imaging Equipment, RSNA, Oak Brook, IL, 161-172 (1996).
rl
6. A system under development for airline baggage inspection is reported on by Richard C. Smith and Patricia R
LJ
Krall, "Full volume dual energy high speed computed tomography (CT) explosives detev'iion development," 2nd
Explosives Tech. Symp. & Aviation Security Tech. Conf., Nov 12-15, 1996, Atlantic City, NJ.
7. H. Kudo and T. Saito, "Feasible cone beam scanning methods for exact reconstruction in three-dimensional
tomography," J. Opt. Soc. Am. A, 7,2169-2183 (1990).
8. H. Kudo and T. Saito, "Three-dimensional helical-scan computed tomography using cone-beam projections,"
Journal ofthe Electronics, Information, and Communication Society (Japan), J74-D-ll, 1108-1114 (1991).
9. X.H. Yan and RM. Leahy, "Cone beam tomography with circular, elliptical and spiral orbits," Phys. Med. Bioi.,
37, 493-506 (1992).
10. G. Wang, T.-H. Lin, P.-C. Cheng, D.M Shinozaki, "A general cone-beam reconstruction algorit.hm," IEEE
Trans. Med. Img., 12,486-496 (1993).
11. J. Eriksson and P.E. Danielsson, "Helical scan 3D reconstruction using the linogram method, " in Proc. ofthe
1995 Int. Meeting on Fully 3D Image Recon. in Rad and Nucl. Med., Aix-Ies-Bains, France, 287-290 (1995).
12. S. Schaller, T. Flohr, P. Steffen, "A new approximate algorithm for image reconstruction in cone-beam spiral
cr at small cone-angles," reprint of presentation at IEEE Nucl. Sci. Symp. and Med. Img. Conf., Anaheim, CA
(November, 1996).
13. A.V. Bronnikov, "X-ray cone-beam tomography with nonplanar orbits," in Proc. ofthe 1995 Int. Meeting... ,
ob.cit., 299-302. Although the paper has only simulations, the poster at the conference showed an experimental
I i
[)
platform.
14. Michael D. Silver and Kyung S. Han, "Helical cone-beam CT for fast throughput inspection," Paper Summaries ofASNT's Industrial Computed Tomography Topical Conforence, ASNT, Columbus, OR, 75-79 (1996).
15. Willi.A. KaIender, "Spiral CT angiography," in Goodman and Fowlkes, eds., ob. cit., 627-640.
16. Bruce D. Smith, "Image reconstruction from cone-beam projections: necessary and sufficient conditions and
reconstruction methods," IEEE Trans. Med. Img., 4, 14-25 (1985).
17. D.L. Parker, "Optimal short scan convolution reconstruction for fan-beam CT," Med. Phys., 9, 254-257 (1982).
18. A.C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, IEEE Press, New York, (1987),
Section 3.4.3.
I
11997 International Meeting on Fully 3D Image Reconstruction
_I
Exact Cone Beam CT with A Spiral Scan
K.C. Tam, S. Samarasekera, and F. Sauer
Siemens Corporate Research, Inc.
755 College Road East
Princeton, NJ 08540
The condition for complete cone beam data was first formulated by Tuy [1], namely, that each plane intersecting
the object should intersect the scan path. Mathematically exact cone beam image reconstruction algorithm via the
computation of the radial derivative of the Radon transform of the object was derived by Grangeat [2]. The radial
derivative was computed from a pair of closely spaced parallel lines, referred to as the lines of integration, on the
detector plane. The algorithm was later generalized by the author [3]. The pair of lines of integration no longer have
to be parallel, they can be at an angle to each other; and they can be divided into any number of segments.
The completeness condition assumes that the object is completely covered by the detector at all view angles.
Some objects which are of interest in medical as well as industrial inspections, however, are very long, relatively
speaking. Imaging such long objects is challenging for 3D CT systems because it requires the use of very large area
detectors to cover the entire length of the objects. Furthennore, in many such cases only a relatively small sectional
region of the long object is of interest. Even if the image of the entire object is needed, it can be obtained by
stacking up such sectional regions. It is therefore more practical to employ a detector just big enough to cover the
sectional region rather than to cover the entire object. However, such arrangement poses serious difficulties for the
image reconstruction problem. From the perspective of reconstructing the entire object, some of the cone beam data
penetrating portions of the object other than the region-of-interest are missing because of the insufficient size of the
detector. This situation is usually referred as thS truncat~£2!1e beam problem. From the perspective of
reconstructing the region-of-interest, some of the x"ray paths penetrate other portions of the object as well as the
region"of~interest, and thus the cone beam data coll~cted no longer represent the regionwof.. interest exclusively but
are corrupted by the overlaying materials.
Chen [4] published a region-of-interest cone beam algorithm, but the method requires a detector large enough to
cover the entire object. In [5,6] we reported a method to reconstruct a sectional region within an object from cone
beam x-ray data collected on a detector just big enough to cover the section~l region-of-interest, there being no
contamination from the rest of the object. The x-ray source scans the region"of-interest along a path consisting of
two circles and a connecting curve. With this method the requirement on the detector is reduced from covering the
entire object to covering just the height of the region-of-interest.
In this paper we are going to show that the height requirement on the detector can be further relaxed with a two~
circle and spiral scan path. The height requirement on the detector is reduced from covering the entireregion&of~
interest to~~g the distance between ad,illcent tums in.1lliuI2h:a!:. Consequently the height of the detector no
longer limits' e helglit of the region that can be scanned. The method is mathematically exact. Cone beam
reconstruction methods with spiral scan path have been reported 'in the literature earlier [7]. They are mostly
adaptations of the Feldkamp algorithm to the spiral scan path, and therefore mathematically approximate.
Furthermore it can be shown that compared to the Feldkamp approach using the same detector, the new algorithm
can aocommodate a spiral more than two times the pitch, and thereby cuts down the x-ray dosage to the object by
more than a factor of 2.
The x-ray source scans the object along a spiral only scan path. If only a region-of-interest of the object is to be
imaged, a top circle scan at the top level of the region-of-interest and a bottom circle scan at the bottom level of the
region-of-interest are added. The ohl height requirement on the detector is that it should be longer than the distance
.c~tween adjacent turns i~e spiral. To reconstruct the object, the ra ial Radon derivative for each plane
intersecting the object is computed from the totality of the cone beam data. This is achieved by suitably combining
the cone beam data taken at different soUrce positions on the scan path.
SPIRAL97 .ABS
11997 International Meeting on Fully 3D Image' Reconstruction
481
!rl
I
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In Figure 1 the object is circumscribed by a cylinder, which we will call the object cylinder. The object cylinder
is enclosed by a larger cylinder, which we will call the scan path cylinder, defined by the top and bottom circles and
the connecting spiral; in other words the spiral wraps around the cylindrical surface connecting the two circles.
Consider a plane Q intersecting the region-of-interest. Since a plane intersects a cylinder in an ellipse, the plane Q
intersects the object cylinder and the scan path cylinder in two ellipses, one inside the other. Consider Figure 2,
which lies in the plane of plane Q. Plane Q intersects the object cylinder in the smaller ellipse E 1, and it intersects
the scan path cylinder in the larger ellipse E2. Since the spiral path lies on the scan path cylinder, it intersects the
plane Q in points that lie on the ellipse E2. Label these source positions S 1, S2, and S3, as illustrated in the figure.
Similarly, it is easy to see that the top circle intersects the plane in the two points Tl and T2 which lie at the
intersection between E2 and the top edge of the region-of-interest, and that the bottom circle intersects the plane in
the two points Bland B2 which lie at the intersection between E2 and the bottom edge of the region-of-interest.
r'\
l1
In order to image the region-of-interest, one needs the Radon derivative for the portion of plane Q that lies within
the region-of-interest. This quantity can be obtained by combining the partial results computed from the cone beam
data at the various sources positions on the ellipse E2. This can be achieved as shown in Figure 2. The source
positions that contribute to the Radon derivative are T2, S 1, S2, S3, and B2. The angular range of the cone beam
data used to compute the Radon derivative of the corresponding partial plane using the procedure described in [2,3]
is indicated for each source position in the figure. For example, at T2 we only use the cone beam data within the
angle bound by Tl T2 and S 1T2 to compute the Radon derivative for the portion of plane Q bound by T 1T2 and
S 1T2, and at S 1 we only use the cone beam data within the angle bound by T2S 1 and S2S 1 to compute the Radon
derivative for the portion of plane Q bound by T2S 1 and S2S 1. And so on.
(\
lJ
("1
I I
, I
L_
[J
Since the five partial planes in the ab<?ve steps do not overlap and together they completely cover the portion of
plane Q that lies within the region-of-interest, the Radon derivative for plane Q can be obtained by summing the
Radon derivatives for the five partial planes. It should be noted that the method of obtaining the Radon transfbml' of
the object by combining cone beam data from the different source positions that intersect the plane is possible if the
operation which computes the function of the Radon transform from the cone beam data is linear and local, such as
Radon derivative computation.
r·
I
L
In order to cover the appropriate partial plane on the plane of integration; it is necessary set limits in weighted line
integral computation. This task can be accomplished through a masking process. The mask consists of a top curve
and a bottom curve. For each line integration, only the segment of the line bound between the two curves contribute.
The equation for the top curve for the spiral scan is given by:
h
y=-tan
21t
-l(a)(
X2)
- 1+a
X
2
x~o
(1)
x<o
(1
LJ
r
"i
1
j
\..--~
where a is the radius of the spiral, and h is the distance between adjacent spiral turns (the pitch). The bottom curve
is the reflection of the top curve about the origin. The shape of the spiral mask is shown in Figure 3.
For ROI imaging, circular arc scans are needed at the top and bottom levels. The top circle scan starts at the
angle (1t + a) before the start of the spiral scan, and the bottom circle scan ends at the angle (1t + a) after the end of
the spiral scan, where a is the fan angle of the x-ray beam. The detailed geometry of the mask depends on the
location of the source in the scan path. For this purpose divide the spiral scan path into 5 regions, as illustrated in
Figure 4: (1) the last (1t + a) turn of the top circle; (2) the first (1t + a) tum of the spiral; (3) the interior portion of
the spiral, i.e. after the first (1t + a) turn and before the last (1t + a) tum; (4) the last (1t + a) turn of the spiral; (5) the
first (1t + a) tum of the bottom circle. The masks for these cases are modifications of the basic spiral mask given in
(1).
SPIRAL97.ABS
11997 International Meeting on Fully 3D Image Reconstruction
491
The spiral scan algorithm has been successfully validated with simulated cone beam data. The technique
presented in this paper was first disclosed in [8,9].
REFERENCES
[1]
Tuy, H.K., (1983). "An Inversion Formula For ConesBeam Reconstruction SIAM J. Appl. Math., Vol. 43
( 1983) 546~552.
[2]
Grangeat, P., "Mathematical framework of cone beam 3D reconstruction via thc first derivative of the Radon
transform." Mathematical Methods in Tomography, an, Louis, Natterer (eds), Lecture Notcs in Mathematics
No. 1497, Springer 66 97, 1990.
ll
,
9
[3]
Tam, K.C., "Exact Image Reconstruction in Cone Beam 3D CT", Review of Progress in Quantitative Non8
Destructive Evaluation, Eds. D.O. Thompson and D.E. Chimenti (New York: Plenum Press) Vol. 4A, pp.657~
664).
[4]
Chen, J., IIA Theoretical Framework of Regional ConeftBeam Tomography", IEEE Trans.Med. Imag., MI· 11
(1992) 342.
[5]
Tam, K.C., "Method alld apparatus for acquiring complete Radon data for exactly reconstructing a three
dimensional computerized tomography image of a portion of an object irradiated by a cone beam source", US.
Patent 5,383,119, Jan 17, 1995.
[6]
Tam, K.C., "Regionaof·Interest Imaging in Cone Beam Computerized Tomography", presented in the IEEE
MIC, November 3-9,1996, Anaheim, CA.
[7]
Wang, G., Lin, T., Cheng, P., and Shinozaki, D.M., "A General Cone-Beam Reconstruction Algorithm", IEEE
Trans.Med. Imag., MI-12 (1993) 486.
[8]
Tam, K.C., "Three-dimensional computerized tomography scanning method and system for imaging large
objects with smaller area detectors", US. Patent 5,390,112, February 14, 1995.
[9]
Tam, K.C., "Helical and circle scan region of interest computerized tomography", US. Patent 5,463,666, Oct
31,1995.
SPIRAL97.ABS
11997 International Meeting on Fully 3D Image Reconstruction
501
L_~..-"
---
\
~
.---.."
i
' - . -.....------
.~
"-- __ J
r-----
1--
1
,---!
'~------,'
.~--,
-----,
I
,
.
T=l~--'
---_/
-'--;
B .. Bottom circle
S=SpiraJ
Long
object
Top circle
~
co
co
........
CD
3
~
(;-
t> =F -. -:- - ~
~Re;~on
:;Spiral
interest
t
::::l
~
S2
Detector
s::
CD
height
~
~
::::l
4-:::::
Bl
B:
Bottom circle
CD
F-r .. i
::D
CD
o
o
s-f.
::::l
!a.
2
Sl
f)
AJ
cf
::::l
Circle scan ends
Helix scan starts
--..
L
FlI'It (x+«) helix
/
Projection of the helix tum below
~
I~A-r
3
EjLf-
ENERGY-BASED SCATTER CORRECTION FOR 3-D PET:
A MONTE CARLO STUDY OF "BEST POSSIBLE" RESULTS
David R. Hayno!', MD
Robert L. Harrison, MS
Thomas K. Lewellen, PhD
Imagulg Research Laboratory
University of Washington
Seattle, WA 98195
INTRODUCTION
Direct 3 D acquisition (i.e., without the use of between-plane collimator septa) is becoming increasulgly popular
in clinical applications of positron emission tomography (PET) because of increased detection efficiency, which is
typically 2.5 to 5 times that of 2-D (septa-in) collection schemes. Typically, the increase in noise-equivalent count
rate is not as high as the increase in raw count rate. This is largely because, as the transaxial angle of acceptance
for a 3-D tomograph increases, the ratio of scattered events to true events increases. This has only a moderate
effect when the clinical goal is hot-spot detection, because the added scatter events constitute a low-spatialfrequency background, and the conspicuity of hot spots is little affected. Scatter has a much greater effect on
quantitative PET, because it adds an unknown bias to the estimated emission rate for each pixel. This becomes
particularly important for accurate nletabolic modeling or for dosimetry estimates.
H
Several researchers have proposed different scatter-correction schemes. These schemes can be divided into
purely energy-based schemes and spatially-based, or mixed spatial- and energy-based, schenles. The first
category, with which this paper is exclusively concerned, includes the dual energy window proposed by
Grootonk et a1 [IEEE Med Imaging Conference 1991; 1569-73] and the schemes proposed by Bendriem et al [IEEE
Med Imaging Conference 1993; 1779-83] and Shao et al [IEEE Trans Med Imaging 1994; 13:641-48]. These
methods may be abstractly characterized by formulae of the following form, where A is a 3D line of response
(LOR):
T(A)=:
f f W(E,E') n(E,E' ,A) dE dE',
(1)
where T(?v) is the estimate of the trues-only event rate along the LOR A, n(E,E',A) is the attenuation-corrected
observed event rate along ?v of all events in which the two detected photons have energies E and E', respectively,
and W(E,E') is a weighting function that depends on the detected photon energies but does not depend on the
LOR. In the dualHwindow schemes, W(E,E') will be piecewise constant, taking on a positive value if E and E' both
exceed a certain threshold (upper energy window), a negative value if one of E, E' is in the upper energy window
and the other falls within a specified lower energy window, and zero otherwise. More complex schemes can
easily be imagined, and the question then arises as to what the best possible performance of such an estimator is.
For example, if it were possible to greatly improve upon the performance of dual-window methods, this might
justify more complex front-end event~proce$sing hardware for better 3D scatter rejection. One can also ask
whether any additional benefit can be gained by spatial smoothing of the estimate T(A); put another way, does the
noise, or the bias, in the estimate (1) dominate the error at typical count rates?
A second class of methods are exemplified by the work of Bailey et al. [Phys Med BioI 1993; 39:411-424], which is
based on the older work of Bergstrom et al UCAT 1983; 7:42-50], in which a nonstationary, empirically derived
filtering of n(A) is performed, purely in the spatial (A) domain, to yield an estimate of T(A). Another important
method of this type is the work of Ollinger et al.[Phys Med BioI 1996; 41:153-76], who reconstruct the initial value
of n(A) to derive an estimate of the emission intensity, then use this estimate to derive an estimate of the firstorder scatter, which is then subtracted from n(A) and the process repeated. Empirical corrections are used for
119971nternatlbnal Meeting on Fully 3D Image Reconstruction
521
il
I I
l
I
I
!
higher-order scatter. Since the overall process is linear, this could be viewed as a rational method for deriving a
spatially-varying filter to n(A) to derive T(A). We are not aware of any direct comparison of these spatially-based
methods to energy-based methods for 3D PET. Spatially-based methods cannot, however, correctly account for
out-of-FOV scatter, and the empirical methods are derived from phantom measurements which may not be
accurate models of patient studies.
(
We approached the study of methods represented by equations of the form (1) by simulation. We studied a
variety of phantoms and two different energy resolutions, with a simulated tomograph based on the CE Advance
PET scanner. For each configuration and choice of energy windows, we calculated the "best possible" weighting
function, in the least-squares sense and examined the ability of a single weighting scheme to remove scatter across
the whole series of eleven phantoms.
L
I
METHODS AND RESULTS
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Tomograph geometry was modeled on a GE Advance Scanner [Miyaoka RS et al., IEEE Med Imaging Conference
1995; 3:1771-75] in high-resolution (NaI=47 KeV FWHM) and low-resolution (BCO=101 KeV FWHM) modes.
Axial FOV was 15.2 cm. A cylindrical 20 cm x 45 cm phantom was simulated, with a point source (phantom 1),
uniform activity (phantom 2), anci with all activity outside the axial FOV (phantom 3). Phantoms 4 and 5 were an
elliptical phantom with minor/major axes of 20 and 40 cm, with uniform activity inside the FOV and outside the
axial FOV, respectively. Phantoms 6, 7, and 8 consisted of 15.2 cm of the Zubal anthropomorphic phantom with
activity in the heart only, liver only, and all soft tissues, respectively. Phantoms 9, 10, and 11 consisted of 45.6 cm
of the Zubal phantom with activity in the heart, liver, and soft tissues, respectively. For each phantom, tht;:ee
slices, each 5.2 cm in thickness, were collected: one central slice, one off-center slice, and one edge slice. The
simulation data was binned by distance (21 bins) and byE, E' (10 KeV increments, from 100 KeV to 511 KeV). To
simulate energy response of the detector, Gaussian noise with the correct FWHM was added to each of E, E' to
obtain the final blurred, or observed, energies. Finally, the data was sorted into a number of specified window
pairs (one for E, one for E'), using up to 12 windows for the "detected" energy. Each of the 33 data sets thus
contained 63 different (E/E') spectra and contained 10-60 million simulated events.
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For the 47 KeV FWHM detectors, we analyzed three window schemes: the triple energy window (TEW) of Shao et
al. with 305-450, 385-450, and 450-575 KeV windows, but no dependence on phantom size or LOR position, a
7-window scheme (380-440,440-475,475-495,495-510,510-525,525-545, and 545-575 KeV windows), and a
12-window scheme (350-390,390-440,440-460,460-475,475-488,488-500, 500-510, 510-520,520-532,532-545,545560, and 560-575 KeV). For the BCO-detector simulation, we studied the schemes of Bendriem (250-550 and 550850 KeV windows), Grootonk (200-380 and 380-850 KeV windows), a 7-window scheme (340-390,390-440,440480,480-510,510-540,540-580,580-620 KeV) and a 12-window scheme (300-360,360-405,405-445,405-420,420-445,
445-470,470-490,490-510,510-530,530-550, 550-575, 575-600, 600-640 KeV). The 7- and 12-window schemes were
designed to sample the main photopeak approximately uniformly and to include one or two windows
representing lower-energy scatter just to the left of the photopeak. We present in the Table the residual RMS
error in estimating the true events after performing scatter correction by each of these schemes, expressed as a
proportion of the total counts in the window from 382-640 Ke V for the BCO scanner and as a fraction of the
counts between 450 and 575 Ke V for the NaI design. For each window scheme, a single optimized set of weights
was chosen by regression analysis for all 33 slices (11 phantoms x 3 slices/phantom). Errors are averaged over
the 11 phantoms and given separately for the center, off-center, and edge slices. Note that RMS errors are not
comparable between the NaI and BGO machines because the denominators are different.
!]
11997 International Meeting on Fully 3D Image Reconstruction
RMS ERROR BY ENERGX RESOLUTION, WINDOW SCHEME, AND SLICE LOCATION
.-
energy
resolution
window scheme
RMS relative
error-center slice
NaI
NaI
NaI
rEvV
"V.100
... ,,"
7-window
12-window
0.049
0.045
BGO
BGO
BGO
BGO
Bendriem
Grootonk
7-window
12-window
0.138
0.213
0.066
0.065
RMS relative
error-off-center
slice
RMS relative
error-edge slice
0.247
0.111
0.110
0.258
0.100
0.096
0.195
0.297
0.134
0.134
0.174
0.233
0.132
0.127
DISCUSSION AND FURTHER WORK
The results in the Table are interesting in several respects. A noise analysis, not shown here, demonstrated that
the aggregate error measures reported were not significantly influenced by Poisson noise at the count rates
specified. Mean relative L1 errors demonstrated identical trends to the relative L2 (RMS) errors, as did the
residual error when expressed as a proportion of true events for the simulations in which not all of the activity
was outside the axial FOV. The scheme of Bendriem fares better than that of Grootonk, although only the latter
has seen significant clinical application. This discrepancy is most likely due to the fact that detector variability in
energy threshold, a known problem with the Bendriem method, was not simulated. Errors in the center slice of
the tomograph are consistently smaller than errors in the offNcenter and edge slices. Since the latter are most
strongly influenced by activity outside the FOV, and since estimating the outside~FOV contribution to the scatter
is difficult for all known scatter correction methods, this is not surprising. Finally, these preliminary results
suggest that use of more complex window schemes (i.e., the 7 window schemes) yields a significant improvement
in scatter correction, compared to dual- or tripleNwindow schemes. A 7..window scheme has 28 free weight
parameters, whereas a dual-window scheme has just three. The results presented here ate based on fitting a
single weight scheme to hundreds of spectra and consequently the coefficients are overdetermined. Going
beyond 7 windows to 12 windows, however, does not significantly reduce error further.
w
Further work on this problem, some of which will be presented at the meeting, will include the effects of allowing
the weights for the LOR A. to depend on the distance of A. from the central tomograph axis, as suggested by
Grootonk and Shao and their co-workers. We are examining the artifacts introduced by scatter correction into
reconstructed images. Finally, plots of the optimal weights for each of the window scheme will be displayed.
ACKNOWLEDGMENT
This work was partially supported by NIH CA 42593 to TKL.
Correspondence to:
David R. Haynor, MD
Imaging Research Laboratory
Box 356004
University of Washington
Seattle, WA 98195
e-mail: [email protected]
telephone: (206) 543-3320
11997 International Meeting on Fully 3D Image Reconstruction
fax: (206) 543-3495
AlgoritlUl1S for calculating detector nonnalisation coefficients in 3D PET.
R.D.Badawi and P.K.Marsden
The Clinical PET Centre. Division of Radiological Sciences. UMDS. Guy's and St. Thomas' Hospital. London.
[email protected]. tel +44 171 9228106, fax +44 171 6200790
1. Introduction
Most methods for calculating detector normallsation coefficients in 3D PET involve the
calculation of individual detector efficiencies from direct-plane data. and the estimation of crossplane line-of-response (LOR) coefficients from the product of the detector efficiencies involved. The
efficiencies may be calculated following the method of Defrise (1991) or by means of the "fan-sum"
approximation described by Hoffman (1989). Ollinger (1995) uses the method suggested by Casey
(1986) and implemented by Hoffman (1989) in 2D to calculate LOR coefficients in 3D directly
without reference to individual detector efficiencies. We have developed and assessed a method
based on Hoffman's fan-sum approximation which utilises the full 3D dataset. We find that using this
method it is possible to reduce normalisation times by an order of magnitude with no loss of signalto-noise ratio. We also describe how Casey's exact method can be extended to utilise the full 3D
dataset.
2. Theory
2.1. Normalisation model.
The new 3D algorithms described here are an extension of the component-based 2D model
described by Hoffman (1989) in which each LOR in the 3D dataset is the product of 2 detector
efficiencies and various geometric correction factors. This may be written as follows:
11 iujy = CUyE jut jvg uvijl
[]
[]
[J
u
11
LJ
where Tljujv is the normalisation coefficient for the LOR joining detectors i in ring u and j in ring v. Cav
is the "plane efficiency" - a global scaling factor for each sinogram containing LORS between rings u,
and v. £iu, Ejv are the individual detector efficiencies for detectors i in ring u and j in ring? w
respectively. and guvljl is a geometric factor dependant on the radial position of the LOR defined by i
and j. the rings u and v. and an index I governing the position of i and j in their respective block
detectors.
2.2. Geometric corrections.
All systematic in-plane and axial geometric variations described in the model may be corrected for
using the following procedure:
i) A uniform scatter-free dataset is acquired by means of. for example. a scanning line-source
(Bailey et al 1995).
ii) The in-plane radial profiles are approximated for each sinogram by summing the sinogram
columns and calculating the ratio of each sum to the mean for that sinogram (Hoffman 1989).
iii) The crystal interference pattern (the variation in the radial profile with the position of the
individual crystals in their respective block detectors) is accounted for by selectively sampling
the columns used in the radial profile calculation. If there are D crystals in a block, then D
profiles should be generated, by summing column elements in every Dth projection. starting
from projections 0.1 .... ,D-1 (Casey et aI1995).
iv) The plane efficiencies are obtained by summing all the elements in each sinogram and taking the
ratio of each sum to the mean of all sums. If the acceptance angle of the scanner is large. it may
be necessary to apply an analytical correction for the fact that cross-planes will see a thicker
source than direct planes. It may also by necessary to apply a similar correction in-plane prior to
radial profile calculation, as cross-plane LORs at the projection edge lie at a greater angle to the
plane of the detector ring than those at the projection centre.
Correcting for geometric effects in this way makes no assumptions about the separability of
transa~ial and axial effects. Assuming that the model is accurate. then after correction there will be
unique efficiency for each detector such that all LORs in the 3D dataset are products of the two
relevant detector efficiencies.
2.3. Detector efliciellcy calculations.
Consider P rings of N detectors, to which all geometric and crystal interference corrections
have been applied. Using the notation of Defrise (1991). the number of counts in an LOR joining
detectors i and j in a particular ring is njj. If <njj> is the noise free value of njj, then we write:
<njj> = EjEjC
(1)
where C is a constant dependant on the duration of the scan and. the intensity of the source.
Henceforth we will assume that the data has been scaled so that C becomes unity, and we write
11997 International Meeting on Fully 3D Image Reconstruction
551
Algoritiuns
rOI'
calculating detector J1ol"lnalisation coefficients in 3D PET.
R.D.lladawi and P.K.Marsden
(2)
111j :::: 818j
Now define a group of detectors A such that there are M detectors 8j in A which are all in coincidence
with 81 • Also define a group B of M detectors such that all detectors 8j in n are in coincidence with
all detectors in A. We can then state the following:
(BiJ;B X(BJ~Bi)
j)
~iJ = €i J =
B
(~€ jJ;>I)
(3)
This is ule method described by Casey (1986). and gives an exact solution when applied to noise-free
data. The quantities in the brackets are readily calculated from the sinogram data. If we make the
following assumption:
~M
I,:€;
-J)
we can redefine A and B with the one condition that all detectors in A are in coincidence with 8j and
This is the fan-sum approximation method described by Hoffman (1989). It allows one to increase
the size of A and D. thus improving the signal-townoise ratio of the efficiency estimates. Since it
estimates the individual detector efficiencies themselves; it also allows- one to calculate efficiencies
for LORs not illuminated by the source. However. it does increase the susceptibility to systematic
errors. particularly if the number of illuminated LORs per detector is small.
The way to include cross-plane data in both these methods is straight-forward the groups A
and B are extended to include detectors from all rings and the detector efficiencies are indexed by
ring as well as detector position.
a
3. Method
3.1. AC(IUisitioll of data for geometric corrections.
Data for geometric corrections were obtained from a series of scanning linewsource
acquisitions using the equipment and method des~ribed by Bailey et al (1995) with a few small
modifications. The scanner used was an ECAT 951R (Siemens-Cn, Knoxville Ten) with 16 rings
and 512 detectors per ring. giving rise to 256 sinograms each containing 192x256 elements. 8 line
positions were used instead of 6. and the data was conected for "rod.. dwell", that is the sinusoidal
variation in the speed with which the rod crosses a given LOR as the angle of the LOR to the normal
increases. The central 162 (out of the possible 192) elements in each sinogram projection were
illuminated uniformly. The eight acquisitions each consisted of four passes of the line source and
took approximately 13 hours each. and the number of counts per illuminated LOR in the fmal
sinogram was roughly 275. The rod source contained 3.5 MBq of activity.
3.2. Assessnlellt of 2D and fully 3D fan ..sum algorithms.
M
Systematic errors in the 'algorithms were assessed by means of comparison of performance
when applied to a 3D scanning line source acquisition as described in section 3.1. The data was selfcorrected for geometric effects as described in 2.2. The data was then selfwcorrected for detector
efficiency variation using the 2D and the 3D fan-sum methods. The deviation of the resultant
.sinograms from uniformity was used to assess the performance of each method. This deviation was
e)tamined by calculating the maximum deviation and the standard deviation of individual sinogram
elements before and after 8x8 element box-car smoothing.
Performance with respect to noise in the normalisation data was assessed by means of a
series of 3D scans of a uniform 20 em cylinder containing,Ge68. Acquisition times varied from 150
seconds (4.6 xl06 total counts, 1.1 counts per illuminated LOR) to 4800 seconds (147 x106 total
counts. 35.1 counts per illuminated LOR). A further 19200-second scan was acquired. All scans were
corrected for crystal interference and geometry using data acquired from the scanning line source.
Plane efficiencies were calculated from each scan. as were detector efficiencies u'sing both
algorithms. These were then used in turn to complete the normalisation of the 19200 sec. scan. It
11997 International Meeting on Fully 3D Image Reconstruction
561
Algorithll1s for calculating detector nornlalisation coefficients in 3D PET.
[1
R.D.Badawi and P.K.Marsden
should be noted that plane efficiencies calculated in this way will incorporate scatter. The resulting
sinograms possessed a non-uniform radial profile due to differing activity and attenuation in each
LOR. These radial profiles were rendered uniform by means of a correction derived from the 19200
sec. scan. The standard deviation of the fully illuminated LORs was then measured.
Finally the 19200-second scan was self-normalised using both algorithms. The resultant
sinograms were then reconstructed using a measured attenuation correction.
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4. Results
Table 1 shows the results of the assessment using the scanning line source data. The values
for the smoothed data are a measure of large-scale systematic error. The fully-3D algorithm is
slightly more accurate than the 2D algorithm.
% max. diff from
algorithm
% standard dev % max. diff from
mean
3D Fan-sum
2Dv Fan-sum
50.95
56.49
7.45
7.83
mean (smoothed)
15.81
22.44
% standard dev.
I(smoothed)
2.61
3.37
Tahle 1. The maximum difference from the mean and the standard deviations for the fan-sum algorithms, calculated on
smoothed and unsmoothed data.
[1
Figure 1 shows the standard deviation of single elements for the cylinder data as a function
of scan time. The 3D method produces significantly better results with short scan times. There is no
real advantage in scanning for longer than 20 minutes when using the 3D technique. Fig 2
demonstrates that even a 10 minute scan (4-5 counts per illuminated LOR) can produce an adequate
normalisation. Figure 3 shows the end and central planes of the reconstructed 19200 sec. scan self,.
normalised for efficiencies using the two algorithms. There are no significant differences between th~<,
images, demonstrating that the 3D method does not introduce new artefacts.
5. Discussion and conclusions
We have shown that normalisation scan times can be significantly reduced and accuracy
slightly enhanced by extending detector efficiency algorithms to include cross-plane data. Using such
a method with a scan of a uniform cylinder it,is possible to normalise an EeAT 951R in 3D mode
using just 40 x 106 counts. This corresponds to around 5000 counts per detector, and in practice. half
of this number may be sufficient.
A cylinder scan introduces scatter into the normalisation, potentially producing artefacts
(Bailey et al 1995, Ollinger et a1 1995). A better source for obtaining detector efficiencies would
therefore be a fast scanning line source. Even with a line source containing just 5 -10 :Ml3q of activity,
the fully-3D fan-sum method could produce an adequate normalisation in about an hour.
The authors are currently implementing the fully-3D Casey-type algorithm. Early
indications are that the method will have improved systematic accuracy compared to the fan-sum
method, at the expense of a slight reduction in noise perlotmance.
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Standard Deviation of illuminated
LORs from 20 cm cylinder data
25
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~2Dfan-sum
20
-0-- 3D fan-sum
15
1ii 10
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5
0
0
1-\
L~_I
20
40
60
BO
1()()
Nonnallsatlon scan time (mlns)
Figure 1. % standard deviations for illuminated LORs from the normalised
20 cm cylinder data after radial profile flattening.
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11997 International Meeting on Fully 3D Image Reconstruction
571
AlgoriUUllS fo .. calculating detector Ilol'lnalisatiol1 coefficients in 3D PET.
R.D.Badnwi and P.K.Marsden
(n)
(b)
(c)
(d)
Figure 2. Sll1ogl'Hlll dnlH nClJlllrod using 1\ 20cIll cylinder
(n) 20 ell! cylindel" unprocessed, sClln time 19200 sec. There nrc opproxlmotely 140 counts per 1I1umlnoted LOR.
(b) 20 COl cylindcl', unproccllscd, senn timo 600 Rec. Thoro nrc npproxlmntely 4·5 counts per ilIum/nnted LOR.
(c) SCIII1 (II) Ilormnliscd wUh (b) nnd the 2J) fnn"surn nigorlllun.
(d) sClln (n) normnliscd wUh (h) /uH.llho 3D fnll-sum nlgorilhm.
(a)
(b)
Figure 3. End (upper row) and central (lower row) planes from reconstructions oCthe 19200 sec scnn.
'l11e reconstruction used was the projection/rcprojection algorithm Implemented by Dyars Consulting (Knoxville, Ten.), with a
romp filter and 1\ cut-off frequency of 0.5 Nyq,
(n) sclf-normnlised using 2D fan-sum
(b) self-normnllsed tlsing 3D fan-sum
Acknowlegements
The authors would like to thank Dale Bailey for the loan of the scanning line source equipment, and Larry
Byars and Martin Lodge for useful discussions.
References
ME Casey, H Gadagkar and D Newport 1995 "A component based method for normalisation in volume
PET"Proceedings of the 3rd International Meeting on Full 3D Reconstruction,
ME Casey and EJ Hoffman 1986 "Quantitation in Positron Emission Computed Tomography: 7. A technique to
reduce noise in accidental coincidence measurements and coincidence efficiency calibration" J Comput Assist
Tomogr 10,845-850
M Defrise, DW Townsend, D Bailey, A Geissbuhler, C Michel and T Jones 1991 "A normalisation technique
for 3D PET data", Phys Med BioI 36(7),939-952
EJ Hoffman, TM Guerrero, G Germano. WM Digby and M Dahlbom 1989 "PET system calibrations and
corrections for quantitative and spatially accurate images" IEEE Trans Nuc Sci, 36(1),1108-1112
JM Ollinger 1995 "Detector efficiency and Compton scatter in fully 3D PET" mEE Trans Nuc Sci, 42(4) 11681173
DL Bailey and T Jones 1995 "Normalisation for 3D PET with a Translating Line Pseudo Plane Source." J Nuc1
Med 3692P
11997 International Meeting on Fully 3D Image Reconstruction
581
i)
Normalization for 3D PET with a low-scatter planar source:
Technique, Implementation and Validation
I
T.R. Oakes, V. Sossi, and T.J. Ruth
University of British ColumbialTRIUMF PET Centre
ij
1\.BSTRACT
ri
For 3D PET normalization methods in general, a balance must be struck between statistical accuracy and
individual detector (or LOR) fidelity. Methods with potentially the best LOR accuracy (such as ratios of single LORs)
tend to be statistically poor, while techniques to improve the statistical accuracy (such as calculating the average of a
grOliP of similar LORs) tend to reduce the individual detector fidelity. We have developed and implemented a 3D PEr
normalization method for our CTI/ECAT 953b that determines the detector Normalization Factors (NFs) as a product of
a 4-climensional matrix of accurately measured Geometric Factors (GFs) and single detector Efficiency Factors (EFs).
The GFs are specified by the two detector rings for each LOR, the radial distance of the LOR from the tomograph
center, and the position within the detector block of the two crystals which defme the LOR. An accurate set of
Geometric Factors (GFs) is crucial; inaccurate NFs result if LORs with similar but not identical geometric symmetries
are grouped together in an attempt to improve the statistics of the GFs. The dimensionality of the GF matrix may be
scanner-specific, although the general method may be extended to other tomographs; the key is to determine the optimal
number of dimensions in the GF matrix. Our normalization technique obtains an axial uniformity of +/-1.4 % and a
radial uniformity of +/-2.0% in reconstructed images from a 20cm uniform phantom (excluding the two end planes).
The impoqance of a uniform low-scattering source for this technique cannot be overstated. We have used a
moving line source [Bailey, 1995a] as well as various plane sources developed by our group. All sources have some
degree of non-uniformity and introduce some degree of scatter. We have examined the effect of various magnitudes and
types of source non-uniformities on the quality of the NFs. We have also addressed the practical aspects associated"with
making a uniform plane source, including developing criteria for a "uniform low-scatter source"·for this purpose. 'The
effects of various alterations to the algorithms on the NFs and the accuracy of the normalization have been examined;
results indicate that the implementation of the technique plays a crucial role in obtaining an accurate normalization.
I :
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INTRODUCTION
Casey & Hoffman proposed a fan-shaped collection of LORs connecting a single detector to a group of other
detectors as a method for acquiring a statistically robust data set by increasing the number of counts associated with
each detector. Defrise et aI. addressed the problems of 3D PEr normalization, adding an elegant refmement to Casey &
Hoffman's technique by splitting the NFs into two components: a set of GFs, related to the geometric relationship of one
detector to another, and a set of EFs, related to the change in response over time of one detector to the others. The GFs
and EFs are combined to calculate the actual NFs. By making this distinction, a data set with very high counts may be
acquired once in order to calculate the GFs. and a lower-count data setmay be acquired as needed to calculate the EFs.
Stazyk et al. modified Casey & Hoffman's original2D fan-beam to become a 3-dimensional fan. relating LOR
fans from several detector rings related to a single detector by using axial geometric factors. By using a uniform lowscatter planar source for 3D PET normalization, a collimated (2D) normalization scan is not required; this has led.
Using data with a large scatter fraction (30-40%) to calculate the NFs will underestimate the activity in the center of the
PET scanner's Field of View (FOV) if some type of Scatter-Correction (SC) is not applied. IT SC is applied, the scatter
component must be clearly distinguished from the normalization component; this becomes increasingly difficult as the
scatter fraction increases. The inclusion of a large number of scattered events is particularly deleterious to the
calculation of GFs, necessitating a low-scattering source.
Casey [1995] presented a method for performing a completely 3D normalization using rotating rods present in
most modem PEr scanners, and examined the contribution of the crystal interference pattern to the overall GFs. Casey's
work indicates that there may be numerous symmetries within a PET scanner which must be taken into ,consideration.
The present work demonstrates that there is an optimal number of parameters for the GF matrix; using too few
parameters introduces image artifacts. while too many parameters reduces the statistical power of the GFs.
(-I
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METHODS
r-l
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-
A planar configuration has been'found to be a favorable source geometry [Stazyk, Kinahan, Bailey 1995b]. The
plane-source is scanned in six different positions (Fig. 1). Corrections related to geometric considerations of scanning a
rectangular plane in a cylindrical detector array must be performed, such as to correct for the varying effective thickness
of the plane-source as viewed in clifferent angular views and non-axial planes.
11997 International Meeting on Fully 3D Image Reconstruction
591
1
2
2
3
4
5
6
3
4
5
6
Figul'e 1. Construction of Composite Sinogram (CS). The 6 positions of the planeMsource are shown at the top left a.nd UIO resulting
sinograms are shown below. The LORs which are "'perpendicular (90 +/M15 degrees) to the planeMsource are shown in the boxes. The
six areas are pieced together to form the CS. as shown at the right.
Calculation of Geometric
Factors
-.-The OFs provide an estimate of the response of a LOR based on the geometric orientation of the LOR within
tlle PEl' scanner. Since in theory the OFs only need to be calculated once and then may be reused with each new set of
EFs. it is vital that the GFs be as accurate as possible; the OFs for a particular scanner geometry must take into account
all important geometric aspects of the LOR efficiencies. Due to axial and radial symmetries within the PEr scanner.
every LOR may be placed into a group so that all members of the group originate from detector pairs with the same
geometric relationship to one another. The four parameters used to establish these groups for our specific PET scanner
(CfIlECAT 953B) are described in Table 1.
.-
-
Parameter
Range (for EeAT 953B) Description
Ring 1
(1 .. 16)
the crystal ring associated with Detectol'A of the LOR
Ring2
(1-16)
ilie crystal ring associated wiili DetectorH of the LOR
k
(Ow42)
the radial distance of the center of an LOR from ilie center of the FOV
b
(1s8)
the position within the blockHstructure of the detector t>air
Table 1. Parameters describing the- GFs. The GFs are calculated as a 4·dimensional matrix with the functional dependence: GF(Ringl, Ring2, k, b).
The crystal rings influence the efficiency of a LOR tllrough two major mechanisms: the predominant
mechanism is related to the block structure and a minor mechanism is the changing orientation of the detectors in
various rings to one another. We found that sinograms with the same ring..difference but different rings cannot be placed
into the same OF group. since each ring has a different efficiency; each sinogram must have its own unique set of GFs.
The only exception is that complementary sinograms (RingA-RingB and RingB-RingA) may be combined. For our
particular scanner, each OF is comprised of 48 individual LORs; each LOR typically contains ",,60 counts.
An early attempt was made to improve the statistics of each OF by taking advantage of further symmetries
inherent to the PET scanner. There are groups of planes whose original crystal rings are separated by ilie same ringdifference. which could be expected to have similar geometric properties. The GFs from all sinograms with the same
ring':'difference were read from the GF array, and ilie values weJ;e normalized to account for the efficiencies of the crystal
rings contributing to each LOR in order to eliminate the axial block"position dependence. The average of ilie normalized
OFs with the same k and block..posltionfrom different planes was determined, and ilie GFs from each plane were
calculated by putting ilie individual plane..efficiencies back into the individual GFs. Although this operation conserved
the average GFs for each plane as well as fpr each group of GF(k;b), individual GFs were changed erratically. Despite
various attempts to correct for known asymmetries. we could create an accurate set of GFs only by grouping LORs
which had the same ring (i.e. sinogram). k. and block"position. While using a GF matrix with fewer dimensions would
have increased the statistical accuracy of ilie GFs, the introduction of ring-artifacts negated any potential benefit.
Calculation of Efficiency and Normalization Factors
The EFs provide an estimate of ilie relative efficiency of each individual detector. These efficiencies change
over time. mainly due to drifting PMI' gairts. To calculate the EFs. two inputs are needed: the Composite Sinogram (CS)
and the GFs. The CS provides information about ilie current state of each detector, and the GFs are needed to remove
geometric dependencies from the EPs when the fanMsums are created. The calculation of the EFs is simply the process
of summing all LORs associated wiili each detect()r and normalizing the Suttls to an average mean of 1.0.
A unique NF is calculated for each LOR. The NFs are written in sinogram format. and typically range from 0.5
to 2.2. For each LOR. ilie EFs for the detectors which defme tlle LOR are combined with the GF for iliat LOR:
Eq.l
NF(LOR) =[ EF(RingA. DetectorA) * EF(RingB. DetectorB) ] / GF(LOR)
11997 International Meeting on Fully 3D Image Reconstruction
RESULTS
Composite Sinogram
.
A series of diagnostic tests is performed on every CS data set. All sinograms are summed to create a single ,
sum-sinogram. The LaRs in this sum-sinogram are further summed over the angular views to create a plot as a function
of the radial bin (Fig 2). One effect which becomes apparent from these plots is if the plane-source is non-uniform. with
e.g. alarger amount of radioB;ctivity on one side than the other. It is just as important to check a moving line-source as' a
solid plane for this effect. since the carriage may not move uniformly throughout its range.
Sum Angular View
(LORs summed over a'ngular vieWS)
~ ~--------------~
..
11
~
,.,~"
J
311
Ii
§ ~
6!
2.85BHl6
2.80BW6
eg ~'"
1
I,
Sum Angular View
Sum Angular View
(LORs Bummed over angular views)
j
,
.' I
It"
1"'1'11'"
•
J
i
~
2.70BHl6
2.6SB+06
~+---l~-~-H----I
11!+-~-+---t----fo-==--I
o
0
40
80
160
120
2.75Btll6
0
40
Radial Bin
80
120
2.6081-06
160
30
SO
90
70
110
130 -:
RodlaIBin
R.diaiBin
Figure 2. Left: Sums of all LORs belonging to the same radial bins (a sum-angular view). Middle: Detail of sums of all LORs
belonging to the same radial bins. Note small left/right asymmetry Right: Comparison of CSs from two plane-sources, with one
!source relatively weaker in the center. This small difference does not contribute to significant differences in uniformity in the
reconstructed data.
,
;
:
Geometric Factors (GFs)
GFs calculated from a non-uniform plane-source lead to image artifacts in the reconstructed data. prim8rily
ring-artifacts. Artifacts can become more pronounced when GFs based on an earlier set of non-uniform plane~source
data are used to calculate EFs and NFs from a different plane-source. Since there is a unique set of GFs which is correct
for each PEr scanner; we feel the best approach is to measure the GFs as accurately as possible and to use the same set
of GFs for all further NFs.
The average GF as a function of block position (Fig.3b) shows the importance of including this parameter. The
excursion in the range of values as a function of block position may be observed in Fig.3d; comparison with Fig.3a
emphasizes the importance of including the block structure in the NF calculation. The dependence of crystal response on
block position causes a larger change in the magnitude of the GFs than the dependence on k.
If the block-position dependence is ignored, the k-dependent GFs are actually the average GFs for all blockpositions. as shown in Fig.3a. This GF distribution averaged over all planes and block-positions is in good agreement
with previously published data [Defrise], both in magnitude and in shape. The average GF smoothed over block
positions ranges from 0.96-1.03. which for most LORs is incorrect. The GF sums for each crystal ring (Fig.3c) likewise
underscore the need to calculate GFs for each sinogram or to otherwise explicitly account for the different ring
efficiencies. Our GFs (as opposed to GF averages) range in magnitude from 0.60 - 1.55. We found that averaging over
any of the four dimensions of the GF matrix introduced image artifacts.
[I
[]
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('1
L-!
3b
AaIg!<F1oI. Hodquilm
U.r-------.
[)
3c
3d
s..dGollbicHdDir~
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Iwo block·podUono
.
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o
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12
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:n
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1', 1I
1
L,'
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~
k l
I
1
".;
Il:tcda ~ #
Figure 3. GFs summed as functions of each of the parameters. The graphs vs. k and block-position plot the average GF, while the
graph vs. Ring # plots the GF sums. The data come from a PEG plane and a moving-rod simulated plane. The two sets of GFs are
quite similar. Note that the average GFs for each k (left plot) have a significantly smaller range than the actual GFs.
lJ
I
L
11997 International Meeting on Fully 3D Image Reconstruction
611
Normalized Sinogrrul1
Our general approach for comparing two sets of NFs is to perform a series of comparisons between two
normalized data sets which are based on PET data from the same phantom but were normalized with different NFs. To
test our llormalizatiollmethod, four different sets of NFs were calculated and applied to a single data set acquired ill 3D·
mode of a 20cm uniform cylinder (see Table 2). The axial and radial uniformity of the reconstructed images were
examined, and the noise in the sinognuns and the recollstl'Ucted images was estimated and compared betweell the
various NFs. To date, we have concentrated on improving tlle uniformity; further investigation is in progress regarding
noise fIlld contrasH'esolutioll aspects.
Il.
NFDescriJ!Jion GFs
EFs
1
Standard
Best OFs (moving line)
current (from aqueous plane)
2
Old EPs
Best OFs (moving line)
out~ofNdate (from moving line)
3
Poor GFs
current GFs (aqueous plane) current (from aqueous plane)
4 _ LORNb MLOR_
N.A.
curt~llt frolU a ueous lane
Table 2~ DescrIptIon of originS-of various NFs. The bestGFs to date were obtained using a movingvline source. The current EFs
(Le., relevant to the PET data at hand) were obtained using a slightly non-uniform aqueous planevsource ("'5% thicker in center).
The LORMby"LOR method simply calculates the ratio of each LOR value to the average of all LaRs.
Reconstructed Image Tests
The goodness of a set of NFs is ultimately measured by the absence of artifacts in the reconstructed images. As
a word of caution, a poor set of NFs will always produce image artifacts, although the artifacts can be masked under
certain circumstances by other compensating processes; for instance, increasing the a priori value of the scatter fraction
used by the Iterative Convolution Subtraction method (a.k.a Pristine, [Bailey 1994]) can remove a "coldMspot" in the .
center of the image that may be an artifact due to the NFs.
Of the four NF sets in Table 2, Method #1 produces reconstructed images with the lowest level of visual and
quantifiable artifacts (see Fig. 4). This method uses what we consider to be the most accurate set of GFs (obtained with a
moving.. line source) and the current set ofEFs (in this case, obtained with an aqueous planar phantom). Using this
method, we are able to achieve axial and radial uniformities of 1.4% and 2.0%, respectively, in a 20cm uniform
cylindrical (or elliptical) phantom.
Of the other 3 sets of NFs studied, two (#2. #3) are variations on our "standard" method (#1), while the LOR~
bYaLOR method (#4) was considered a goOd basis for comparison because it is computationally simple and should have
good detector fidelity, although it is statistically sparse. Nevertheless. this method was found to produce reconstructed
images which are nearly as uniform as those produced by method #1. although nonNulliformities in the planar source are
more readily reflected in method #4. The similarity between these methods is credited mainly to the 10wMscattering
uniform planar sources used throughout these experiments. A detailed comparison of the statistical properties of these
twg meth~s i~ currentl bein
rf'ormed.
A x I a I Un Ito r m Ily
Radial Uniformity: Slopo
Fracllonlll S IlIndlrd D ovlallon
0025
I
I
I
0.0007
0.0006
0.0 I
00 "
-t----
0.0005
of----
0.0004
+-=:::---
0.0005
001
0.0002
0.0001
000'
H .'hI.UIIII •• M .Ib ••
1I. . 4u4
Oll(aq
LOR.by-
pluo)
La R
old BI'.
Figure 4. Axial and mdial Wlifonnities obtained f()reach of the NF sets described in Table 2. The axial unifonnity is for the central 29 of 31 image pl8ijes.
The mdlaluniformity was quantified by calculating the slope of a llrte fit through totalB of annuli of increasing radius placed on images of a 20cm unifonn
cylindrical phantom. The slope has units of O.t.CI/ml)/cmj a slope of 0.0 would be ideal. Our "standard" NF method (#1) perfonns the best on the basis of
image WlITonnity, although the other NFs yield usable results. A recent detector setup and accompanying new nonnalization has resulted in further
improvements of the axial and radial unifonnity, to 1.4% and 2.0 %, respectively.
This work was supported by a post-doctoral fellowship grant from the National Institute of Health (CA67486)
REFERENCES
Bailey DL and Jones T,J. Nucl. Med., vo1.36pp. 92P, 1995 (a).
Bailey DL, et al,1995lEEEIMIC Conference Record, pp. 997,1995(b).
Bailey and Miekle, PhysMed.Biol., 39:3, pp.411-424,1994.
Casey ME and Hoffinan FJ, J.Comp.Assist.Tom., 10(5):845-850, 1986.
Casey ME, et al, Proc.lntn' I Mtng on Fully 3D Image Reconstructio, 1995.
11997 International Meeting on Fully 3D Image Reconstruction
Defrlse M, et al, Phys. Med. Bioi., 36(7), 939·952, 1991.
Kinahan PE et al., Proc.1995IEEEIMIC, San Francisco, 1995.
Spinks n, et al,PhysMed.Biol., 37:8, pp. 1637-1655,1992.
Stazyk MW, et al, Abstract, 1994 SNM meeting, Orlando, FL, June 1994.
(,
AXIAL SLICE WIDTH IN 3D PET:
POTENTIAL IMPROVEMENT WITH AXIAL INTERLEAVING
I
I
!
i
ME Daube-Witherspoon, SL Green, and RE Carson
PET Dept., National Institutes of Health, Bethesda, MD
INTRODUCTION
The axial slice width for the GE Advance PET scanner has previously been reported to be worse
for 3D acquisitions than for 2D "high sensitivity" data [1, 2, 3]. This difference has been attributed
to the combined effects of septa removal and axial smoothing effects introduced during the 3D
reconstruction process [3]. There are four main differences between 2D and 3D image data on the
Advance: (1) the presence of septa in 2D mode and their absence in 3D acquisitions, (2) the larger
maximum axial acceptance angle (maximum ring difference acquired) in 3D, (3) the
approximations for coincidences between different detector rings in the acquisition and
reconstruction processes, and (4) the reconstruction algorithms themselves, especially due to
limited axial sampling and interpolation. We hypothesized that axial interleaving to improve the
axial sampling would lead to a better 3D axial slice width. The goals of this study were to
investigate the source(s) of the observed difference in axial slice width between 2D and 3D and to
assess whether the 3D axial slice width would be significantly improved by acquisition and
simultaneous reconstruction of axially-interleaved data.
[1
[\
Li
The physical characteristics of the GE Advance PET scanner have been previously published'
[1]. Briefly, the system comprises 18 rings ofBGO detector blocks (crystal dimensions =4.0 mm
x 8.1 mm x 30 mm), covering an axial field of view (FOV) of 15.3 cm, with a center-to-center ring
separation of 8.5 mm and a slice separation in the reconstructed image of 4.25 mm. The scanner
can acquire data in three possible modes: (1) 2D "high resolution" mode (RR), with a ring
difference of 0 (direct slices) or ±1 (cross-slices), (2) 2D "high sensitivity" mode (HS), with ring
differences of±2,O (direct slices) or ±3,±1 (cross-slices), and (3) 3D mode, with a maximum ring
difference of ± 11. HS mode is the standard 2D mode on this scanner. The 2D data are
reconstructed using the 2D filtered back-projection algorithm. For both RR and HS modes, crosscoincidences between different detector rings are treated as parallel to the direct coincidences (no
ring difference), with an axial location that is the average of the axial positions of the two detector
rings. The 3D data are reconstructed using the reprojection algorithm [4]. Cross-coincidence data
between different detector rings are positioned properly in the 3D back-projection, with the
exception that lines of response (LORs) centered on a cross-slice (Le., with an odd ring difference,
2n+1) are approximated to have an even ring difference (2n), centered at the same axial position, in
the same way as conventional2D cross-slices are processed. No data reduction schemes (e.g.,
"mashing") are applied to 3D data.
[!
[]
METHODS
Axial Slice Width
{
I
\
;
I I
,/
The axial response functions were acquired on the GE Advance PET scanner using point
sources ofF-18 «1 mm axial extent), sandwiched between two 1.5-mm aluminum disks and
suspended in air from the end of the patient bed at radial positions r = 0, 5, 10, 15, and 20 cm. To
reduce the number of scans, data were acquired at 1-mm axial intervals for half of the axial fieldof-view (90 scans), since the scanner is symmetric about the axial center. Axial response functions
were measured in all three acquisition modes with the septa extended (usual configuration for 2D) ,
and retracted (usual configuration for 3D), in order to assess separately the effects of the axial
acceptance angle, septa, and handling of cross-coincidences on the axial slice width. The data were
reconstructed prior to calculation of the axial response functions. Corrections were made for
11
l,
11997 International Meeting on Fully 3D Image Reconstruction
Daube-Witherspoon, et al.
deadtime, randoms, detector nonnalization, and slice sensitivity, but not for scatter or attenuation.
For all data, a transverse ramp filter at the Nyquist cutoff frequency (0.25 1nmol)owas used; for 3D
elata, the axial filter was also a ralup at the Nyquist cutoff frequency (0.118 lnln }). Regions of
interest were drawn in the reconstructed images around the five point sources and the total counts
for each radial location determined for each slice and axial bed position. The axial response
functions for each slice and radial position were created from the total counts as a function of axial
source position. The axial slice width, parameterized by the full-width-at-half-Inaximmn (FWHM),
was detennined froln the axial response functions by linear interpolation.
Effects ofAxial Interleaving
One possible method to improve the 3D axial slice width is to increase the axial sampling of the
data by acquiring data at two axial positions, separated by one-half the slice separation (Le.,
approximately 2 mm apart) and then reconstlucting these data as one 3D set. A similar technique
for handling 3D whole..body PET data has been investigated as a way to reduce the axial noise
non-uniformity arising from differences in slice sensitivity for whole-body scanning in 3D data [56] and to improve axial resolution [7]. To assess the utility of axial interleaving during acquisition
in reducing the 3D axial slice width, the contribution of 3D reconstruction effects (FWHM,eco,) was
estimated from the axial slice width of 3D data (FWHM3Doou,) by
FWHM;D_out == FWHM~R_out + FWHM~con
(1)
where FWHMHRoout is the axial slice width ofHR data with the septa retracted~ Since HR data
effectively have no cross-coincidences and, therefore, no axial mispositioning effects, the
difference between HR-out and 3D-out data is primarily in the 3D reconstruction process,
including (1) the axial filter applied, (2) axial "smoothing" introduced by interpolation during the
back-projection step, and (3) approximations in positioning the crossncoincidence LORs in 3D.
The first two contributions to the axial slice width will be affected by the axial sarnpling. If the
third effect is small (e.g., near the center of the FOV), then interleaving axially will approximately
halve the contribution of reconstruction smoothing (FWHM,eco,,)' since a sharper axial filter could
be used. The best-possible value for the axial slice width with axially-interleaved data
(FWHM3Do/llte,) would then be given by
FWHM;D_illter
:=
FWHM~R_out + (FWHMrecoll /2)2
(2)
or
(3)
RESULTS
Axial Slice Width
Table 1 summarizes the axial slice width results for direct and cross-slices, averaged over slices
18-33 (i.e., one-half of the axial FOV, not including the edge slices). The edge slices (34 and 35)
have consistently better values for the axial slice width than the other slices, due in·part to
differences in the number of cross-coincidences contributingo The 2D results with septa extended
("septa in") are somewhat worse (-0.5 mm) than the values given in [1]; the 3D results with septa
retracted ("septa out") are comparable to those reported in [2].
When cross-coincidences are not acquired (direct-slice HR), the septa have minimal effect on
the axial slice width. However, when cross-coincidences are accepted (HS), then the septa have a
narrowing effect on the axial slice width by collimating some of the cross-coincidences. At r=20
11997 International Meeting on Fully3D Image Reconstruction
641
Daube-Witherspoon, et al,
r~)
cm, however, thls collimation can actually remove some LaRs that would have contributed to the
center of the axial response function, so the HS axial slice width is larger than HR (see direct
slices). When the septa are removed, the axial slice width values for HS data are close to the HR
results out to -10 cm. At larger radial positions, axial mispositioning of the cross-coincidences
results in larger axial slice width values for HS than HR, directly analogous to the blurring results
seen with the single-slice rebinning algorithm [8].
I
I
(
J
Table I
Average Axial Slice Width Results
FWHM(mm)
Acquisition Mode
[]
r=Ocm
5.7 ± 0.6
r= 10cm
5.8 ± 0.5
r=20cm
5.6 ± 0.6
...~.~ ..:....~~P.~.~..!~............ ....·····s·:6. f . o:·2·····..· . · ·········s·:'1..f ..o:·i..··......·.. ····....·S·:S··±··O:'2·..·..······..
DIRECT
SLICES:
Septa out
4.2 ± 0.2
4.3 ± 0.2
7.7 ± 0.3
...~.~..~....~~p.!.~.i~............ ·..·....·s·:'7··f··o:·I·
..··....·.... ..·..·..·s·:6..f··o:·2···· ..······.. ..·..·..·'1·:5..:£··0:·1'..····. ··..·
Septa out
3D- Septa out
6.0 ± 0.1
6.9 ± 0.1
7.2 ± 0.2
5.3 ± 0.3
6.2 ± 0.4
..............................................
Septa out
5.6 ± 0.2
5.8 ± 0.3
6.5 ± 0.3
4.2
±
0.2
4.2
±
0.2
5.8
± 0.4
...~.~..~....~~p.~~..p............. .............................................. ..............................................
Septa out
5.8 ± 0.1
5.7 ± 0.2
8.9 ±O.J'
3D- Septa out
6.0 ± 0.1
7.2 ± 0.1
7.7 ± 0.3
4.6 ± 0.4
...!!~. :. . ~~P.~.~)E............ ..............................................
CROSSSLICES:
• • • • • • • • • • • • • • • • • • 11 • • • • • • • • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 11 • • • • • •
Effects of Axial Interleaving
Table II summarizes the estimated maximum benefit of axial interleaving by one-half the slice
separation on the axial slice width, as calculated from equation 3.
[]
[J
I]
Table II
Estimated 3D Axial Slice Width Improvement with Axial Interleaving
FWHM(mm)
Radius (cm)
0
5
10
15
20
[J
r-'
,J
[)
II
DIRECT SLICES
No interleave
Interleaved
5.7
6.0
5.8
6.7
6.0
6.9
6.1
7.0
6.2
7.2
Near the center, where HR and 3D data have comparable axial slice widths, the benefit of axial
interleaving is small. Farther out, the 3D axial slice width can be improved, but the decrease is only
-1 mm across the FOV. The slight improvement in axial slice width with interleaving would also
be accompanied by an increase in noise that would arise with the sharper axial filter required to
achieve this better resolution, since the Nyquist frequency is higher with increased axial sampling.
LJ
rI
I
i.
I
J
1
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CROSS-SLICES
No interleave
Interleaved
6.0
5.7
6.9
5.8
6.2
7.2
7.5
6.4
6.8
7.7
11997 International Meeting on Fully 3D Image Reconstruction
Daube-Witherspoon, et al.
DISCUSSION
The results fronl this study confirm the worsened axial slice width for 3D data, when compared
to the standard 2D (HS) data. The difference between 3D and HR data is much smaller. The Inain
reason for the slightly degraded axial slice width in 3D compared with HR is axial filtering and
interpolation effects, not the septa. The primary reason for the difference between the axial slice
widths for 3D and standard 2D acquisitions on the GE Advance is that the axial slice width results
for I-IS data are better than expected because the septa effectively narrow the slice width of crosscoincidences. When the septa are relnoved, the axial slice width in HS mode worsens to that seen
in I-IR mode. At large radial distances, cross-coincidences are also mispositioned axially, further
blurring the axial response function. While similar results for axial slice width have not been
reported for 3D acquisitions on the Siemens/eTI BeAT EXACT HR or HR+ scanners, it is
pos$ible that some combination of Inaximum ring difference and "span" (or summing of crosscoincidences to create fewer LORs) could be found that would reproduce these results (Le., large
span in 2D, with little or no span in 3D and comparable axial acceptance angles to those on the
Advance). I-Iowever, the shorter septa on the Siemens/eTI scanners (6.55 cm [9], compared with
12 cm on the GE Advance [1]) may make the differences in axial slice width between 3D and 2D
smaller than are seen on the Advance.
We conclude that axial interleaving, followed by 3D reconstruction of the combined data set
with finer axial sampling, would provide only a small reduction in axial slice width. The increase
in noise that would ensue from the sharper axial filter that would be used to achieve this better
resolution is likely to outweigh any advantage of improved resolution.
REFERENCES
1. DeGrado TR, Turkington TO, Williams JJ, Stearns CW, Hoffman JM, and Coleman RE. Perfonnance
characteristics of a whole-body PET scanner. J Nucl Med 1994; 35:1398-1406.
2. Lewellen TK, Kohlmyer SO, Miyaoka RS, Kaplan MS, Stearns CW, and Schubert SF. Investigation of the
perfonnance of the General Electric ADVANCE positron emission tomograph in 3D mode. IEEE TrailS Nucl Sci
1996; 43:2199-2206.
3. Pajevic S, Daube-Witherspoon ME, Bacharach SL, and Carson RE. Noise characteristics of 3D and 2D PET
images. Submitted to IEEE Trans Med Imag.
4. Kinahan PE and Rogers JG. Analytic 3D image reconstruction using all detected events. IEEE TrailS Nucl Sci
1989; 36:964-968.
5. Dahlbom M, Cutler PD, Digby WM, Luk WK, and Reed J. Characterization of sampling schemes for whole
body PET imaging. IEEE Trans Nucl Sci 1994; 41:1571 .. 1576.
6. Cutler PD and Xu M. Strategies to improve 3D wholeBbody PET image reconstruction. Phys Med Bioi 1996:
41: 1453~ 1467.
7. Dahlbom M, Chatziioannou A, and Hoh CK. Resolution characterization of continuous axial sampling in whole
body PET. 1995 Nuclear Science Symposium and Medical Imaging Conference Record, pp. 1011-1015.
8 . Daube-Witherspoon ME and Muehllehner G. Treatment of axial data in three-dimensional PET. J Nucl Med
1987; 28:1717 I724.
9. Wienhard K, Eriksson L, Grootoonk S, Casey M, Pietrzyk U, and Heiss W-D. Performance evaluation of the
positron scanner ECAT EXACT. J Comput Assist Tomogr 1992; 16:804-813.
w
11997 International Meeting on Fully 3D Image Reconstruction
661
Implementation and Evaluation of Iterative Three-Dimensional
Detector Response Compensation in Converging Beam SPECT
[-I
E.C. Frey*\ S. Karimi\ B.M.W.Tsui*\ and G.T. Gullberg:t
*Department of Biomedical Engineering, The University of North Carolina at Chapel Hill
tDepartment of Radiology, The University of North Carolina at Chapel Hill
:tDepartment of Radiology, The University of Utah.
[i
[I
[I
~
1
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Converging beam collimator geometries offer improved tradeoffs between resolution and
noise for single-photon emission computed tomography (SPECT). The major factor limiting the
resolution of these images is the detector response blurring. One method of detector response
compensation is the use of iterative reconstruction algorithms with the modeling of the 3D detector
response function (DRF) in the projector-backprojector. Previous studies of DRF compensation in
parallel beam tomography have indicated that the ability to recover resolution depends on a variety
of factors including the size and position of the object [1]. It was also observed that the resolution
recovery is best in the longitudinal direction, i.e., along the rotation axis of the collimator. In this
study we investigate resolution recovery in SPECT imaging with converging beam collimator
geometries.
In order to apply iterative resolution recovery, one must be able to model DRF, including
both the collimator and intrinsic components. Previously methods have been proposed to compute
the collimator point response function (CPRF) for both parallel [2] and converging beam. [3]
collimators. The method begins by computing the probability that a photon emitted at a particular
point in the object and detected at a particular point in the imaging plane will pass through a
particular collimator hole. The value of the response function for this source and detection position
is then computed by averaging the collimator hole over all possible positions. It turns out that this
can be done quickly and compactly using convolution in the spatial domain.
In the theoretical analysis of converging beron geometries, the holes are moved such that
the center of the holes are always aligned with the focal point. A limitation of this analysis is its
assumption that, when this averaging process takes place, the collimator holes can be described by
the same aperture function. This assumption allows the CPRF to be computed using a convolution
similar to the case for parallel holes. However, this assumption is invalid. In manufacturing cast
collimators, the shape of the pins used to cast the collimator holes remains the same, but the pins
are angled differently so that they focus to a common line (in fan-beam) or point (in cone-beam).
This gives rise to an elongation of the hole apertures along the fan and radial directions for fan and
cone beam collimators, respectively. Since, the elongation changes as a function of the distance
from the focal line or point (i.e. , the collimator holes cannot be described by the same aperture
function), the averaging over hole position cannot be performed by a convolution.
In this work, we have derived expressions that include the elongation of the hole apertures
in a cast collimator. To overcome the difficulty mentioned previously, we note that the hole
aperture shape, while varying across the face of the collimator, is essentially constant over the
support of the point spread function. As a result, the CPRF can once again be estimated by
convolution. The expression for the CPRF, ~(r;Fo), for a source whose projected position on the
image plane is Fa
= (xo,Yo)
evaluated at position
r = (x,y)
on the image plane is given by:
~(r;Fo)=K[af(8)af](-rT)'
(1)
In this expression, K is a factor that is essentially constant, af( ij) is the aperture function on the
[I
l.
'
I)
r
\
l
J
I
face of the collimator for a hole lying under the focal line or plane, (Le., af(ij) is zero if ij lies
outside the collimator hole and one inside the hole), and rT describes the offset of the projected
aperture functions with respect to each other. Let F be the distance from the focal line or point to
the front face of the collimator, Z be the height of the point source above the collimator face, L be
,
11997 International Meeting on Fully 3D Image Reconstruction
671
the thickness of the collitnator, and B be the distance f1'01n the back face of the collhnator to the
hnage plane. Then, for fan beatn collitnators, the components of the offset parallel (x.. direction)
and perpendicular (y-direction) to the focal line are:
rT.t:
.
rry =
L
= Z+L+B
. (x - xo), and
CF+L)(~+L+B)[ ~i :C;~::B)2 }CF-Z)Y-CF+L+B)Yo).
(2)
(3)
In Equation (3), the tenn in square brackets 1110dels the elongation of the holes. Note that this
expression l11akes use of the approxllnation that the shape of the holes does not change over the
width of the response function. The expression for cone beam collitnators is silnilar, but the
elongation occurs in the radial direction.
A second difficulty in converging beam tomography is the fact that DRF is spatially variant
in planes parallel to the face of the collimator. As a result, a direct llnplementation would require
cOlnputing or storing a different DRF for each point in the plane and would not allow using the
cOlnputational efficiency that can be gained by convolution. However, Tsui and Gullberg
previously derived a linear transfonnation that restores the spatial variance of the response
function. First the activity distribution in a given source plane is transformed linearly and then
convolved with a spatially invariant DRF. This blurred unage is then transformed linearly to give
the detection~plalle image due to activity in the source plane. However, when the assumption of
constant hole shape is removed, this linear transfolmation no longer produces a DRF that is
spatially invariant in planes parallel to the collimator. In this work we have derived an expression
for an additional nonlinear transformation that can be used to account for the elongation of the
holes and restore the spatial invariance in these planes.
To validate the CPRF fonnulas, we computed the full width at half maximtUll (FWHM)
resolutions and compared these to experimental measurements. The effects of the hole aperture
elongation are nIost noticeable for large fan angles. As a result, we used an asymmetric fan beam
collimator having a focal length of 50 cm n1easured from the face of the collimator. The hexagonal
hole size (flat to flat) and length were 0.18 cm and 3.8 em, respectively. This collimator was used
on a GE Optima camera system having a 35 cm size detector. Line sources were placed at various
distances from the central plane of the fan and from the face of the collimator. Detector line source
response functions (DLSRF) were measured and compared. The FWHM of the DLRF was
estimated by fitting the experimental data with a Gaussian function. Similarly, the CPRF was
computed numerically, collapsed to form the line response function, convolved with a Gaussian
representing the intrinsic response of the detector, and fitted with a Gaussian to give the width of
the DLSRF. Table 1 shows the results. In both cases the FWHM was scaled to represent the
resolution in the source plane.
Table 1. Experimental measurements and theoretical calculations of the resolution in tenns of FWHM in the source
for1
'
variOUS
source _pOSl'fIOns.
Distance from fan beam axis
5.8 em
10.8 cm
15.8 em
20.8 cm
25.8 em
Distance from
Theory, Exp
Theory Exp_
Tl1eory E~p
Theory Exp
Theory
face of collimator Exp.
5.3 mm 5.4 mm 5.4 rom 5.5 mm 5.6 nun 5.7 mm 5.9 mm 5.8 mm 6.5 mm 5.9 mm
5cm
7.2 mm 7.3 nun 7.5 nun 7.5 mm 7.7 mri:1 7.7 nim 7.72 nul 8.05 mn
10 em
9.4 mm 9.5 mm 9.9 mm 9.7 mm .10,6 mn 10.2 mn 10.4 mn 10.8 mn
15 cm
12.0 mn 11.7mn 12.3 mn 12.2 mn 13.0 mn 13.0 mn
20 em
14.4 mn 14.1 mn 14.9 ron 14.9 mn
25 em
PiI ane
There are several things to note. First, there is ,good agreement between the experimental
and theoretical results. Second, there is a noticeable difference in the width of the DLSRF for
points away from the axis of the fan. For example, at 10 cm from the face of the collimator, the
11997 International Meeting on Fully 3D Image Reconstruction
681
r1
l I
resolution changes by as much as 10% as we move from the fan beam axis to the edge of the
collimator.
To investigate the properties of the iterative resolution recovery, we implemented a model
of the CPRF for cone and fan beam collimators in a projector-backprojector pair. The method is
based on the rotation-warping method proposed by Zeng et al [4]. These were incorporated into the
iterative ordered-subsets expectation-maximization (OSEM) algorithm.
To study the properties we computed the reconstructed local 3D modulation transfer
function (MTF) using the method described by Wilson [5]. Projection data were simulated with
and without point sources at various positions in a uniform 3D background. The simulation was
carried out at a pixel size of 0.15 cm, and the resulting projection data collapsed to a bin size of 0.3
cm. Data were simulated at 128 angles over 360°, A collimator 4.1 cm thick having a 0.19 em
diameter round holes and a radius of rotation of 15 cm was used. We reconstructed the images
using 20 iterations of the OSEM. The difference of the image reconstructed with and without the
point source gives the point response function. The magnitude of the Fourier transform of this
image gives the reconstructed local 3D MTF.
Sample results are shown in Figures 1a-c. In these figures, profiles through the 3D MTFs
for a point source at the center of rotation are shown in the longitudinal direction (along the axis of
rotation) and in the transaxial plane containing the point source. This is shown for 2 (Figure la),
10 (Figure 1b) and 20 (Figure 1c) iterations. These curves exhibit several interesting features.
First, note that the MTF exhibits a relatively sharp cutoff and the cutoff frequency increases with
the number of iterations. This would suggest the existence of ringing in the reconstructed image,
and this was indeed observed. Second, the cutoff frequency shifts outward and becomes steeper
with increasing iteration. Finally, the MTF is wider in the longitudinal direction. This indicates'tliat
there is better resolution recovery in that direction, and agrees with previous observations for
parallel beam geometries.
.
In conclusion, we have derived expressions for the CPRF for converging beam collimators
that take into account the variation of the hole aperture shape seen with cast collimators. We have
verified this expression by comparing the theoretical predictions with experimentallIleasurClnents.
A model for the DRF was implemented in a projector-backprojector pair of the iterative OSEM
algorithm. The resolution properties of this algorithm were investigated for fan beam SPECT. It
was found that the reconstructed frequency response, as measured by the reconstnlcted local MTF,
exhibits a relatively sharp cutoff. This cutoff frequency increases with iteration and the cutoff
becomes sharper. Resolution recovery was faster in the longitudinal direction, where computed
tomography is not taking place, than in the trans axial direction of the image plane.
Li
[I
[]
[I
[J
[J
1.2 -H+++H+I-+f+i-++f++t-+j-H-H-/~:::B-ral-Hnsf-tiaxic+:f-al+t-Transaxial
/., ............... ::. ,,\Longitudinal
.... _,..:.:•..... Longitudinal
.~ 0.8
[]
:l
~0.6
:e 0.4
~
1=
[]
~
0.2
...
-2
-1.5
/
-1
.\.
\
:e
~0.4
\.
"-
\
~ 0.2
\..........
1
1.5
2
\
{0.8
/ \
~0.6
\
-0.5
0
0.5
Frequency (cm· l)
/
{ 0.8
\,
-1.5
"-
~ 0.2
.........
......
-2
~0.6
:e~0.4
-1
-0.5
0
0.5
Frequency (cnil)
1
1.5
2
-2
-1.5
-1
-0.5 0
0.5
Frequency (cm· l)
I
1.5
2
(a)
(b)
(c)
Figure 1. Plot of reconstructed local MTF in the trans axial and longitudinal directions for a point
source at the center of rotation for after (a) 2, (b) 10, and (c) 20 iterations.
[J
[1] B. M. W. Tsui, X. D. Zhao, E. C. Frey, Z.-W. Ju, and G. T. Gullberg, "Characteristics of reconstructed point
response in three-dmensional spatially varland detector response compensation in SPECT ," in ThreeDimensional Image Reconstruction in Radiology and Nuclear Medicine, Computational Imaging and Vision, P.
Grangeat and J.-1. Amans, Eds. Dordrecht: Kluwer, 1996, pp. 149-162.
[2] C. E. Metz, F. B. Atkins, and R. N. Beck, "The geometric transfer function component for scintillation camera
collimators with straight parallel holes," Phys Med Bioi, vol. 25, pp. 1059-1070, 1980.
(-1
\l I
j
11997 International Meeting on Fully 3D Image Reconstruction
[3] B. M. W. Tsui and G. T. Gullberg, "The Geometric Transfer Function for Cone And Fan Beam Collimators,"
Pllys Med BioI, vol. 35, pp. 81-93, 1990.
[4] O. 1. Zeng, Y.-L. Hsieh, and G. T. Gullberg, "A rotating and squashing projector-backprojector pair for fanbeam and cone-beam iterative algorithms," presented at IEEE Nuclear Science Symposium and Medical Imaging
Conference, San Francisco, CA, 1993.
[5] D. W. Wilson, "Noise and resolution properties ofFB and ML-EM reconstructed SPECr images," : University
of North Carolina at Chapel Hill, 1994.
11997 International Meeting on Fully 3D Image Reconstruction
701
Minimal Residual Cone-Beam Reconstruction
with Attenuation Correction in SPECT
[I
Valerie La and Pierre Grangeat
LETI (CEA - Technologies Avancees)
DSYS - CEA/G - 17 rue des Martyrs
F 38054 Grenoble Cedex 9 - France
E-mail: [email protected], pierre.grangeat@ceaJr
[i
1
Introduction
This paper presents an iterative method based on the Minimal Residual (MR) algorithm for tomographic
Ci
attenuation compensated reconstruction from attenuated Cone-Beam (CB) projections given the attenuation
distribution. The attenuation map is obtained by a preliminary reconstruction from transmission measurements.
[:
[]
[]
[I
[J
[J
The development of a CB reconstruction algorithm is motivated by the idea of replacing the external radioactive
source with an X-ray point source to perform the transmission scan. The X-ray source may be used either
simultaneously with the emission acquisition or in a flash mode before each emission acquisition step. However,
for more convenience, the collimator ought to be fixed. Hence the use of an X-ray point source implies that
the acquisition be performed with a CB collimator. A CB reconstruction method is then required since this
defines a CB geometry.
An iterative reconstruction method for CB attenuation compensation using the Conjugate Gradient (CG)
algorithm has been derived for non-planar orbits [1]. Unlike CG-based reconstruction techniques, the proposed
MR-based algorithm solves directly a quasi-symmetric linear system [2]. Thus it avoids the use of normal
equations, which improves the convergence rate.
This work introduces two main contributions. First, a regularization method is derived for quasi-symmetric
problems, based on a Tikhonov-Phillips regularization applied to the factorization of the symmetric part of
the system matrix. A regularized MR-based algorithm is then obtained. Second, our existing reconstruction
algorithm for attenuation correction in PB geometry [2] is extended to CB geometry.
Experimentations will be done in the case of a full circular orbit. However the algorithm principle can be
extended to more complex trajectories. The effect of the shadow zone in the Radon domain will also be
analyzed.
[I
2
Theoretical background
The inverse problem in tomographic reconstruction consists of reconstructing the unknown object / given the
projection data m.
1-]
2.1
Non-attenuated case
Let X be the operator modeling the attenuation-free projection process. The direct problem is written as :
X/=m.
[I
An exact inverse operator
x- 1 can be derived for PB geometry.
In CB geometry, only approximate inverses
are available for a circular trajectory; however if the orbit satisfies the completeness condition, then there is
r- 1
l j
11997 International Meeting on Fully 3D Image Reconstruction
an exact inverse. If X-I exists, then
f is given by :
f =
X- 1 m.
In practise, because of the ill-posedness of the inverse problem, the obtained solution! is very noisy and hence
not satisfactory. A common stabilization technique is to apply a mollifier W to low pass filter the measurements
m before reconstructing. The regularized solution is then:
!=X-1Wm.
Note that in the case of indirect reconstruction [3] such as Grangeat's algorithm, an additional flltcring
operation is applied within X-I to invert the intermediate Radon transform or its derivative.
2.2
Attenuated case
The attenuation phenomenon arising in emission tomography results in a perturbation of the non-attenuated
direct problem, where the operator X is replaced with the attenuated projection operator XI-' . In the general
case of a nonuniform attenuation distribution, there is no exact inverse operator X;l. The object f is then
recovered by solving the linear system corresponding to the direct problem:
(1)
XI-'f=m
where X/J' f and m are now in a discrete form. (1) can be solved iteratively using e.g. the CG algorithm.
However the convergence rate is fairly low. It is improved using preconditioning techniques. The system is
left preconditioned by an operator P as follows:
PX/Jf
= Pm.
P is chosen as an approximate inverse of XI-" The closer P XI-' is to the identity operator within a normalization
factor, the faster the convergence rate is. A possible expression for P is the exact inverse, when it exists, or
an approximate inverse of the un attenuated projection operator X [2]. As in the non-attenuated case, a
mollification technique is applied to prevent noise amplification. The resulting system is the following:
(2)
This system can also be solved using CG. This requires that (2) be symmetrized, which squares the condition
number of the matrix. Hence, the convergence rate is degraded. The approach we adopted in [2] aimed at
avoiding the symmetrization in order to preserve the initial convergence rate. (2) was solved directly using
the MR algorithm described in the next section. The results we obtained demonstrated that, in the case of
a PB geometry, MR was at least twice as fast as preconditioned CG (10 vs. 20 to 40 iterations to converge).
The approximate inverse chosen as the preconditioner was the filtered backprojection formula, i.e. the exact
inverse in the PB non-attenuated case.
2.3
The MR algorithm
MR ([4]) is a conjugate residual type algorithm for solving a system:
Ax = b
(3)
where A is a quasi-symmetric matrix, i.e. a symmetric positive semi-definite matrix perturbed by a small
skewsymmetric matrix. It minimizes at each iteration the squared residual error
lib - AxW, where
11.11 is the
quadratic norm. It solves directly the system (3) instead of the following normal equations:
I19971nternational Meeting on Fully 3D Image Reconstruction
721
Hence, MR yields a faster convergence rate than CG, as indicated above. The difference between these two
algorithms is the conjugacy property between two successive residual vectors rk = b - Axk. The latter are
conjugated with respect to A for MR whereas they are conjugated with respect to AAt for CG. This difference
causes MR to converge much faster than CG.
[-I
The application of MR to invert (2) supposes that the matrix P XJ.l be quasi-symmetric. This assumption may
be made since P XJ.l is chosen to be close to the identity matrix, within a normalization factor.
3
Regularization
When applied on (2), MR yields images with increased artifacts at high iteration numbers. These artifacts
[)
[]
[]
are related to numerical noise and measurement model errors coming from the use of standard reprojection
and backprojection routines instead of an explicit computation of the matrix coefficients. This unstability
is overcome by applying regularization techniques, in particular by imposing smoothness constraints on the
object. For a symmetric problem, a popular method is Tikhonov-Phillips regularization, for which it is easy
to interpret the regularized system as a penalized quadratic criterion. For an unsymmetric system, such an
interpretation is not straightforward. However, in the case of a quasi-symmetric matrix, it may be done by
neglecting the skewsymmetric part of the matrix. Let us write the symmetric part of the matrix P XJ.l using
a Cholesky factorization :
For clarity reasons, let
[]
[J
-1
[.J
-1
(4)
b= PWm.
Since P X J.l is assumed to be quasi-symmetric, we may state that :
(5)
Then (2) becomes
(6)
t
Solving (6) is equivalent to minimizing the squared error IIQI - (Q )-lbI1 2 ; since QtQI - b is the gradient of
that error term. Then the penalized Tikhonov-Phillips criterion becomes:
J
[
[]
where C represents some differentiation operator and
oX
is the regularization parameter. Setting the gradient
of J (I) to 0 leads to solving the system :
(7)
[]
Using (4) and (5), (7) becomes:
(8)
MR is then applied on the regularized system (8). Note that no explicit decomposition of the matrix P XJ.l is
needed.
[-I
lJ
A spatially adaptive version of the proposed regularization scheme can be derived by making
oX
depend on the
position in the object [2].
We will illustrate this regularization approach on simulated and experimental data in PB geometry.
1
I
)
11997 International Meeting on Fully 3D Image Reconstruction
731
4
Application of MR to CB reconstruction with attenuation correction
From a theoretical standpoint, the MRMbased method for PB data derived in [2] should also be applicable to
CB data. Therefore, the principle of the algorithm remains the same. The only change to be made lies in the
choice of the preconditioner P. As for the PB case, we refer to the non-attenuated case to get some approximate
inverses to X", We will consider the case of a circular trajectory. The two main approaches for cone-beam
reconstruction are a direct reconstruction and an indirect one via the Radon domain [3J. We use Feldkamp's
and Grangeat's inversion formulae respectively. The main difference between the two reconstruction formulae
concerns the shadow zone, which can be filled in by interpolation in the indirect approach to smooth the CB
reconstruction artifacts. The attenuation corrected reconstruction method applied to CB data is then to solve
system (8), where XI' models the CB attenuated projector, P is either Feldkamp's or Grangeat's inversion
formula. ,\ is set experimentally.
The proposed regularized attenuation compensation method will be evaluated on the following simulated data:
a heart phantom, a MTF phantom consisting of small spheres fitted into each other and the Defrise phantom.
The last two phantoms will allow to compare the two approximate inverses mentioned above, in particular to
evaluate the effect of processing the shadow zone and the contribution of a better approximate inverse to the
solution of the attenuation compensation reconstruction problem. We will also compare the reconstructions
between non-attenuated data and attenuation compensated data, and between PB and CB data for attenuated
data.
The MR-based CB reconstruction algorithm with attenuation correction can be extended to more complex
trajectories such as the helicoidal orbit, since direct and indirect inversion algorithms for non-attenuated CB
projections have been proposed.
References
[1] Y. Weng, G.L. Zeng, G.T. Gullberg "Iterative Reconstruction with Attenuation Compensation from
Cone-Beam Projections Acquired via Non-Planar Orbits", Conference Rec01'd of the iEEE 1995 Nuclear
Science Symposium and Medical Imaging Conference, San Francisco, California USA, Oct 21-28, 1995.
[2] V. La, P. Grangeat, S. Iovleff, A. Mallon, P. Sire "Evaluation of Two Conjugate Gradient Based Algorithms for Quantitation in Cardiac SPECT Imaging", Conference Record of the IEEE 1996 Nuclear
Science Symposium and Medical Imaging Conference, Anaheim, California USA, Nov 2-9, 1996.
[3] P. Grangeat, P. Sire, R. Guillemaud, V. La, "Indirect Cone-Beam Three-Dimensional Image Reconstruction" in C. Roux, J.L. Goatrieux Eds, Contemporary Perspectives in Three-Dimensional Biomedical
Imaging, lOS Press, to appear.
[4] O. Axelsson, "Conjugate Gradient Type Methods for Unsymmetric and Inconsistent Systems of Linear
Equations", Linear Algebra and its Applications, vol. 29, 1-16, 1980.
11997 International Meeting on Fully 3D Image Reconstruction
741
Simulation of dual-headed coincidence imaging using the SimSET software package
RL Harrison, SD Vannoy, WL Swan Costa, MS Kaplan, TK Lewellen 1
Department of Radiology
University of Washington Medical Center
Seattle, WA 98195, USA
[I
Abstract
We are currently testing extensions to the Simulation System for Emission Tomography (SimSET)
software package that facilitate simulation of positron volume imaging (PVI) imaging using dual-headed
gamma cameras. Coincidence photons may be tracked through the tomograph field-of-view, to the face
of the detector assembly, through graded absorbers (e.g., to reduce low-energy photon counts), and
finally through the detector. Detected coincidences may be binned in single-slice rebinning (SSRB),
multi-slice rebinning (MSRB), or 3D reprojection (3DRP) format. Events may also be binned by energy
and number of scatters.
[]
Introduction
[I
PET imaging is becoming a standard clinical tool,
creating a need for lower cost tomographs (e.g. [1],[2]).
One solution being investigated is the use of dualheaded gamma cameras capable of both single photon
and coincidence (positron) imaging [3],[4].
[).
The SimSET software package was first released to the
public in August 1993 [5]. At that time, the package
would only track photons through the tomograph fieldof-view. Over the years, we have added collimator and
detector simulations to SimSET. We are currently
adapting the package to facilitate simulations of PVI
imaging using dual-headed gamma cameras.
[I
II
Software description
[\
1
[
J
[]
Figure 1: Overview of the SimSET software.
[\
To execute the SimSET program, the user first creates
parameter files that describe the object being imaged,
the tomograph being simulated, and the desired output
format. SimSET has four main computational modules:
the photon history generator (PHG), the collimator
module, the detector module, and the binning module
(Figure 1). The PHG generates decays and tracks the
resulting photons through the tomograph field-of-view.
The collimator module provides a geometric transfer
function [6] approximation for SPECT collimators
(including cone- and fan-beam) and a Monte Carlo
simulation of PET collimators. The current general
release of the detector module only performs energy
blurring. The current release of the binning module
histograms events by position, energy, and number of
scatters.
We are currently performing in-house testing of extensions to the detector and binning modules. We
have added Monte Carlo tracking through flat detector heads or dual flat detector heads, and some
binning options to facilitate the use of PVI reconstruction algorithms.
IThis work was partly supported by PHS grant CA42593.
11997 International Meeting on Fully 3D Image Reconstruction
The flat detectors can consist of multiple layers of different materials. The user specifies the dhnensions
of each layer, whether each layer scintillates or not, the radius of rotation of the detector, the ntunber of
detector positions, and the angular range of detector positions. (This allows the for layers of lead,
copper, or tin, which are often used as graded absorbers to reduce low-energy photon counts when using
dual-headed cameras for coincidence imaging [7].) Two importance sampling features are offered to
itnprove simulation efficiency: photons can be forced to interact at least once in the detector asselnbly;
and the detectofwhead position can be chosen on a per-decay basis to maximize the chance of detection.
For each photon entering the flat detector head simulation, we produce a list giving the location and
energYMdeposited for each interaction in the scintillating layers of the detector. The detector module
calculates the centroid of the interactions and the total energy deposited. A Gaussian blur can be applied
to the deposited energy.
The new binning options Blake it easier to use SimSET data as input to PVI reconstruction algorithlns.
Previously, detected position information for coincidences was binned by radial distance, azimuthal
angle, and the detected axial positions of the two photons. It was left to the user to transform these data
into coordinates appropriate for the reconstluction. The binning module will now create the arrays
needed by the SSRB, MSRB, and 3DRP algorithms automatically. The coordinates are calculated using
tnethods from [8].
Future directions
We expect to implement several other SimSET extensions before the meeting. These include better
modeling of physical processes (including coherent scattering, positron range, and annihilation photon
non-collinearity), improved importance sampling for cone-beam collimation, and binning for the FORE
reconstruction algorithm [9].
References
1.
Muehllehner, G., J.G. Colsher, and R.M. Lewitt, A Hexagonal Bar Positron Camera: Problems
and Solutions. IEEE Trans Nuc Sci, 1983. NS ..30(l): p. 652-660.
2.
Townsend, D., et al. Design and performance of a rotating positron tomograph, RPT-2. in
Nuclear Science Symposium and Medical Imaging Conference. 1993. San Francisco: IEEE.
3. . Nellemann, P., et al. Performance characteristics of a dual head SPECT scanner with PET
capability. in Nuclear Science Symposium and Medical Imaging Conference. 1995. San Francisco:
IEEE.
4.
Miyaoka, R., et al. Coincidence imaging using a standard dual headed gamma camera. in
Nuclear Science Symposium and Medical Imaging Conference. 1996. Anaheim: IEEE.
5.
Harrison, R.L., et aI., Preliminary experience with the photon history generator module of a
public-domain simulation systemfor emission tomography. IEEE NSS-MIC Conf. Rec., 1993.2: p.
1154-1158.
6.
Tsui, B .M.W. and G.r. Gullberg, The geometric transfer function for cone and fan beam
collimators. Phys Med BioI, 1990.35(1): p. 81-93.
.
7.
Muehllehner, G., R. Jaszczak, and R. Beck, The reduction of coincidence loss in radionuclide
imaging cameras through the use of composite filters. Phys Med BioI, 1974.19(4): p. 504-510.
8.
Kinahan, P., Image reconstruction algorithms for volume-imaging PET scanners. PhD thesis,
1994, University of Pennsylvania.
9.
Defrise, M., A factorization methodfor the 3D x-ray transform. Inverse Problems, 1995. 11: p.
983-994.
11997 International Meeting on Fully 3D Image Reconstruction
QUANTITATIVE CHEST SPECT IN THREE DIlVIENSIONS:
Validation by Experimental Phantom Studies
Z. Liang, 1. Li, 1. Ye, 1. Cheng and D. Harrington
/i
I, II
l
Departments of Radiology, Electrical Engineering and Computer Science
State University of New York, Stony Brook, NY 11794
j
EXTENDED ABSTRACT
(1). Introduction
SPECT has been shown to provide useful information on the metabolic and physiologic functions of organs through
the reconstructed images of radiopharmaceutical uptake distributions [C2l. Currently available SPECT protocols support
only qualitative uptake images. Although the diagnosis based on the qualitative images has succeeded in many cases, the
sensitivity and specificity have not yet met our expectation, especially for diagnosis of heart and lung diseases, where the
thoracic heterogeneity and the cardiac and respiratory motion render a very challenging and currently unsolved problem.
[I
It has been well understood that quantitative SPECT will improve sensitivity and specificity of patient diagnosis [310]. A quantitative reconstruction of the uptake image requires a simultaneous compensation for (a) attenuation of primary
photons inside the body, (b) inclusion of scattered photons from the body in the measured data of photopeak-energy window, and (c) variable detector resolution at different depths from the detector, as well as suppression of noise propagation in
the image reconstruction, in addition to corrections of patient motion, detection dead time, and isotope decay.
[j
Many quantitative approaches have been proposed in the past years [3-10]. Some of them addressed the degrading
effects individually, and others addressed the effects simultaneously. The major obstacle in implementing those simultaneous compensation approaches is the extremely heavy computing burden. This work develops, implements, and~validates an
efficient simultaneous compensation approach for quantitative SPECT reconstruction of the ~hest.
(2). Method
This section describes the efficient compensation approach and details its implementation by the following steps. The
new ideas were described by more words and old ones were mentioned briefly.
fAJ. DATA ACQUISITION
[J
Three sets of data are required to reconstruct the quantitative image: (a) the point-source measurements for the
detector-response kernel, (b) the transmission scans for the attenuation map, and (c) the du~l-energy-window emission data.
[I
(A.l). Point Source Measurements:
Point-source measurements are necessary to construct the system-specific spatial resolution kernel for compensation
of the depth-dependent resolution variation. ,These measurements are also necessary to design a collimator-speci fic filter to
remove the unwanted frequency components, which are embedded in the measurements due to the dual samplings of collimator holes and PMTs.
[J
A point source of 37 MBq Tc-99m activity and 1.5 mm radius size was imaged in air at depths of 1, 5, 10, 15, 20, 25,
30,35, 40, and 45 cm, respectively, from a low-energy, high-resolution, parallel-hole collimator by a SPECT system. The
photopeak window was centered at 140 keY with 20% width. The acquired image array at each depth was 1282 on a FOV of
45 cm~ i.e., the pixel size was 3.5 mm. The counts of each image were 10 thousands.
[]
11
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In the Fourier space, we discovered that the frequency distribution of the point-source measurements follows a
specific pattern depending on the collimator geometry, i.e., the hole shape and septum thickness. A filter was then designed
to remove the unwanted frequency components, while preserving the useful information in the low-frequency domain. The
point-spread function (PSF) at each depth was then completely determined by the useful information. These PSFs reflect
approximately a Gaussian function with variance depending on the depth. The collimator-speci fic filtering is very important
to remove the effect of dual samplings, especially for high-energy collimation. The resolution kernel at 160 depths (by 3.5
mm increment from collimator surface) was constructed by interpolation of the PSFs via cubic fitting, so the kernel was an
array of 1282 x160 size.
Since both the kernel and the filter are system specific, they are constructed only once for the chosen radiotracer and
collimator/detector system. They were stored in the computer for all later applications.
(A. 2). Transmission Scans:
Transmission scans are necessary for an object-speci fic attenuation map of each patient, although some ad he
approaches based on segmentation techniques and emission data have been investigated.
11997 International Meeting on Fully 3D Image Reconstruction
A flood source of 1110 MBq Tc-99m activity with the same size of the FOV was used. A chest phantom filled in with
water was placed on the patient bed with marks on both the phantom and bed for later repeating measurements. The chest
phantom consists of three parts: (a) a cylindrical elliptical tank; (b) lung inserts; and (c) a spine insert. A cross section of the
phantom is depicted on the top left of the figure below (without the cardiac insert). A circular orbit of scans was employed
with 128 evenly spaced stops by the same SPECT system. The photopeak window was set at the center of 140 keY with
20% width. At each stop, an acquisition of 10 seconds was taken. The total counts were approximately 28.5 millions.
The data were reconstructed by a FBP method [4] with a lowMpass Butterworth window on the Ramp filter at half
Nyquist frequency cutoff and power factor of 5. A slice of the 3D reconstructed objcct-speci fic attenuation map is shown on
the top second left of the figure. In order to see the effect of different body characteristics on the attenuation compensation,
the chest phantom was modified to simulate a female model. The reconstructed attenuation map is shown on the top second
right of the figure.
(A. 3). Emission Data Acquisition:
Emission data were acquired using the same SPECT system after the phantoms were filled in with radiotracer solutions as specified below. The chest phantom was modified by including a cardiac insert (see the top left of the figure). The
cardiac insert simulates the left ventricle of the heart. The insert consists of two concentric cylinders. The inner cylinder
fonns the ventricular chamber with 8 cm length and 4 cm diameter. The space between the two cylinders fonns the myocardial wall of 1 cm thick, filled in with solution of 278 MBq per cc (referred as 100% concentration). 1\vo defects of 50% and
25% concentration, respectively, were placed inside the myocardial wall. The bullseye display of the myocardial activity is
shown on the top right of the figure. The space inside the inner cylinder (or the ventricular chamber) and the "soft-tissue"
region across the FOV had a tracer solution of 5%. The "lungs" and "spinal bone", as well as the outside regions (Le., the
cylindrical tank walls) had no activity. The above tracer distribution represents a typical extraction fraction of clinical myocardial perfusion studies using TC 99m.
The emission scans had 128 stops evenly spaced on a circle of 20 cm radius. Each scan matrix had a sample size of
128 2 • The photopeak-energy window remained the same (i.e., from 126 to 154 keY). The offMphotopeak or scatter-energy
window ranged from 90 to 126 keY. The scanning time was 14 seconds at each stop. The total counts were approximately
7.1 millions from the photopeak-energy window and 4.3 millions from the scatter-energy window.
a
[BJ.IMAGERECONSTRUCTION
Reconstruction of the acquired emission data for the uptake image requires compensation for all the statistical and
physical degradation effects associated with the photon emission, transportation, and detection. The major ones are
described below.
(E. I). Collimator-Specific Filtering of Dual-Sampled Data
The constructed collimator-speci fic filter from point-source measurements was applied to both the photopeak and
scatterHwindow samples to remove those unwanted frequency components, as discussed before. These components do not
carry useful information, except for the alias and other artificial effects. This step is very important to suppress noise propagation in the foll?wing compensation process.
'
(B. 2). Treatment of Signal-Dependent Poisson Noise:
The Poisson noise embedded in both the dual energy-window samples was treated by the square-root transformation.
Each of the transformed data is nearly signal-independent Gaussian distributed with a constant variance (=0.25). The mean
of each transformed datum was then very satisfactorily estimated by a Wiener filtering approach. The estimated means were
finally square"transformed back to the projection space. These modified projection data are the estimate of the means of the
photopeak and scatterHwindow data.
(B.3). Correction for Isotope Decay:
Following the treatment of Poisson noise, a correction for the isotope decay was applied tdthe modified photopeak
and scatter-window data, given the half life of the isotope and the acquisition time per stop.
(B.4). Estimation of Scattered Contribution:
In order to adequately estimate the scatter contribution to the photopeak measurement, the modified scatter-window
data were further smoothed by a low-pass Hann window at 0.25 Nyquist frequency cutoff[4]. The approach proposed in [5]
assumes a fraction k 0.5 of the scatter-window data to be the scatter contribution; The fraction factor k is a variable
depending on the window size and location and so its application is limited in clinic. Another approach uses the subwin M
dows inside the photopeak and estimate the scatter contribution by fitting the subwindow samples [6,9]. This later approach
is very sensitive to the window settings and, therefore, the results vary when the energy spectrum shifts time-:by-time. We
propose a new approach which takes the advantages of both the photopeak (free of adjustable parameter) and the offphotopeak (easy implementation) approaches. The new approach uses the counts and width of the scatter window to
=
11997lnternational.Meeting on Fully 3D Image Reconstruction
..
.
781
[1
compute the height (their ratio) of a triangle inside the energy spectrum of the photopeak window, where the triangle has the
same width of the photopeak window. By analyzing the SPECT energy spectra of a point source in air and water, the area of
the triangle reflects the scatter contribution to the photopeak window.
The estimated scatter contribution was subtracted from the photopeak data bin-by-bin in the projection space. This
subtraction approximately removes the tails of the photopeak-window data profiles of a small object in attenuating media.
An underlying assumption is made for the scatter subtraction that the tails are the consequence of the scatter. The subtraction also minimizes the background contribution to the photopeak window.
(B.5). Restoration of Detector Resolution:
After the scatter subtraction, the primary-photon contribution to the photopeak window was modeled as a depthdependent convolution of the attenuated source with the detector-resolution kernel. The depth-dependent deconvolution
was then performed via the depth-frequency relation [7].
ri
(B.6). Attenuation Compensation and Image Reconstruction
[-I
After the resolution restoration, the projection data were specified by the attenuated Radon transform [3]. Inversion of
the attenuated Radon transform is generally performed by iterative algorithms due to the activity spread over the nonuniform attenuating media (lungs, liver, myocardium and surrounding soft tissues). The attenuation compensation was included
in the projector and backprojector. One iterative algorithm among those efficient ones is the iterative FBP [8], which was
employed here.
[1
(3). Results
[1
We evaluated the efficient quantitative approach by the following means: (a) compare the quantitative accuracy
between the reconstructions of the efficient compensation approach and the conventional FBP method [4], and (b) study the
performance of the approach for different 'body characteristics, such as the thorax of men and women. Two experiments
were performed.
.
In the first experiment, we used the thorax phantom of men (see the top second left of the figure). The emission phantom is depicted on the top left of that figure. On the top right is the bullseye display of the phantom myocardial activity.
Quantitative analysis on the bullseye representation is a standard means for myocardial perfusion SPECT study. Three ROls
were selected on the bullseye display. The first ROI was selected for defect 1 (25% concentration) on the bottom left of the
bullseye display. The second one was chosen for defect 2 (50% concentration) on the top right. The whole myocardium,
except for the two defects, was selected for the third ROI. In the second experiment, the thorax phantom was modified by
adding two breasts to simulating a female's chest (see the top second right of the figure). The emission configuration
remained the same, as shown on the top left of that figure. So is the bullseye display.
[1
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[1
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A slice of 3D reconstructed images is shown by the middle row of the figure. From left to right are: the conventional
FBP image of the male model, the conventional FBP image of the female model, the quantitative reconstruction of the male
model, and the quantitative reconstruction of the female model. These 4 images show qualitatively the performance of the
reconstructions. By visual inspection, it can be concluded that the conventional FBP result varies for different body models
(left 2 images). The quality in the target areas of lungs and myocardium are poor, as compared to that of the quantitative
reconstructions (right 2 images). The quantitative reconstructions are similar, as expected, for the two different body
models.
The bullseye display of the myocardial activity from the 3D reconstructions in the middle is shown by the bottom row
of the figure. The variation and lower quality of the conventional FBP reconstruction are again observed. The quality
improvement by the quantitative approach is significant. The consistent performance of the quantitative approach for
different body characteristics is again clearly seen. The quantitative ROI measures vary from 12% to 30% for the conventional FBP reconstructions between the male and female models, while the variation for the quantitative reconstructions is
less than 1% between the male and female models.
The reconstruction time of 128 3 image array from 128 projections of 1282 size by the conventional FBP was less than
a minute on an HP1730 desktop computer. The reconstruction by the quantitative approach finished in less than 20 minutes.
Further reduction of computing time can be achieved by optimizing the program code.
(4). Conclusion
An efficient simultaneous compensation approach to quantitative chest reconstruction was described. Its implementation was detailed. Its validation by different body characteristics was documented. Extended studies will be presented.
11997 International Meeting on FUlly 3D Image Reconstruction
(5). References
(1)
(2)
Alavi A, "Perfusion"Ventiiation Lung Scans in the Diagnosis of Pulmonary Thromboembolism/' Applied Rad 6: 182·188 (1977).
Bennan D, Kiat H, ct aI, "Tc-99m Scstamibi in the Assessment of Chronic CAD," Scmin Nucl Med 21: 190"212 (1991).
(3)
Gullbcrg G, The Allellllatet/ Radoll 7hmsform: Theory and applicatioll ill medicille alld biology, (Ph.D. Thesis, Donner Laboraa
tory, University of California, CA, 1979).
(4)
Huesman R, Gullbcrg G, Greenberg W & Budinger T, Donner Algorithms for Recollstructioll Tomography, (Lawrencc Bcrkeley
Laboratory, University of California, Berkeley, 1977).
[5]
Jaszczak R, Greer K, Floyd C, Harris C & Coleman E, "Improved SPECT Quantification Using Compensation for Scattered Photons," JNM 25: 893-900 (1984).
[6]
King M, Hademcl10s G & Glick S, "A Dual·PhotopctLk Window Method for Scatter Correction," JI':M 33: 605·612 (1992).
[7]
Lewitt R, Edholm P & Xia W, "Fourier Method for Correction of Depth"Dcpcndent Collimator Blurring," SPIE Med Imag III
1092: 232"243 (1989).
[8]
Liang Z, "Compensation for Nonuniform Attenuation, Scattcring, and Collimator Rcsponse with an Iterative FBP Method for
SPECT Rcconstruction," Radiology 181: 186 (1991).
[9]
Ogawa K, Ilarata Y, Ichihara T, et aI, "A Practical Method for Position-Dependent Compton-Scatter Correction in SPECT," IEEE
TMI 10: 408-412 (1991).
[10]
Tsui B, Gullbcrg G, ct aI, "Correction of Nonuniform Attenuation in Cardiac SPECT Imaging," JNM 30: 497 507 (1989),
M
11997 International Meeting on Fully 3D Image Reconstruction
801
Submitted to 1997 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and
Nuclear Medicine, June 25-29, 1997, Pittsburgh, Pennsylvania, USA.
3D Reconstruction from Cone-Beam Data using Efficient Fourier Technique
Combined with a Special Interpolation Filter
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Maria Magnusson Seger, Image Processing Laboratory, Dept. of Electrical Engineering,
Linkoping University, S-581 83 Linkoping, SWEDEN, email: [email protected]
Introduction
We here present a method for 3D-reconstruction from cone-beam data. The method has a strong relation
to the LINCON method presented in [Jac96]. LINCON is an exact 3D reconstruction method from conebeam projections. The method is based on Grangeat's result [Gra91] which claims that the derivative of
the Radon transform of the object function can be obtained from the cone-beam projections. Grangeat's
original method has the complexity O(N4), whereas LINCON has lowered the complexity to O(N31 0gN)
by using Fourier techniques. The method suggested here has also O(N310gN)-complexity, but it is even
faster than LINCON. In LINCON, the chirp z-transform is frequently used for computation of Fourier
transforms. Unfortunately, the chirp z-transform is at least a factor of four more computation intensive
than the FFT. Here we basically replace the chirp z-transform with an interpolation step followed by FFT.
By using this we can reduce the computation cost of a factor between 2 or 3. The interpolation filter belongs to a special class of filters and must be carefully designed to keep a good image quality.
The Interpolation Filter
We have developed a new class of filters that are well suited for interpolation. The filter equation is
.f
()
J Mh
x
= sin(nx/1) cos(nx/M1) [h
Ml11 sin (nx/ M1)
+
(1 _
h).cos (21rx)]
MT '
(1)
elsewhere
where 111 is the sample distance i~ the input sample grid, M is the size of the filter measured in sample
points, and h can be used to adjust the shape of the filter. The filter with h=1 is similar to a class of filters
independently developed by [Be195]. For our new algorithm we have chosen a filter of size M=8 and the
parameter h=O.59.
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1~----~------~----~----~
...:.... "I'"
1
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I
1 '\ :
1 \:
1
1
I:
1
1
1
1
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h=0.59
0.5 ......... "..... .
""""""
.... " ... " ..
-
1
. 1
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1
0.5 ....I.... · .... I .. " .:" ... ~" .. " ....I.. ·
h-0.5
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b) 0
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:
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:'
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... ~Qnel" an, ". ~ zone ~
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Fig. 1 Three versions of the fMh (x)-filter, M=8. a) Spatial domain, b) Fourier domain.
Interpolation can be seen as convolution followed by resampling. Convolution with a filterfMh(x) applied
to a function g(x) corresponds to multiplication in the Fourier domain i.e.
(2)
n
where" * " means convolution, 9= means the Fourier transform and 9=-1 means the inverse Fourier
transform. When applying an interpolation filter we normally want no influence of the interpolation function in the Fourier domain. It can be shown that the sinc function is an optimal interpolation filter to use
[1
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11997 International Meeting on Fully 3D Image Reconstruction
811
Submitted to 1997 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and
Nuclear Medicine, June 25-29, 1997, Pittsburgh, Pennsylvania, USA.
if the function is bandMlimited. The Fourier transfonn of the sinc function is a rectangle function. Multiplication of a bandwlimited function with a rectangle function will not change the band-limited function
provided that the rectangle function is broad enough. The disadvantage of using the sinc function as interpolation function is that it has infinite length. The filter family in (1) is designed to have limited length
in the spatial domain but yet be similar to the rectangle function in the Fourier domain, i.e.
•
The filters are designed to be as flat as possible in the pass-band and to have as steep edges as possible.
Other useful properties of the filters are ...
•
The DC-level is equal to one, i.e. FMh(O)::::l, where FMh(X)=g:ffMh(X)].
•
The filters have the value one in the origin,.JMh (0)=1, and pass through zero at the other samplewpoints,
i.e. f(i7)-:::.O, where i is an integer ~ and T is the sample distance.
•
The filters are symmetric and continuous and have continuous derivatives in every point except for
the two points Ixl=MI1112, where only the first derivative is continuous.
•
The filters are easy to design since there is only one parameter h which can be adjusted.
°
When a filter is used in an algorithm, the data points can be arranged so that the pass-band of the filter
coincide with important data points, whereas the edge zone of the filter coincide with an unimportant
"safety zone" of the data. "Safety zones" can be obtained by for example zero-padding of the data. For
explanation of the words "pass-band" and "edge zone", see Fig. 1.
Grangeat's result
In figure Fig. 2 the arrangement of an X-ray cone-beam source S and a detector with dataXf(p,q)=Xfa(s,t)
is shown. The coordinate system (p,q) is fixed on the detector and the coordinate system (s,t) is a rotated
version of (p,q). We denote the rotation angle a.
Fig. 2 XHray source and detector arrangement.
Grangeat showed that the derivative of the Radon transform of the object function can be obtained from
the cone-beam projections [Gra91]. With the notations in Fig. 2 this can be written
00
a (F\
1
a
ap 9bj p~J = cos 2 f3 as
f ISA(s,
ISOl
F
t),Xja(S(P':l)' t) dt
(3)
-00
where 9bjis the Radon transform of the object functionj and gis a 3D unit vector. Xj denotes the X-ray
transform i.e. line integrals ofjtaken along a line starting from the source point S and moving towards
a point (p,q) or (s,t) on the detector. The data Xf(p,q)=Xfa (s,t) is simply the cone-beam projection detector
values. The basic meaning of (3) is that th~ derivative a/ ap of the Radon transformed object can be obtained by differentiating ( a/as) and taking projections (line integrals) on the detector. The equation also
11997 International Meeting on Fully 3D Image Reconstruction
821
Il
I,
Submitted to 1997 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and
Nuclear Medicine, June 25-29, 1997, Pittsburgh, Pennsylvania, USA.
:
)
contains pre-weighting with ISOI/ISA(s, t)1 and post-weighting with
11 cos 2 f3, where ISOI is the distance
11
between the source S and the origin 0 and ISA(s, t)1 is the distance between the source S and the pointA(s,t)
on the detector.
[-1
3D reconstruction phase 1
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The reconstruction proposed by Grangeat is performed in two phases, where the first phase is computation
of the derivative of the Radon transform of the object according to (3). To do this we will utilize the projection slice theorem which is stated as follows.
Theorem: The ID Fourier transform p(R,e) of a parallel projection p(r,e) of an objectfix,y) taken at
angle e is found in the two-dimensional transform F2(X, Y) on a line subtending the angle e with the
(4)
X-axis i.e. P(R, e) = F 2 (R cos e, R sin e).
This means that the projections of X.f(p,q) can be obtained by taking the 2D Fourier transform of X.f(p,q)
and then taking ID inverse Fourier transform along the radial directions. Moreover, instead of differentiating with aI as we can multiply withj27rS in the Fourier domain. In summary, all this gives the computation
scheme shown in Fig. 3. The interpolation in the 2D Fourier domain is performed in one dimension using
fMHO from equation (1), M=8, h=O.S9.
derivative of Radon transfom, ..2....- <3bf
detector data,
X.f(p,q)
q
Q
~,/a
l(a
s
preweighting
p
p
p
.........
ap
Q
zeropadding,
2DFFT
s
\}.J "
multo
withj27rS
ID
interp.
with fMh 0
ID radial
I&1
Fig. 3 Reconstruction Phase I: Computation of the derivative of the Radon transform of the object
function from cone-beam projection detector data.
3D reconstruction phase 2
From phase 1 we now have the derivative of the Radon transform (alap)r:1b! Unfortunately the data is
arranged in a rather complicated pattern [J ac96]. However, interpolation can be performed so that the data
points lies along radial lines in (a I ap )r:1bf(p~).
There exists a similar theorem to (4) for three dimensions, namely
(5)
where ~is a 3D unit vector, ~e denotes the Fourier transform in the radial e-direction, and F3 denotes the
three-dimensional Fourier transform of the object! From (5) it is clear that if we take the one-dimensional
Fourier transform in the radial direction of r:1bfwe get data along radial lines in the 3D Fourier domain of
the object! The object itself can then be obtained by simple 3D inverse Fourier transform. The procedure
is illustrated in Fig. 4. For the first cube in the figure only a third part of the data points are indicated. For
the second and third cube, the inner points are not visible, but the sampling in these two cubes are simply
cubic. An additional important thing must be emphasized regarding Fig. 4. Hopefully it can be imagined
that the sample density is proportional to lIR2 in the 3D Fourier domain. This overloading of data value
contribution in the center remains after interpolation, since the interpolation is performed by so called
gridding, see for example [SuI85]. Each data point in the in-data volume spread its values according to
!l
lJ
11997 International Meeting on Fully 3D Image Reconstruction
831
Submitted to 1997 International Meeting on Fully Three~DimensionalImagc Reconstruction in Radiology and
Nuclear Medicine, June 25-29, 1997, Pittsburgh, Pennsylvania, USA.
the shape of the interpolation filter in the out-data volume. Multiplication must be performed with R2 to
cOlnpensate for the factor lIR2. However, the differentiation of~6jwith a/ ap corresponds to multiplication withj2nR in the Foul'ier domain. Therefore we mUltiply with -jR/2:rc instead of R2 in Fig. 4. The interpolation in the 3D Fourier domain is performed in two dimensions by two one-dimensional steps using
fMlIO from equation (1), M=8, h=O.59.
reconstructed
object,j
3D IFFT
ID
2D interp.
radial
with !Mil 0
FFT
mult. with -jR/2:Jr
Fig. 4 Reconstruction Phase 2: Fronl the derivative of the Radon transform of the object, via 3D inverse Fourier transform, the object is reconstructed.
Complexity Calculation
The LINCON method frequently utilizes the chirp z-transform, whereas for the new method we avoid it
by using interpolation followed by FFf instead. Hereafter follows the computation complexity for the
most demanding operations in both methods. A more careful derivation is given in [Mag93] or [Jac96].
Computation of FFT on N real data points: (5/2)N log N FLOP
(6)
Computation of FFT on N complex data points: 5N log N FLOP
(7)
Computation of the chirp z-transform on N complex data points: > 20N log 2N FLOP
(8)
Interpolation cost for a complex data point: 4n - 2 FLOP
(9)
where n is the filter size and FLOP means "FLoating point OPerations". Here we have not space enough
to do a careful comparison between LINCON and the new method but it can be shown that the new method
is 2-3 times faster than LINCON.
Acknowledgement
The support for this work from CENIIT (Centre for Industrial Information Technology, Linkoping University) and from the Swedish Council for Engineering Sciences, grant No. 95~470 is gratefully acknow
ledged.
N
References
[Jac96]
[Mag93]
[Gra91]
[BeI95]
[SuI85]
C. (Axelsson-)Jacobson,. Fourier Methods in 3D-Reconstructionfrom Cone-Beam Data., Dissertations NoA27, Linkoping studies in Science and Technology, S-58183 Linkoping, Sweden,
April 1996.
M.B.Magnusson. Linogram and Other Direct Fourier Methods for Tomographic Reconstruc~
tion., Dissertations No.320, Linkoping studies in Science and Technology, S-581 83
Linkoping, Sweden, 1993.
P. Grangeat. Mathematical framework of cone beam 3D reconstruction via the first derivative
ofthe Radon transform. Mathematical Methods inTomography, Herman, Louis, Natterer(eds.),
Lecture notes in Mathematics, No. 1497, pp.66-97, Springer Verlag, 1991.
P.L. Bellon, S. Lanzavecchia. A direct Fourier method (DFM) for X-ray tomographic reconstructions and the accurate simulation of sinograms. International Journal of Bio-Medical
Computing 38, pp. 55-69, 1995.
J.D. O'Sullivan. A Fast Sinc Function Gridding Algorithmfor Fourier Inversion in Computer
Tomography. IEEE TransactionsonMedical Imaging, VoC MI..4, NoA, Dec. 1985.
11997 International Meeting on Fully 3D Image Reconstruction
Iterative reconstruction for helical CT: a simulation study.
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J Nuyts, P.Dupont, M.Defrise, P. Suetens, L.Mortelmans
Nuclear Medicine,K.U.Leuven and Vrije Universiteit Brussel
Herestraat 49, B3000 Leuven, Belgiums
[email protected], fax:32-16/34.37.59
Abstract. This study was undertaken to examine the potential of iterative reconstruction algorithms
for helical CT. Simulations have been carried out to compare the performance of two iterative
algorithms (iterated filtered backprojection and maximum likelihood reconstruction) to that of
interpolation followed by filtered backprojection (linear and nearest neighbor interpolation between
projections separated by 180 0 ). The simulated object consisted of two concentric homogeneous
cylinders configured to produce a single sharp gradient in the axial direction. Noise propagation was
controlled by stopping the iterations and/or filtering in the x-y plane. Reconstruction bias versus noise
sensitivity curves were calculated for each algorithm. For this simulation, maximum likelihood
reconstruction produced superior noise-bias curves. Iterated filtered backprojection did not
outperform linear interpolation followed by backprojection.
Introduction In helical CT, only one projection angle is measured for every table position.
Currently, helical CT scans are reconstructed by filtered backprojection (FBP) of synthetic data. A
sinogram for the selected slice is c9mputed from the measured sinogram by interpolating between data
obtained at the same projection angle but at slightly different table positions. This approximation
introduces inconsistencies in the sinogram, resulting in reconstruction artifacts. The severity of the
artifacts depends on the interpolation strategy and on the pitch. The current standard approach is linear
interpolation between projections separated by 1800 (LINI80).
Because of their higher computation times, iterative reconstruction algorithms have not yet received
much attention for this application. However, different from the FBP approach, they can handle a
very accurate model of the acquisition process. In addition, iterative algorithms have been shown to
outperform FBP in some PET and CT applic.ations [1,2]. Ever increasing computer power is bringing
iterative reconstruction for helical CT within reach.
Algorithms. Two iterative algorithms have been implemented and their performance compared to
the current standard LIN180, and to nearest neighbor interpolation between projections separated by
1800 (NNI80). The algorithms have been implemented for a parallel beam configuration because of
the availability of parallel projection / backprojection software. However, adaptation to fan-beam
geometry is straightforward. Because the difference between a fan-beam and parallel beam geometry
vanishes with decreasing object size, the findings from a parallel beam geometry are relevant for a fan
beam configuration.
All four algorithms use a three-dimensional image to represent the current estimate of the
reconstruction. The z-axis is chosen parallel to the table feed direction. The two iterative algorithms
are based upon a forward model of the acquisition process, denoted "spiral projection" (SP). SP is a
straightforward extension of projection in 2D, using linear interpolation in the z-axis to compute the
slice at a particular z-position. SP has been extended to take into account finite slice thickness, by
convolving the image with a slice sensitivity profile prior to projection. The transpose operation will
be denoted "spiral backprojection" (SBP).
'
Notations
J1j
The linear attenuation coefficient at position j, represented by voxel j.
Yi
The measured value at i, where i indicates both the angle of the projection line and its distance
to the center of the field of view. The angle is proportional to the z-position.
bi
The blank scan value at i.
C ij
The normalized contribution of voxel j to the projection line i, with a maximum value of 1.
N
The number of samples (voxels) along the x or the y-axis.
Ly (f.1) The likelihood function of the reconstruction J..L, given the measured sinogram y.
11997 International Meeting on Fully 3D Image Reconstruction
Note that the Inodelling of the slice thickness in SP and SBP is only approxiInate: convolution in the
itnage d01nain is not equivalent to convolution in the sinogrmn d0111ain, since the acquisition is a non-12 ·c"tt·
linear operator: Y i ::::: b i e . J IJ J
N N 1.8 0: Nearest neighbor interpolation FB 12. A sinogrmn for axial position z is constructed by
extracting froln the 111easurecl CT scan the set of projections over 1800 , acquired closest to position z.
This set is reconstructed with FBP.
LIN180:LinearinterpolationFBP A sinogrmn for axial position z is constructed by extracting from
the lueasured CT scan a set of projections over 3600 , acquired closest to z. This sinogrmn is reduced
to a 1800 set by interpolating between opposite projections, linearly weighting each projection
according to its distance fr01n z.
MLTR: Maxitnmll liJ(elihoocl for tl'anslnission t01nogr~nhy. A gradient based InaxiInmn likelihood
(ML) algorithm, originally developed for PET transillission applications [3] was adapted for parallel
bemn helical CT by substitution of the projector and backprojector operators by SP and SBP
respectively. The algorithln can be written as
new _ u ( y .
~
a
-/l~ + Iv
aL),(p) -
p~ - tl~ + fJ (Il)' ap
~
[1 - ~.:.. LiCi~Yi-I/ijJL 1
L ..N ci~bie
c.
J
2
fJ ( )::: _" a LyCtL)
~ /l
~j atL~atLj
a is a relaxation factor, and fl;(IJ.) a normalization factor designed to
obtain convergence when
which was used in the study. Ly is the log-likelihood function for transmission tomography:
Ly {Jl} ==
a
=:
1,
r{
-hie-L jCijllj + Yi In(hie-LjCijllj )-In(Yi!))
This algorithln is based on a Poisson model for the noise. This is a good approximation, since in CT
the noise is dominated by the statistical fluctuations in the number of detected photons [4].
ILIN180: Iterative LINI80. /lnew =tL+ aLIN180(y -SP(p)) .The relaxation factor a was set to 1.
Constraining of axial resolution The finite thickness of the slices (the collimator opening) causes axial
sluoothing of the sinogram. The iterative programs will aut01natically attempt to compensate for this
slnoothing. When no constraints are applied, this results in excessive amplification of high spatial
frequencies along the z .. axis. Both algorithms have been constraint by applying the- metnodof sieves
[5], implelnenting the sieve as a convolution with a Gaussian in the z-direction only. The sieve is
applied prior to projection and after backprojection. The resulting reconstruction is convolved with the
sieve to produce the final image.
Simulation. The simulation was designed to study the performance of the algorithms in the
presence of a z-gradient, because z-gradients produce artifacts in FBP of helical CT scans. The
simulated phantom consisted of an attenuating elliptic cylinder (diameters 36x28 cm) oriented parallel
to the z-axis, with infinite length, with a linear attenuation coefficient of .15 cm- I . This cylinder
contains a smaller, circular cylinder (diameter 12 cm) with a higher attenuation coefficient (.18 cm- 1),
positioned excentrically. The base of that smaller cylinder is 'located at the center of the field of view,
the other edge is outside the axial field of view. Consequently, there is a strong z-gradient in the
center (fig. 1).
The slice thickness is assigned the normalized value of 1. The simulations have been done for a table
feed of 1.8 (table motion for 3600 orbit). The Gaussian sieve had a FWHM of 0.8. Plane distance 'in
the reconstruction was 0.5.
.
11997 International Meetirig on Fully 3D Irriage Reconstruction
861
A single noiseless sinogram and 10 different noise realizations were computed, using a Gaussian
approximation for the Poisson distribution. Reconstruction bias was computed as the mean squared
difference with the original phantom, only considering voxels inside the elliptic cylinder. Noise
sensitivity was computed as the mean squared difference between each of the 10 noisy
reconstructions and the noiseless one. The number of photons per detector in the blank scan was set
to 1e6. A smaller value than in clinical practice (1 e7 [4]) was selected to amplify the noise-effects.
All backprojections were carried out with a ramp filter. Different points in a bias-resolution graph
were obtained by varying the number of iterations, and by applying Gaussian smoothing (standard
deviation of 0 to 2.5 pixels) in the x-y plane for the three methods using FBP (equivalent to varying
the cut-off frequency of the reconstruction filter). Consequently, there are two parameters varied for
ILINI80, and one for the other three algorithms. Up to 20 ILIN180 iterations and up to 50 iterations
MLTR have been computed.
The simulations were carried out for 40 detectors x 58 angles per 180 0 , pixel size = 1cm, and again
for 150 detectors x 150 angles per 180 0 , pixel size = 2.7 mm. The coarser simulation was repeated
with a higher attenuation in the central cylinder (.3 cm- 1).
Results. Fig.2 presents the bias vs noise curves for the simulation with finer sampling. The results
of the coarser simulation were very similar. For the coarser simulation with increased attenuation in
the small cylinder, the relative location of the curves was not changed. However, all curves were
shifted towards higher bias values, with a slightly higher shift for NN180 relative to LIN180.
Fig. 1 shows the three central slices of the LIN180 and MLTR images.
All curves have also been recalculated with a scale factor ~ minimizing the bias:
[I
bias = min
fJ
I'
lJ
o
[:
fi
Ll
[]
11
I
I
LJ
il
L
j
lJ
N2 , where ris the original
~ ~2 (r.J _f3p.)2/
J
£...,;j=l
phantom and p is
the reconstruction. This was done to eliminate the effect of a possible scaling due to implementation.
The relative position of the bias-noise curves remained unchanged. The high bias values decreased,
but for the low bias values the scale differed less than .3 % from unity and the influence on the bias
values was negligible.
Discussion. The aim of this study was only to examine the potential benefit of iterative
reconstruction as compared to standard interpolation followed by filtered backprojection, not to prove
the superiority of one of the algorithms. Because of study limitations, the results must be treated with
care. The software phantom has been designed specifically to mnplify the effects due to gradients in
the z-direction and to noise. The relative importance of noise and possibly also of z-gradients may be
lower in a typical clinical image. Obviously, similar simulations should be carried out with more
complex phantoms. For small objects, the difference between a fan beam and parallel beam geometry
is negligible, but it may not be for typical clinical images. To quantify bias the mean squared
difference was used. This criterion selects for "optimal" absolute quantification. In clinical routine,
however, helical CT images are used for visual inspection. The use of the scale factor minimizing the
bias only alleviates this problem to some extent. The mean squared difference also penalizes small
digitization effects due to fast fourier transform for the ramp filter and to the finite number of
projections. The iterative procedures tend to suppress these effects, resulting in a better bias value,
even in the absence of inconsistencies in the projections. The number of parameters is very large, and
the influence of the relative values of sieve size, slice thickness and table feed remains to be studied.
On the other hand, the simulation results indicate that linear interpolation is superior to nearest
neighbor interpolation to produce the synthetic sino gram, as expected. The fact that the bias-noise
curves stay in the same relative position when the sampling grid or the attenuation values are changed
indicates that the performance differences are significant.
In this study, iterating filtered backprojection did not improve the quality of the bias-noise curve.
However, it allows to extend the curve. LIN180 cannot produce a lower bias than that obtained with a
ramp filter. ILIN180 allows to further decrease the bias, at the cost of increased noise. The problem
of ll.JN180 is probably the difference in axial and transaxial convergence rates. Convergence in the xy plane is very fast because of the similarity with FBP. However, in axial direction ILIN180
resembles more an ML-algorithm (using the transpose instead of the inverse in the reconstruction
step), with typical slow convergence. As a result, noise in the x-y directions is already increasing
considerably while axial convergence is still poor.
11997 International Meeting on Fully 3D Image Reconstruction
871
MLTR has a l110re unifonn (but slow) convergence. In addition, it uses an accurate noise tl1ode!. As a
result, iI11ages with siInilar bias but lower noise are obtained. Sitnilar to ILINI80, MLTR allows the
cOlnputation of hllages with bias values lower than those obtainable with LIN 180. An hnportant
problel11 of MLTR is the slow convergence. Acceleration techniques designed for ML expectation
l11axitnisation in PET and SPECT can with success be applied to MLTR. Although not used in this
study, we have obtained considerable acceleration with ordered subsets [6].
Conclusion The siInulations results indicate that maximum likelihood reconstruction 111ay increase
the quality of the helical CT reconstruction itnages as compared to the linear interpolation followed by
filtered backprojection.
Fig.l. Top row: Three slices of the software phantOln. BottOln left: the corresponding ML
reconstruction. Bottoln right: the corresponding LIN180 reconstluction.
0.0040
~
x
MLTR
<>
ILlN180
*
LlN180
+
NN160
0.0030 .
ttl
:S
0.0020
0.0010 L--_....L-.~-L,....._....1-_-L-_--1-_--L~--'-_---1_---I'---_L...-_L---l
0.0002
0.0004
0.0006
0.0000
Fig.2. Noise vs bias curves for the four algorithms. Iterations 50, 30, 20, 15, 10, 6 forMLTR,
iterations 20, 10 and 5 for ILIN180.
References
[1.] E. Mumcuoglu et aI. IEEE Trans Med Imaging 1994; 13: 687-701
,[2] G. Wang et aI. IEEE Trans Med ImagingJ996; 15: 657-664
[3]· ... J: Nuyts et aI. EUr J Nucl Med 1995; 22:876
[4] H. Guan et al. Phys. Med. BioI 1996; 1727-1743
[5] D. Snyder et aI, IEEE Trans Med Imaging 1987; MI-6: 228-238
[6] H Hudson et aI, IEEE Trans Med Imaging 1994; 13: 601-609
11997lnternati()hal Meeting on Fully 3D Image Reconstruction
881
~l
:1
J
I
I
I
\
t ..
Iterative Reconstruction of Three-Dimensional
Magnetic Resonance Images from Boron Data
J
F. Rannou and J. Gregor
I
L __
Department of Computer Science
University of Tennessee
Knoxville, TN 37996-1301
!
J
1
[J
c
In this paper, we address sampling and numerical aspects pertaining to a 3D image reconstruction
algorithm for use in Boron Neutron Capture Therapy (BNCT) which is a potential technique for
cancer treatment. A patient is first injected with a boron compound designed to specifically be
absorbed by tumor cells and then irradiated with low-energy neutrons. Ideally, the tumor is destroyed
thereby while the surrounding tissue is left intact. Magnetic Resonance Imaging (MRI) is used as a
tool for determining the tissue-selectivity of a particular boron compound and for patient monitoring.
Due to very short relaxation times of the boron MR signal, e.g., B-11 has a T1 of 0.78 msec and
a T2 of 0.65 msec at a field strength of 2.0 T, traditional proton sampling techniques are inadequate
and a spherical scheme must be employed instead [1]. The standard approach for reconstructing
3D images sampled this way is to use filtered back-projection or, alternatively, interpolation onto
a Cartesian grid followed by direct Fourier-inversion. We consider instead using an iterative reconstruction algorithm for the eventual purpose of incorporating prior knowledge to help guide the
computation. Such an approach, however, carries with it a significant computational burden in
terms of extensive memory and CPU time usage.
The work presented here concentrates on how to compute and store the vast system matrix that
arises when modeling the image formation process as a linear system of equations, as is done in
iterative image reconstruction algorithms. In particular, we describe a spherical sampling scheme
and how certain symmetries introduced thereby can be used to substantially reduce the storage
requirements. We also derive the 3D Radon transform of the spherically symmetric Kaiser-Bessel
basis functions to facilitate the computation of the elements of the system matrix. On-going research focuses on the actual reconstruction of boron images constrained by anatomical information
extracted from high-resolution proton images.
2
II
\ I
(J
11
LJ
Introduction
Obtaining projection data
A 3D projection reconstruction method for species with short T2was described in [1]. With reference
to Fig. 1 (a), the method is based on sampling data along spherical trajectories (e,cp) of the Fourier
spectrum where angles e and cp define latitude and longitude positions, respectively. No is preset
and remains fixed. On the other hand, N fjJ is proportional to sin e and is thus reset for each value of
e. The objective is to sample the Fourier spectrum evenly in all directions. To create appropriate
projection readout gradients, all three magnetic field gradients are turned on simultaneously after a
nonselective pulse has been applied. Due to a short T2, sampling is begun even before the gradient
fields have stabilized. The resulting nonuniform radial sampling of the low-frequency components i~
corrected by means of interpolation.
-oU'
We impose the following two constraints on the spherical sampling scheme. First, to obtain
true planar integral projections, the full 3D sampling rays must form straight lines that go through
the origin of the Fourier spectrum as illustrated in Fig. 1 (b). Second, to introduce geometrical
symmetries that will be exploited below, NfjJ must beoa multiple of four for each value of e. The
connection between the sampled projection data, S (e, cp, t), and the planar integral projections of
the object, p(e, cp, r), is established by the projection theorem [2]:
(1)
p((}, ¢, r)
= .1'1 1 {S(e, ¢, t)}
r-'
:L_JI
,
where t and r denote radial sampling distances and
.1'1 1 is the inverse,
11997 International Meeting on Fully 3D Image Reconstruction
ID Fourier transform.
891
z
z
y
x
(a)
(b)
Figure 1. The Fourier spectrum is sampled spherically along rays that go through the
origin. The number of rays and thetr orientation are chosen to cover the spectrum evenly.
3
lInage basis representation
In the finite serieB-oxpansion approach to image reconstruction [3], the image
linear combination of translated versions of a basis function, say 'I/J:
f is assumed to be a
m-l
(2)
f(x)
= L Cj"p(x -
Xj)
j=o
where {Cj} is a set of coefficients and Xj :::: (Xj, Vj, Zj) is the position of a particular instance of the
basis function. Lewitt [4] introduced a family of spherically symmetric Kaiser-Bessel window basis
functions (called blobs) which depend only on the scalar Sj IIx-xjll. Specifically, 'l/J(Sj) £. "p(x-Xj)
where:
=
(3)
and zero otherwise. 1M is the modified Bessel function of the first kind order M, a is a Bcalar that
controls the blob shape, and R is the blob radius.
Lewitt [4] also derived the 3D X-ray transform of a blob. Here, however, we need the 3D Radon
transform, denoted
because we deal with planar rather than line integral projections, i.e.:
n3,
(4)
Since
p(O,¢,r)
:=:
na{f(x)}.
na is a linear transform, we get the following two equations:
m-l
'R.a{f(x)} ::::
(5)
pee, cp, r) ==
(6)
L: c/R.a{"p(Sj)}
m-l
L cja(O'j).
j=O
The 3D Radon transform of "p( Sj) computed along the projection plane defined by view direction
(0, </J) and radial distance r'is a function only of the distance O'j between the projection plane and
the center of the blob. Hence:
(7)
a(O'j)
= a~:~:)
[Jl- (O';/R)·t+1 I M +1
hit - (O'j/R)2)
11997 International Meeting on Fully 3D Image Reconstruction
901
:1
LJ
n
I
I
l
i
where aj = r - Xj sin B cos ¢ + Yj sin 0 sin ¢ + Zj cos B.
In practice, each projection plane has a fixed, nonzero width, say w, which is defined by the
radial sampling rate in the Fourier spectrum. For M = 0, this leads to the volume given by:
27r R3
(8)
r----o
I
rIP
.2
a(aj) = aIo(a) } e II (a sin I) sin '"1 dl
I
l
e
where = cos- 1 ((aj + w/2)/R) and <p = cos- 1 ((aj - w/2)/R). We note that the integration must
be done numerically as the integral does not have a closed form solution.
4
Iterative reconstruction
Let indices i and j refer to a specific projection plane and blob, respectively. Then a discrete model
of the image formation process given by Eq. (6) can be written as:
(9)
Cr
p=Ac
where A E jRmxn is the system matrix whose elements aij are given by Eq. (8), p E jRm is the
projection data vector whose elements Pi are defined by Eq. (1), and c E jRn is the coefficient vector
whose (unknown) elements Cj produce the image through Eq. (2).
The above linear system of equations is solved using the Richardson-Lucy [5, 6] iteration scheme
which is identical in form to the well-known EM-ML algorithm used in emission tomography:
(10)
[j
r~',
L;
[]
[]
j=O ... n-l
where a i denotes the ith row of A and c k denotes the kth estimate of c. Two important properties
are preservation of mass between iterations, i.e., L:i(ai , c k ) = L:i Pi, and nonnegativity in the
solution, i.e., c k ~ O. Also, unlike methods such as the Landweber iteration scheme, storage and/or
computation of A's transpose, which would be prohibitive, is not required.
Matrix A constitutes a significant computational problem. For instance, the boron application
calls for a sampling geometry of 1,645 views each consisting of 64 planar projections, and an image
volume of 64 x 64 x 64 voxels, which results in A having on the order of 28 billion elements. Even when
considering only the approximately 800 million nonzero elements, about 8 Gbytes is required for a
single-precision floating-point implementation; this exceeds the memory capacity of most computers.
In the same spirit as in [7, 8], we therefore introduce geometrical symmetries that reduce the memory
requirements by a factor of eight. This allows us to embed the problem on a small network of regular
workstations.
Recall that N¢ is divisible by four for each value of B. This leads to rotation symmetries as
follows. Let Ao,<p denote the block of rows in A that contain all the aij elements associated with
the spherical trajectory defined by (B, ¢ ). Then
(11)
It
L,J
Figures 2 (a) and (b) illustrate the relation between a (B,¢) view (black circle) and (B,¢ + 7r/2) ,
(B, ¢ + 7r), and (B, ¢ + 37r/2) views (open circles). A memory reduction factor of four is obtained.
We also exploit the following refle~tion symmetry. Let Ao,<p,r denote the subset of elements in
Ao,¢ associated with the particular planar projection for direction (0, ¢) and radial distance r. Then
(12)
This relation is illustrated in Fig. 2 (c). An additional memory reduction factor of two results.
Finally, we note that further memory savings can be achieved by discretizing O'j as this allows
representing each aij by an integer index that points to a look-up table where the actual floatingpoint value is stored.
(1
I j
L~
11997 International Meeting on Fully 3D Image Reconstruction
911
z
; - - - J.....
I
y
I
x
x
(u)
(b)
Figure 2. (a) Rotation symmetry for the halfspace z
(c) Reflextion symmetry.
5
x
~,
-
\
... I.
\
I
+1:
0,
--
- - - -1:
(c)
> 0 and (b) as seen from the z-axis.
Current Status
The algorithm is being implemented in C on a network of ATM-connected Sun Ultra workstations,
each equipped with 250 Mbytes of memory. The MPI standard for parallel and distributed computing is used for the inter-processor communication. MRI data is most kindly being provided by
Dr. C. Tang, Department of Radiology, University of Tennessee Medical Center, Knoxville. Actual
image reconstructions and other experimental results will be available by the time of the meeting.
Acknowledgement
This research was supported in part by the National Science Foundation under grant CDA-95-29459.
References
[1] G.H. Glover, J.M. Pauly, and K. Bradshaw. Boron-II imaging with a three-dimensional reconstruction method. Journal of Magnetic Resonance Imaging, 2:47-52, 1992.
[2] F. Natterer. The Mathematics of Oomputarized Tomography. Wiley & Sons, 1980.
[3] Y. Censor. Finite series-expansion reconstruction methods. Proc. IEEE, 71(3):409-419, 1983.
[4] R.M. Lewitt. Multidimensional digital image representations using generalized Kaiser-Bessel
window functions. Journal of the Optical Society of America. A, 7(10):1834~1840, 1990.
[5]
w.n. Richardson.
Bayesian-based iterative method of image restoration. Journal of the Optical
Society of America, 62(1):55-59, 1972.
[6] L.B. Lucy. An iterative technique for the rectification of observed distributions. The Astronomical
Journal, 79(6):745-765, 1974.
[7] L. Kaufman. Solving emission tomography problems on vector machines. Annals of Operations
Research, 22:325-353, 1990.
[8] C.M. Chen, S.-Y. Lee, and Z.H. Cho. Parallelization of the EM algorithm for 3-D PET image
reconstruction. IEEE Transactions on Medical Imaging, 10(4):513-522, 1991.
11997 International ~eeting on Fully 3D Image Reconstruction
Adaptive Inverse Radon Transformer
r
A. F. Rodriguez
l
W.E. Blass;
J. Missimer
F. Emert
K.L. Leenders t
[]
l]
[J
ri
LJ
lJ
Overview
Artificial Neural Networks (ANN) are massively parallel connected systems that are modeled after
biological neural networks [Sim90J [Rod92J [SZ91J [Kos92J. One of the most important features of
ANN is the ability to learn to perform a given computational task. Supervised Artificial Neural
Networks (SANN) are taught to reproduce a user-provided database. This database consists of
pairs of input and output facts that represent the response of a system. Among SANNs, the
Backpropagation Supervised Artificial Neural Network (BSANN) has proven successful in dealing
with a great variety of problems [Sim90J. They generalize well and are robust in dealing, with noisy
and incomplete data. Learning is incorporated in a matrix memory that relates input and output
through connection weights (i.e., synapses). The memory matrix of a trained ANN is able to
reproduce a given target when its related input is fed to the network. Successfully trained SANNs
could generalize from the training database, and thus these systems are able to solve problems of
the same nature as those included in the training process. Constructing an SANN can be divided
in three stages: building a database which appropriately characterizes the transformation, training
the network with a subset of the database, and testing it with the remainder. Training consists
of repetitively presenting the network with input-output pairs until the weights converge. Testing
involves presenting the network, whose connection weights are fixed in this step, with inputs from
the database not in the original subset, and observing the deviation from the t.argets. If the SANN
is able to reproduce the expected output, then the SANN should be able to deal with problems of
the same nature as those encoded in the database
..
It!
Vile present an investigation of the ability of ANN systems to perform an inverse Radon transform [KB95J. ANN can provide the robustness to deal with the ill-posed problem of finding an
original object from a limited set of projections as well as the speed to reconstruct quickly. In
previous work, researchers have applied ANNs to image reconstruction in SPECT [SSG95J [KB95]
[KB94] [MFBC94]. Our ultimate goal is to develop an ANN that can perform image reconstruction
for 3D PET.
As indicated earlier, ANN processing consists of three stages: database construction, training
and generalization testing. A BSANN with a three layer architecture was used in the experiments
we show below. The number of input Processing Elements (or neurons) matched the number of
elements (LOR) in the sinogram arrays (in Radon space). The number of PE for the output layer
corresponded to the number of pixels in the reconstructed image (in object space). It was our
intention to find the optimal architecture required for learning to invert the Radon transform.
Simulated data were used for the initial stage of experiments. Images of 30 elliptical phantoms
were created. These phantoms had different shapes and densities for each training fact. A total of
• Physics Department, University of Tennessee, Knoxville
fpau! Scherrer Institut, CH-5232 Villigen PSI, Switzerland
11997 International Meeting on Fully 3D Image Reconstruction
931
hidden layer PE
96
144
192
240
288
336
384
432
480
trained rms
0.00199404
0.00199918
0.0-0100782
0.00199927
0.00199-667
0.00198682
0.00199662
0.00198523
0.00197730
test rms
0.00345750
0.00357700
0.00350904
0.00343606
0.00356106
0.0034900'2
0.00358229
0.00343997
0.00354266
iterations
414
328
309
293
278
262
252
237
251
Table 1: Inverse Radon study on elliptical phantoms with BSANN
25 phantoms were used for training, and 5 facts were kept for testing. These 30 phantoms simulated
a 3D scan on a brain-like phantom. The objects (Le., phantoms) were 32X32 discrete functions
(Le., arrays) and we computed their radon transform for 40 elements and 48 angles. The inverse
Radon problem consisted then in mapping 40X48 Radon space arrays to their corresponding 32X32
phantoms in object space. We started out with a maximum structure of 480 PE for the hidden
layer and decreased the size of this layer to 48 elements.
Table 1 shows the rms error for the BSANN for training and testing stages. These errors
measure the ability of the network to obtain and image from its tomogr.aphy. As indicated by the
numbers, the BSANN was able to reconstruct the phantoms from the training set. Also, the test
error shows an acceptable degree of generalization for the BSANN. When comparing the hidden
layer size to that of the input and output layers, we observe that a considerably smaller hidden
layer can perform the inverse Radon transformation. We also investigated on the minimum number
of PE elements for the hidden layer to perform the inverse mapping. Obtaining the minimum (and
thus, most efficient) ANN architecture is important for speeding up the image reconstruction
process. Figure 1 shows the results of the test on the control set for a BSANN with 48 PE for the
hidden layer. As indicated earlier, these sinograms we not known to the BSANN, and the graph
reflects ANN ability to perform as an inverse Radon transformer.
In PET imaging, coincidence counts acquired by the scanner are ordered into sinograms, which
are formally the forward Radon transforms of the planes intersecting the radioactivity distribution
in the object _scanned. The sinogramsare -incomplete sets in two respects~ First, the arrangement
of the detectors in the scanner requires that the plane projections be composed of a discrete number of lines of response, a limitation which introduces artifacts in conventional methods of image
reconstruction [Her79]. Second, the scanner design also introduces gaps and sampling inhomogeneities in the acquired sinogram [PCS86]. The design of the PET scanner is known and can be
used to derive an instrumental response function [BMC95]. This information is difficult, however,
to incorporate in analytical methods of performing the inverse Radon transform, such as filtered
back projection algorithms (FBP) [BS80]. Artifacts, intensive CPU usage and long computing
times are among some of the problems related to current numerical algorithms. When dealing
with filters, the choice of the optimum filter method is not a well defined problem.
Initial experiments indicate that the BSANN promises to be an efflcient method to perform
as an inverse Radon transformer, and therefore to make an important contribut.ion t.o 3D image
reconstruction. ANN could provide an alternative reconstruction technique appropriate for PET
imaging.
We emphasize that ANN are trained off-line. Once an ANN system is trained and tested,
11997 International Meeting on Fully 3D Image Reconstruction
941
\
!
:
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fl
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I
[1
[:
I :
lI
[
]
Figure 1: Testing a BSANN with 48 PE in the hidden layer
its capacity to produce an image from a sinogram is limited only by processor speed since only
simple non-iterative calculations are required. Moreover, migrating from a computer emulation
of an ANN to an integrated circuit is also possible, thus providing additional possibilities for
speeding up image reconstruction. A hardware realization depends on constructing an ANN which
performs the inverse Radon transform independent of the object whose image is to be reconstructed.
Determining the database for training and testing an ANN with this capability is the present focus
of our activity.
References
[J
[I
[J
[BMC95]
W.E. Blass, S.L. Mahan, and Gordon Chin. Convolution connection paradigm neural
network enables linear system theory-based image enhancement. International Journal
oJ Imaging Systems and Technology, 6, 1995.
[BS80]
Harrison
1980.
[Her79]
G.T. Herman, editor. Image Reconstruction from Projections, volume 32 of Topics in
applied physics. Berlin; New York: Springer-Verlag, 1979.
[KB94]
John P. Kerr and Eric B. Bartlett. High-speed reconstruction of spect images with a
tailored piecewise neural network. IEEE, May 1994.
[KB95]
John P. Kerr and Eric B. Bartlett. A statiscally tailored neural network approach to
tomographic image reconstruction. Med.Phys., 22(5):601-610, May 1995.
[Kos92]
Bart Kosko. Neural networks and fuzzy systems. Prent.ice-Hall, 1992.
n.
Barret and William E. ·Swindell. Radiological Imaging. '-'Tiley and Sons,
11997 International Meeting on Fully 3D Image Reconstruction
[MFBC94] Micahel T. Munley, Carey E. Floyd, James E. Bowsher, and R. Edward Coleman. An
artificial neural network approach to qauntitative single photon emission computed
tomographic reconstruction with collimator, attenuation and scatter compensation.
J.led.Phys., 21(12):1889-1899, December 1994.
[PCS86]
Michaei E. Phelps, John C.Mazziotta, and Heinrich R. Schelbert. Posit7'071 emission
tomography and autoradiography: principles and applications f01' the brain and hearl.
New York: Raven Press, 1986.
[Rod92]
Alberto F. Rodriguez. Image restoration using a feedNforward error backpropagation
neural network ensemble. Master's thesis, University of Tennessee, Knoxville, 1992.
[Sim90]
Patrick K. Simpson. Artificial neural systems: foundations, paradigms, apl)/icati01ls
and implementations. New York: Pergamon Press, 1990. Q335.s545.
[SSG95]
T.J. Hebert S. Snjay Gopal. PreQreconstruction restoration of spect projection images
wtih a neural network. IEEE Proceedings in Nuclear Science symposium and AI edical
Imaging conference, 2:1279-1281, Oct 1995.
[SZ91]
Roberto Serra and Gianni Zanarini. Complex Systems and Cognitive Process. SpringerVerlag, 1991.
11997 International Meeting on Fu"y 3D Image Reconstruction
(
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The effect of activity outside the direct FOV on countrate
performance and scatter fraction in the ECAT EXACT3D
TJ. Spinks, M. Miller, D. Bailey, P.M. Bloomfield, T. Jones
MRC Cyclotron Unit, Hammersmith Hospital, London
Introduction
The principal emphasis behind the design of the ECAT EXACT3D (966) 3Donly PET tomograph is the maximisation of sensitivity and hence statistical quality in
image or projection data for a given dose administered to the patient. The large axial
field-of-view (FOV) of 24cm enables, for example, the whole brain plus brain stem to
be imaged. The overall efficiency is about 4 times higher than for a tomograph with
10cm axial FOV (e.g. ECAT 953B) after scatter subtraction. However, statistical
quality is critically dependent on the fraction of random coincidences acquired, and
hence singles rates. Futhermore, increase in singles leads to an increase in deadtime.
The inevitable consequence of the longer axial Fav in the 966 and the requirement for
scanning any part of the body leads to a larger FOV for singles events and thus a
relatively higher randoms fraction. The smaller detector block size relative to earlier
generations gives less deadtime but reduction in the coincidence timing window and
pulse integration time yield no advantage [1]. More fundamental improvements will
only come with a much faster detector. An increase in scattered events also arises due to
activity outside the direct coincidence FOV.
Reduction of administered doses or modification of the tracer input lessens the
problem of random events but there is also scope for introducing additional side
shielding. The effect of such shielding on the fraction of random and scatter events has
been investigated.
r-i
L_
.1
[]
r~'
LJ
Methods
[I
[J
I.
f. J
fl
LJ
Annuli of lead of thickness 8mm and 16mm were used. These reduced the
patient port from 60cm to 35cm diameter. Experience with an EeAT 953B tomograph
has shown that this diameter is compatible with routine scanning of the brain. An
inactive (water-filled; 20cm diameter x 30cm long) phantom was placed in the FOV
with a similar phantom axially adjacent to it. The distance between the edge of the
detectors and that of the active phantom was about 5cm. Total system singles, randoms
and trues rates and deadtime were measured for each thickness of shielding (as well as
no shielding) and for activities up to about 130MBq in the phantom. Comparison was
made with the cold phantom removed from the FOV.
The contribution to scattered events from activity outside the FaV and the
performance of a model-based scatter correction algorithm [2] were tested with the
'Utah' phantom, a 20cm diameter cylinder containing independently fillable internal
cylinders. A I60cc cylinder was inactive (water-filled) and an 80cc cylinder contained
an activity concentration C8F) approximately 6-7 times that of the surrounding
'background'. The precise concentrations were determined from aliquots measured in a
well counter. The phantom was scanned with its end-cylinder alternately active and
inactive (lie), the end-cylinder being placed just outside the FOV. The total activity
within the Fav was about 15MBq and that in the end cylinder varied from 70MBq to
about 1MBq. Images were reconstructed (reprojection algorithm) with and without
scatter correction. Attenuation correction was by measurement with a 137CS point source
[3].
Results
r;
[1
Table 1 summarises the results for 8mm and 16mm shielding. The 8mm
shielding reduces singles, randoms and trues rates, by 650/0, 87% and 92%
respectively. The deadtime (loss correction) factor was reduced, at the maximum
r I.
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11997 International Meeting on Fully 3D Image Reconstruction
971
activities, froln 18% to 5%. The reduction in singles (and thus randOllls) is clue to the
shielding of detectors fro 111 the direct line of sight to the active phantOlll. The large effect
on true coincidences clelTIOl1stl'ates the l11agnitude of prOlllpt scatter events within the
active phantOlTI. The ratios, cOlllpared with no shielding, are fairly constant over the
experhnental activity ranges (up to about 150Mbq), varying by only a few percent.
With an additional 81111n, there is still ahnost a factor of 2 reduction in the rand0111s rate
but the effect on scatters (trues) is negligible.
A proportion of the photons which would not have reached the detectors in the
absence of the cold phantOln are scattered when it is inserted, leading to an increase of
19% in randOlTIS (9% in singles). The scattered (hue) events recorded decrease (by
40%) with the cold phantoll1 inserted due to scattering back out of the FOV.
Table 2 cOlnpares ratios of satnple cps with ROJ pixel cps (lnean over planes
containing each cylinder) for the Utah phantOlTI with and without scatter correction,
without activity outside the FOV. lInage counts in the cold cylinder are close to zero and
the hot/background ratio is about 3% low relative to the smnples. Figures 1 and 2 show
the ratios as a function of activity outside the FOV (in the end cylinder). The scatter
correction gives good results with about 10MBq or less outside the FOV but for higher
activities there is a clear overwcorrection in the cold cylinder. There is a slight downward
trend in the hot/background ratio with out of FOV activity but this is not as clear. Below
the 10MBq level, the Ineasured ratio is some 3-40/0 below the tnle value (similar to
Table 2).
Conclusions
The effect of activity outside the FOV is significantly reduced by 8mm
additional side shielding in terms of the reduction in singles and randoms rates. This
thickness is close to two half-value layers for 511keV photons. It also appears that an
extra 8mm is advantageous but the effect of more shielding would be marginal. The
geometry tested would be appropriate and practical for brain studies but there would be
less scope for additional shielding in body studies. On the other hand, it should be
elnphasised that the experimental set-up was rather extreme and that the activity outside
the FOV would be more distributed in vivo.
The model-based scatter correction performs well without activity outside the
FOV but is increasingly inaccurate as this activity increases. Again, the experimental
conditions were extreme, but the results do point to the, possibly complementary, use
of energy-based scatter correction which has been shown to be advantageous in this
respect [4].
References
[1] Spinks TJ, Bailey DL, Bloomfield PM, Miller M, Murayama H, Jones T, Jones W,
Reed J, Newport D, Casey ME, Nutt R. Performance of a new 3D-only PET scannerthe EXACT3D. IEEE Medical Imaging Conference, Anaheim, 1996.
[2] Watson CC, Newport D, Casey ME. A single scatter simulation technique for
scatter correction in 3D PET. Proc. International Meeting on Fully Three-Dimensional
Image Reconstluction in Radiology and Nuclear Medicine, Kluwer Academic in press,
1996.
W:
[3] Jones
Vaigneur K, Young J, Reed J, Moyers C, Nahmias C. The architectural
impact of single photon transmission measurements on full ring 3-D positron
tomography. IEEE Medical Imaging Conference, San Francisco, 1995, voI.2, pp 10261030.
[4] Grootoonk S, Spinks TJ, Sashin D, Spyrou NM., Jones T. Correction for scatter in
3D brain PET using a dual energy window method. Phys. Med. BioI. vo1.41 (1996)
2757-2774.
11997 International Meeting on Fully 3D 'Image Reconstruction
981
Table 1. Mean % of total system rates relative to no shielding
Shield thickness
(cm)
[]
singles
randoms
trues, (scatters)
0
100
100
100
8
35
13
8
16
27
8
7
Table 2.
phantom
g
Comparison of ratios between compartments in the Utah
no activity outside FOV and no additional side-shielding
,
[I
cold cyl./background
[]
hot cyl.lbackground
a
6.22
uncorrected image
0.16
5.18
scatter corrected image
0.003
6.05
samples
"
[]
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11997 International Meeting on Fully 3D Image Reconstruction
991
I
Figure 1: Ratio of hot cylinder/background
0.4r-------~--~--~~~~~~~-1------~--~~~~O~O~O~O~~
0
00
0.3 I-
o
0.2 f0.1
.
0
0 0
o
0
0
0
0
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o
-
uncorrected
0 0 0 0
0 0
-
~
o
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o. *..*.
o
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0
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•
0
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*0
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* *
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-0.11-
••
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0
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measured
0
•••••
0
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corrected
-
* * *
*
-0.2 "-
-
**
-0.3 0
10
•
I
10
1
MBq in end cylinder (outside FOV)
Figure 2: Ratio of hot cylinder/background
8r-------~--~--~--T--~i~i~i~,~I~------~-*~~i--~~~~i~i~~~
............ *' .... *.....* ............ ,....... ,.....................
71-
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0
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1
102
MBq in end cylinder (outside FOV)
11997 International Meeting on FUlly3D Image Reconstr~ction
1001
Binning List Mode Dual Head Coincidence Data into Parallel Projections
WL Swan Costa * , RS Miyaoka, SD Vannoy, RL Harrison, TK Lewellen, F Jansen t
[]
GE Medical Systems t
Milwaukee, WI 53201
Imaging Research Laboratory
University of Washington
Seattle, W A 98195
Introduction
[j
[I
[]
[]
An acquisition system has been developed at the University of Washington to extract position and
energy signals from the GE Maxxus dual headed SPECT system for positron coincidence imaging (DRCI)
[1]. Lead filters (1 mm thick) have been constructed for the Maxxus to reduce the flux of low-energy
photons in DRCI; these are also described in [1].
The system geometry is given in Figure l(a). Complete sampling is attained by rotating the opposing
camera heads by a view angle a (we use 30 view angles between 0 and 180 degrees for head-sized objects
and 45 view angles for torso-sized object~). The coincidence data are collected for each view angle in list
mode and then binned into parallel projections for reconstruction using the 3D reprojection algorithm
(3DRP) [2]. The parallel projection coordinate system is shown in Figure 1(b). Before the list mode data
are binned, they are corrected for isotope decay, pulse pile-up, camera dead-time, spatial and energy signal
distortions, and sampling nonuniformities due to camera rotation and the incident angle between 511 ke V
photons and the detector heads. These corrections are discussed below. In addition, the transformation
from the LOR detection locations to the parallel projection coordinates is described.
[]
[J
[I
[]
y
+y
[I
(b)
(a)
··1
[_J
Figure 1. Coordinate System for UW Maxxus DHCI. (a) Camera position relative to reconstructed image coordinates.
The W x L rectangular detector heads are separated by a variable distance s. They rotate about the z-axis in the clockwise
direction when viewed by an observer facing the gantry. (b) Parallel projection coordinates. The parallel projection plane
(u,v) is orthogonal to the projection direction w(cp,9), where 4> is the azimuthal angle from x to y and 9 is the elevation angle
from the x-y plane to the z-axis. The transformation between (x,y,z) and (u,v,w) may be found in [2].
[J
* Correspondence to:
Wendy Swan Costa
Box 356004
University of Washington
Seattle, WA 98195
tel: (206)548-4386
email: [email protected]
This work was partly supported by GEMS contract and PHS grant CA42593.
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11997 International Meeti~g on Fully 3D Image Reconstruction
1011
Binning Corrections
ViewuAngle Weights
Lhnited count rate capabilities of DHCI systems necessitate lengthy acquisition protocols to acquire
enough events for hnaging. Because this is on the order of isotope decay, the device must operate in
varying hnaging environments between the first and last view angle; corrections based on the view angle
are applied to compensate for this. An approximate decay correction is performed during acquisition by
adjusting the aInount of time spent at each view angle; however, there is a slight difference in the requested
and actual acquisition tilne at each angle. The events from each view angle are weighted by the ratio of
requested to actual acquisition time. In addition, a dead time correction is computed for each view angle
based on the number of events acquired by the cmnel'a vs. the number of events recorded by a scaler.
Finally, view angles are weighted to compensate for data rejected because of pulse pileup. The pileup
factor for a given view angle is equal to the total nUlnber of events above the lower energy threshold
divided by the total number of events detected in the energy window.
Spatial. Distortion and Energy Corrections
Because our system extracts the position and energy signals from the GE Maxxus before they are
processed by the Starcam electronics [1], these signals require separate position dependent energy and
spatial corrections. A table of energy scale factors as a function of (x,y) positions for each camera head
was computed by aligning the energy spectra from a flood source. This table is interpolated to provide a
scale factor to correct the acquired energy signals for non-uniformity in photon collection across the
detector. The spatial distortion correction is applied in the form of tables of x and y adjustments to the
detection locations for each camera head. These tables were generated using fitted line data acquired from
a slit phantom. In addition, because of sensitivity vaxiations at the edge of the detectors, the detector field
of view is limited in software.
Rotational Weighting
Although rotation of the camera heads enables us to collect complete data, the field of view is not
uniformly sampled. Rotational weighting was previously described by Clack et al [3]. We weight LOR on
an event-by-event basis by the inverse of the number of camera head positions at which they could be
detected. A LOR binned to parallel projection coordinates (u, v, <1>, 9) can be detected by the heads at view
angle ex if:
lui
~ W sin <l>a - ~ Icos<l>o:l and
2
u ctn<l>a sin e -
2
(s2 SIn~itllL coseJ : ; v=::; u ctn<l>a sine + (s ~irjell- L coseJ, where
<l>a 2
2 SIn <l>cx 2
<1>0:
=
<I> -
ex.
Incident Angle Nor111alization
The large area detectors used for DHeI lead to a wide range of angles between incident photons and
the detector faces (up to 44 degrees for the maximum head separation of the Maxxus). The path length
through the lead filters and the NaI crystal for oblique incidence is sufficiently different from that for
normal incidence to merit investigating a correction for different detection probabilities as a function of the
incident angle~ Assuming true events, the photon incident angles can be computed· from the detection
locations. We applied a correction inversely proportional to·the detection probability for the assumed
incident photon angles for each LOR. The detection probabilities corresponding to our lead filters and
crystal thickness were obtained based on a curve fit to simulated data. No correction was applied to
account for detection location variation as a function of incident angle.
We used a recent extension to the SimSET package to simulate 511 keV photons impinging on an
infinite-extent flat layered detector at specified angles. The layered detector consisted of a lead filter and a
9.5 mm thick layer of NaI, where the lead filter was comprised of a 1 rom thick layer .of lead backed by
1.25 mm of Sn and 0.25 mm of Cu, sandwiched between two 1.6 mm thick sheets of AI. We simulated 2.4
million photons incident on the detector for each angle ranging from 44 to 90 degrees in 2 degree
increments and then binned the photons according to the energy they deposited in the crystal. A gaussian
blur of 11.5% at 511 keY was applied to the energy deposited in the crystal before binning. The curve fit
11997 International Meeting on Fully 3D Image Reconstruction
1021
[-1
for detection probability (Le., the fraction of photons in a 450-575 ke V window) vs. incident 511 keV
photon angle is shown in Figure 2.
0.11
% Detected = a + bx
where x = cos (Incident Angle)
0.105
[I
13
0.1
0
0.095
B
t)
~
0.09
[:
+ cx2 ,
0.085 L-I-L....J.....1-J-l-L..LL..J.....LI..-.LL.I....l....J....JL-I-L....l-L....I-L.J.....LL..J.....LI..-J......LL...J.....1-JL..L..L...LJ
1.05
0.65 0.7 0.75 0.8 0d·85 An0.91 ) 0.95
costInC1 ent g e
Value
Error
a
0.17362
0.0041711
b
-0.112
0.0097187
c 0.026876
0.0056016
Chisq
20.728
NA
R
0.99951
NA
lJ
Figure 2. Simlated Detection Probabilities for 511 keY photons incident on UW DHCI lead fIlters and 9.5 mm thick NaI
crystal. An energy window of 450 - 575 keV and 11.5% energy resolution were assumed.
n
Coordinate Transformation
The coordinate transformation from global (x,y,z) coordinates of two points along a LOR to parallel
projection coordinates has been described previously (e.g., [2]). For the DRCI position signals (xl, yl, x2,
y2) at view angle a, showin in Figure 1, that transformation is:
[_-1J
~ = arctan(
[]
[J
s ) + 0,;
x2 - xl
e= arctan(
y2 - yl
~(x2 - xl)2 + s2
J;
u=
-s(xl + x2)
.
2~(x2 - xl)2 + s2 '
s2
(x2-xl)(x2 yl-xl y2)+-(y2-yl)
v-
2
~(x2 - xl)2 + s2 ~(x2 - xl)2 + (y2 - yl)2 + s2 .
Results and Discussion
[1
--I
[J
II
[J
Images of a 3D Hoffman brain phantom were reconstructed using 3DRP with different combinations of
the binning corrections applied to demonstrate the effects of the various corrections. Approximately 3.3 M
events were binned into the parallel projection set. The transaxial and axial angle bins were both mashed
by a factor of approximately 2. A 9 nun FWHM trans axial by 8 nun axial filter was used in the
reconstruction. An energy window of 450 - 575 keY was applied to the data in all cases. No correction
was applied for object attenuation. The binning corrections applied for each case are shown in Table 1.
Transaxial and coronal slices of the images are shown in Figure 3, displayed with no windowing applied.
Difference images are shown in Figure 4. Quantitatively, applying rotational weights resulted in an
average difference of 40% in the images; adding spatial and energy corrections effected a 6% average
difference over the images with rotational weighting (mostly in the form of slight shifts in the images); and
incident angle normalization produced a 12% average difference over the spatial and energy corrected
images with rotational weighting. Rotational weighting as described above as opposed to rotational
weighting in the u-direction produces approximately 10% average difference (not shown).
Though each correction appears to lend an improvement to the contrast in the images, it is difficult to
evaluate this improvement in the absence of attenuation correction, as rotational weighting and incident
angle normalization both tend to downweight LOR passing through the FOV center, which are affected
most by attenuation. It should be noted that although these data sets were reconstructed with the 3DRP
algorithm, the corrections should be applied to the events before binning into other 3D formats as well.
[]
r:
11997 International Meeting on Fully 3D Image Reconstruction
1031
Table 1. Binning Corrections Applied to Example Images.
--
tillage Label
NO_RW
RW
SE
IA
,
Softwai'e
Detector FOV
yes
yes
yes
yes
View Angle
Weighting
yes
yes
yes
yes
Rotational
Weighting
no
yes
yes
yes
Spatial and
Energy
Corrections
no
no
yes
yes
Incident Angle
Nonnalizatiol1
no
no
no
yes
I~.
..;~.:~:,
<
r .,' .' .\, ':.,
(NO_RW)
(RW)
(SE)
Figure 2. 3D Hoffman Phantom Images Binned with Corrections Listed in Table 1.
(IA)
(RW .. NO_RW)
(SE .. RW)
-(IA .. SE)
Figure 3. Differences Images for Binning Corrections Listed in Table 1.
References
1.
RS Miyaoka, W.C., TK Lewellen, SK Kohlmyer, MS Kaplan, F Jansen, CW Stearns. Coincidence Imaging Using a
Standard Dual Head Gamma Camera. in mEE Nuclear Science Symposium and Medical Imaging Conference. 1996. Los
Angeles, CA:
2.
Kinahan, P., Image Reconstruction Algorithms for Volume-Imaging PET Scanners, University of Pennsylvania, 1994.
3.
R Clack, D.T., A Jeavons, Increased Sensitivity and Field of View for a Rotating Positron Camera. Physics in Medicine
and Biology, 1984.29(11): p. 1421-1431.
11997 International Meeting on Fully 3D Image Reconstruction
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Characteristics of an Iterative Reconstruction Based Method for
Compensation of Spatial Variant Collimator-Detector Response in SPECT
B.M.W. Tsui and E.C. Frey
Department of Biomedical Engineering and Department of Radiology
The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599
1-
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INTRODUCTION
A major image degrading factor for single photon emission computed tomography (SPECT) is
the spatially variant collimator-detector response which causes a loss of spatial resolution and image
distortion. There are two general approaches to the problem of compensation for collimator-detector
response in SPECT. The first approach analytically solves the reconstruction problem that includes the
spatial collimator-detector response function. A good approximation to the collimator-detector response
function is a Gaussian-shaped function (Figure 1) and the dependence of spatial resolution on source
distance can be approximated by a linear function (Figure 2). However, the use of the Gaussian'
approximation to the collimator-detector response function results in a complex analytical expression
that is difficult to implement [Hsieh 1976; van Elmbt 1993]. An approximation based on Cauchy
functions results in a much simpler analytical solution [Soares 1994; Pan 1996]. However, the shape of
the Cauchy function is quite different from that of the collimator-detector response function (Figure 1).
Recently, a modification was made to the Gaussian approximation of the collimator-detector
response based on the assumption that the change of the spatial resolution with distance, O"p is small
compared with the average spatial resolution, 0"0' i.e., O",v « 0"0' where v is the spatial distance. The
modification resulted in a simple closed form solution of the reconstruction problem [Pan 1997], The
change in spatial resolution of a typical low energy general purpose (LEGP) collimator as a function of
distance is shown in Figure 2. The method estimates components of the Fourier series of the ideal
sinogram and the reconstruction image is obtained by the conventional filtered backprojection (FBP)
algorithm with an appropriate smoothing function. An interesting result from this work is the prediction
of a critical frequency [Pan 1997]
vc =
1-4pao a 1
16n2 a o2 a?1
(1)
beyond which no information can be recovered in the reconstructed image. In Eqn. (1), !l is the linear
attenuation coefficient of the medium. Also, it was shown [Pan 1977] that the problem has four
solutions. Linear combinations of these solutions yield reconstructed images with different noise
characteristics. This analytical approach is effective in compensating for uniform attenuation and the
spatially variant collimator-detector response as long as the approximations used are met.
An alternative approach is based on iterative reconstruction techniques which achieve
compensation by incorporating an accurate model of the collimator-detector response and attenuation in
the projector-backprojector [Tsui 1988]. This approach does not require approximation of the collimatordetector response function. It has been found to be effective at recovering spatial resolution loss in 3D
SPECT due to the collimator-detector blur [Tsui 1994; Tsui 1996] and has been extended to include
compensation for the more complex spatially variant and asymmetric scatter response function [Frey
1993] that is difficult to analyze through analytic means. The major disadvantage of this approach has
been its intensive computational requirement. Also, ring artifacts were found around edges of structures
in the reconstructed images. The reconstructed spatial resolution remains asymmetric with the rate of
resolution recovery being unequal along the radial, tangential and longitudinal directions [Tsui 1996].
Recent advance in fast iterative reconstruction algorithms and efficient implementation methods have
made it possible to use this approach in practice. Further investigations of the iterative reconstruction
LJ
11997 International Meeting on Fully 3D Image Reconstruction
1051
based collimator-detector compensation method is warranted. The purpose of this study is to investigate
the characteristics of an iterative reconstluction based method for collimator-detector compensation in
SPECT and to compare them to that found in the analytical approach.
METJIODS
In our study t we used the fast Ordered Subset (OS) expectation maximization (EM) algorithm
[Hudson 1994] whose convergence rate can be as high as 30 times that of the conventional maximum
likelihood (ML) EM algorithm. A pixel-driven projector-backprojector pail' and a rotating matrix scheme
for fast sitnulation of the collimator-detector response at planes parallel to the detector face were used in
the investigation.
As shown in Figure 3, the phantom used in the study consists of a circularly-shaped background
with uniform attenuation and radioactivity distribution. A point source was placed at the center 01' at a
distance from the center of the phantOlu. The activity of the point source is 0.1 with respect to that of the
0
background. Projection data from 128 views over 360 around the phantom using a LEGP and a low
energy high resolution (LEHR) using different pixel sizes were simulated. The projection images were
binned into 64x64 matrices with 0.3 cm pixel size for the LEGP collimator, and into 64x64 matrices with
both 0.3 Cll1 and 0.15 em pixel sizes for the LEHR collimator. Two-dimensional (2D) and threediulensional (3D) reconstructed images were obtained using the OS-EM algorithm with models of the
spatially variant collimator-detector responses for up to 500 iterations. Profiles through the point source
in the reconstructed images and their Fourier transforms or frequency spectra were obtained for analysis.
RESULTS
Figure 4 shows profiles through the point source in the reconstructed images obtained from the
LEGP and LEHR data using the filtered backprojection (FBP) algorithm without any compensation and
the OSHEM algorithm with collimator-detector response compensation after 500 iterations. The profiles
show the improved resolution of the iteratively reconstlucted image. Also shown is the ringing in the
tails of the profiles that resulted from the iterative reconstructions. This ringing in the tails of the
reconstlucted response function may lead to the ringing artifacts at the edges of structures in the
reconstructed images.
Figure 5 shows the Fourier transforms or the frequency spectra of the LEGP profile data from
the OS-EM reconstructed images for selected iterations up to 500 iterations. The critical frequency, v C' in
Eqn. (1) and the Nyquist frequency, v N' are indicated in the figure. The results demonstrate that, similar
to the analytical approach [Pan 1997], the iterative compensation method boosts the mid-frequency
response with a similar cut-off frequency. At higher iterations t the shape of the frequency spectra
approachs that of a rectangular function that in the spatial domain, gives rise to the ringing in the tail of
the response function shown in Figure 4 and possibly ringing artifacts at edges of reconstructed image
structures. Note that for the 0.3 cm pixel size used, the Nyquist frequency, v N' is higher than the cut-off
frequency, v c.
In Figure 6 t we show similar results as in Figure 5 except that data from the LEHR collimator
were used. Here, the Nyquist frequency, vN' is lower than the cut-off frequency, v e ' predicted by Eqn. (1)
and becomes the limit of the resolution recovery. When the data from the LEHR collimator with the
smaller 0.15 cm pixel were used, the cut-off frequency, vel again becomes the limiting factor in
resolution recovery as shown in Figure 7. However, the cut-off frequency of the iterative method appears
to be lower than that predicted by Eqn. (1) for the higher resolution collimator-detector.
Figure 8 (a) and (b) show the Fourier transforms or the frequency spectra of the radial and
tangentHll' profiles through the point source in the OS-EM reconstructed images of the LEHR data at
selected iterations. The results indicate the asymmetric' resolution recovery in the two orientations
continues into higher iterations.
t
11997 International Meeting on Fully 3D Image Reconstruction
1061
n
[]
[]
[1
-I
J
[1
[
IJ
[]
[I
[]
I]
CONCLUSIONS
We investigated the characteristics of an iterative reconstruction method for compensation of the
spatially variant collimator-detector response in SPECT and compared them to that found in an analytical
approach. Using simulated data from a point source phantom, we studied the frequency spectra of the
iteratively reconstructed response functions. Similar to that predicted by the analytical approach, a
critical frequency occurs beyond which no information can be recovered. The critical frequency depends
on the spatial resolution of the collimator-detector at the center-of-rotation, the change of spatial
resolution as a function of source distance, and the linear attenuation coefficient of the object. As
iteration progresses, the shape of the frequency spectrum approaches that of a rectangular function as the
midfrequency range of the spectrum is boosted. A result of the functional change is the appearance of
ringing in the tails of the reconstructed spatial response function and in the edges of the reconstructed
image.
We found that for the high resolution collimator-detector, the critical frequency found in the
iterative reconstruction approach is lower than that predicted by the analytic approach. When a larger
pixel size is used, the Nyquist frequency, which is lower than the critical frequency, becomes the limiting
factor in the resolution recovery. Also, we found that as the spatial resolution improves, the asymmetry
of the 3D spatial resolution function persists up to a high number of iterations.
We conclude that compensation of spatially variant collimator-detector response in SPECT is
possible. However, the degree of resolution recovery depends on the spatial resolution characteristics of
the collimator-detector and the pixel size used. Similar characteristics of the resolution recovery are
found in both the analytical and iterative reconstruction based approaches. An important area of future
research is minimizing the ringing artifacts in the reconstructed images while simultaneously
maximizing the resolution recovery of the method. Possibilities include the design of smoothing filters
in the analytical approach and limiting the number of iterations, and postfiltering techniques in the
iterative approach. Other areas of investigation are the asymmetric properties of the reconstructed
response and comparison of the noise characteristics of both the analytical and iterative approaches.
REFERENCES
E.C. Frey, Z.W. Ju and B.M.W. Tsui, "A fast projector-backprojector pair modeling the asymmetric, spatially varying
scatter response function for scatter compensation in SPECT imaging," IEEE Trans Nucl Sci, vol. NS-40(4), pp. 11921197,1993.
R.C. Hsieh and W.G. Wee, "On methods of three-dimensional reconstruction from a set of radioisotope scintigrams,"
IEEE Trans Syst Man Cybern, vol. SMC-6, pp. 854-862, 1976.
H.M. Hudson and R.S. Larkin, "Accelerated image reconstruction using ordered subsets of projection data," IEEE Trans
Med 1m, vol. 13, pp. 601-609, 1994.
X. Pan, C.E. Metz and C.T. Chen CT, "A class of analytical methods that compensate for attenuation and spatially-variant
resolution in 2D SPECT," IEEE Trans Nucl Sci, vol. 43, pp. 2244-2254,1996.
X. Pan and C.E. Metz, "Non-iterative methods and their noise characteristics in 2D SPECT image reconstruction,"
Manuscript submitted to the IEEE Trans Nucl Sci, vol. 43, 1997.
EJ. Soares, C.L. Byrne, SJ. Glick, C.R. Appledorn and M.A. King MA "Implemention and evaluation of an analytical
solution to the photon attenuation and non-stationary resolution resolution reconstruction problem in SPECT," IEEE
Trans Nucl Sci, vol. 40, pp. 1231-1237, 1993.
B.M.W. Tsui, H.B. Hu, D.R. Gilland, and G.T. GuIIberg, "Implementation of Simultaneous Attenuation and Detector
Response Correction in SPECT," IEEE Trans Nucl Sci, vol. NS-35(1), pp.778-783, 1988.
B.M.W. Tsui BMW, E.C. Frey, X.D. Zhao, D.S. Lalush, R.E. Johnston and W.H. McCartney, "The importance and
implementation of accurate three-dimensional compensation methods for quantitative SPECT," Phys Med Bioi, vol.
39(3), pp. 509-530, 1994.
B.M.W. Tsui BMW, X.D. Zhao, E.C. Frey and G.T. Gullberg, "Characteristics of reconstructed point response in threedimensional spatially variant detector response comepnstion in SPECT," In Three-Dimensional Image Recontruction in
Radiology and Nuclear Medicine, P. Grangeat and J-L Amans, Eds. (Kluwer Academic Publishers), pp. 149-162, 1996.
L. Van Elmbt and S. Walrand, "Simultaneous correction of attenuation and distance-dependent resolution in SPECT: an
analytical approach", Phys Med Bioi, vol. 38, pp. 1207-1217, 1993.
f
II
11997 International Meeting on Fully 3D Image Reconstruction
1071
~LE'GP'-'--l-'
1.2
,
I
I
1.8
-t---t---/--+--II---+--t--+--+
- - ·Gausslon. f'W'-'MM1.25cm
1 •• ·...... ·Cnuchy. /1-004
LEGP Collimator
1.6
~
.~t])
!
0.0"
E 0.0·'
~
~
.!9
&!
1.2
4:
0,4-'
0.2"
62 pixels
0.8
0'--"
·3
·2
Q)
2.5
"0
~
c
I
(a)
~-"--f----I--f-.-+--t-~+---+-I
5
em
Figure 1. Comparison between the response
functiOIi of a LEGP coilimator and a cnmera
with intrinsic resolution of 4 mm, and a
fitted Gaussian function und a fitted Cauchy
function indicating the goodness of fit of
the fitting functions.
3
10
IS
20
distance from detector (cm)
(b)
25
Figure 2. The change in spatial resolution
of the collimator-detector shown in Figure
1 as a function of source distance and a
lineal' fit of the variation.
Figure 3. (a) The phantom used in the study
consists of a circularly-shaped background with a
diameter of 62 pixel. A point source is placed at
the center or at a distance from the center. (b)
Profile through the centel' of the phantom
showing a ratio of 1 to 10 between the activity
levels of the point source and the background.
12+--+--r-~-4--~--+--r--+
# of Iterations
I
- - - LEGP 100 lIor
"--LEGPFBP
- - LEHR 100 Iter
""" ... LeHR FBP
Q)
~c:::
2
~
0>
~
1.5
.~
1
>
~
~ 0.5
0::
1U
:to
1.4-
~
QJ
~
..... 1
10
• -- _.5
....... 20
......... 100
8
--50
6
03 4
2
04-----..~
wO.5 +=-=F~I-~~-t----=-+---t=--+
o
16
32
48
64
pixel
Figure 4. Profiles through the point
source in the reconstructed images of the
phantom shown in Fig. 3. The images are
obtained from the LEGP and LEHR data
using the FBP algorithm and 100
iterations of the OS~EM algorithm with
collimator~detector response compen
sation. The profiles show the improved
resolution with the iterative method. Also,
the ringing in the tails of the profiles may
cause ringing artifacts at the edges of
structures in the reconstructed images.
a
-1
0
1
_.,
0
1
Frequency (cm· 1 )
2
Frequency (cm' 1 )
Figure 5. Fourier transforms orfrequency spectra
of the profile data from the OSwEM reconstruction
of the LEGP projection data for selected iterations
up to 500 iterations. The critical frequency Vc as
predicted by Eqn. (1) and the Nyquist frequency VN
are indicated. The results demonstrate that
boosting of the mid~frequency response with a cutoff frequency beyond which no resolution recovery
occurs. At higher iterations, the frequency spectra
are closer to a rectangularfunction that, in the
spatial domain, gives rise to the ringing in the tail
of the response function shown in Figure 4.
2
Figure 6. Similar results as in Figure 5
except data from the LEHR collimator are
shown. Here, the Nyquist frequency, V N• is
lower than the cut-off frequency, Vc.
predicted by Eqn. (1) and becomes the
limit of the resolution recovery.
12+--+--F-~-+--~~-~--+
~
# of Iterations
10
•... '50
w---'100
·······200
••• .. ····500
::J
'2 8
g>
E
Q)
~
03
0:
~ 10
~ 10
8
'2 8
..ec::
g>
----~1000
~
.t:!:
6
:J
4
1U
03 4
2
2
0::
O+-~~~~~~~~~-+
w4
-2
0
2
Frequency (cm·1 )
4
Figure 7. Similar results as in Figure 6 except
that a smaller 0.15 em pixel size was used.
Here the cutwoff frequency, Ve, becomes the
limiting factor in the resolution recovery.
~
# of Iterations
· .... 100
----·200
.. ••• .. 300
·········400
--500
6
~>
.... ·100
6
- - - - ·200
·······300
.. ·······400
--500
~
03 4
0::
2
o
~2
-1
0
Frequency (cm'
(a)
1
1
)
2
O~-+--~~~----~~~~+
-2
-1
0
1
Frequency (cm· 1 )
(b)
2
Figure 8. Fourier transforms or thefrequency spectra of the (a) radial and (b) tangential
profiles through the point source in the OSwEM reconstructed images of the LEHR data at
selected iterations. The results indicate that the asymmetric resolution in the two
orientations continues into higher iterations.
11997 International Meeting'on Fully 3D Image Reconstruction
1081
flj
t
n
AN EXACT 3D RECONSTRUCTION ALGORITHM FOR BRAIN SPECT
USING A PARALLEL-PLUS COLLIMATOR
CHUNWU "VU
Positron Corporation, 16350 Park Ten Place 1 Houston, TX 77084
Abstract
[]
[]
[]
[1
[j
n
[1
[]
[]
A type of parallel-plus (P+) collimators [1] has been
designed to increase the sensitivities of SPECT systems
in brain SPECT studies. The collimator contains a
parallel-hole portion that fully covers the imaging field
of view (FOV) and four parallel-hole portions that slant
toward the FOV and obtain additional data. An exact
3D reconstruction algorithm has been developed for this
P+ SPECT system. The algorithm is similar to the
reprojection algorithm of 3D PET [2] and contains the
following steps: 1) Consolidate the data into two sets of
3D x-ray transforms, one parallel to and the other oblique
to the transaxial plane. 2) Use the parallel data set to
reconstruct a set of 2D images. 3) forward project the
images to fill the missing data on the oblique data. 4)
reconstruct a 3D image by 3D filtered-backprojection.
The difference between this algorithm and the 3D PET
algorithm are: 1) Instead of many sets of x-ray transforms
with continuously varying polar angles in 3D PET, there
are only two sets of x-ray transforms with two largely.
separated polar angles in P+ SPECT, thus, the filter
derived for 3D reconstruction is different. 2) Because
the difference between the polar angle of the oblique
x-ray transform and 7r /2 is large, the discontinuouty in
the reconstruction filter can produce artifacts in the
reconstructed images. The effects of this filter and the
modification of the filter tored.uce artifacts are presented.
The results demonstrate that by using P+ SPECT and
the exact 3D reconstruction algorithm in brain SPECT
studies, we can achieve a sensitivity more than three
times higher than that of parallel-beam SPECT and keep
artifacts to less than 0.5% of the background level.
I. INTRODUCTION
[]
[j
n
lJ
Cone-beam collimators have been investigated to
increase sensitivities for brain and heart SPECT studies,
where the imaged. object is small compared to the gamma
camera [3], [4], [5]. However, when traversing a circular
orbit, cone-beam SPEeT cannot acquire a complete data
set for accurate analytic reconstruction, and thus can
produce image artifacts. We have proposed parallel-plus
(P+) collimators for brain and heart SPECT studies [1],
[6]. We also developed a 3D reconstruction algorithm for
P+ SPECT and showed. that by using P+ SPECT and
the algorithm, we can obtain higher sensitivities than that
of cone-beam SPECT and produce fewer artifacts [6].
The major short-coming of that algorithm is that it is not
based on an exact relationship between the P+ SPECT
data and the imaged object. Here, we develop an exact
3D reconstruction algorithm for SPECT systems that use
a P+ collimator specifically designed. for brain studies.
II.
METHODS
A. A parallel-plus collimator for brain SPECT
Using computer simulation, we studied a P+ collimator
for brain SPEeT studies. The simulation uses a 50 cm
by 40 cm rectangle camera with a 2-cm thick collimator.
The simulation assumes a 20-cm diameter by 20-cm high
cylindrical field of view (FOV). We set the center of the
FOV as the origin of a Cartesian coordinate system and
the z-axis parallel to the cylinder and pointing to the
top of the head. As shown in Figure 1, the collimator is
divided into two upper portions and three lower portions
(Figure Ia). The camera is tilted 30° toward the top of
the head so that the camera can clear the shoulder and be
positioned. close to the head. In the side view (Figure 1b),
the angle between the beams of the lower portions and
the z-axis is 90°, and the angle between the beams of the
upper portions and the z-axis is 30°. In the top view of
two upper portions (Figure Ic), the angles between the
beams of the two portions and the collimator are ±45°.
In the top view of three lower portions (Figure Id), the
angle between the beams of the central portion and the
collimator is 90°, and the angles between the beams of the
two outside portions and the collimator are ±45°. The
camera is rotated arround the z-axis. The distance from
the z-axis to the detector face behind the P + collimator
is 23.9 cm, 1B.1 cm, and 12.3 cm for trans axial planes at
z = -10 cm, z = 0, and z = 10 cm, respectively.
B. The 3D filter
The 3D x-ray transform maps a function 1(51) into a
set of its line integrals [7]. A line integral of 1(if) can be
expressed as
where ({, S) parameterizes a line through s along the unit
vector
The notion E (-1. indicates that is restricted
to the projection plane, P, that goes through the origin
and has ( as its normal unit vector, i.e., s· ( = O. A
coordinate system in the projection plane can be defined
11997 International Meeting on Fully 3D Image Reconstruction
t
s
s
1091
by two orthogonal un~t vectors a and bthat satisfy a· b:::
0,
== 0, b· ( == 0, and a z :;:;;; O. Then, '8 can bJ
expressed as li ::: Baa + Bbb with Ba == 8' and Bb == '8 --.b.
We will call the set of line integrals having an identical ~ a
3D x-ray transform. Note that a 3D x-ray transform can
be uniquely represented by its projection plane P.
The 3D Fourier transform of !(x) is expressed as
a· (
a
F(v)
= ir
o:s 7r/2 + e.
Thus, the filter for 00
.. -) =-sm
IVai. e
H(~,'V
and the filter for 00
.
H(~,f!) =
dx!(x) exp( -i21rv· it)
(2)
(6)
H
4
,
= 90° x-ray transforms is
-
{~7r/2 e ~ () $. 1f/2 +
4
l'Val
2
R3
where if is a Fourier space vector, and R3 denotes the
integration is over the entire 3D space. The 2D Fourier
transform of p((, 8) with regard to 8 is expressed as
= e x-ray transforms
is
8,
(7)
elsewhere.
C. The implementation of the algorith1n
The algorithm is implemented by the following steps:
1) Consolidation: a) Project the data on each l~ortion
of the collimator, having the same unit vector ~, into
its projection plane P. b) On each projection plane,
sum the data from different portions and if> angles.
c) Normalize the data on each projection by dividing
where v:::: vaa + Vb'S is a Fourier space vector restricted on the value on each position by the number of times
the projection plane 'P, and f.i denotes the integration is that the SPEeT data are projected into this position.
on the plane. The 3D central slice theorem relates the two After that, we obtain two sets of x-ray transforms with
Fourier transfo~ms as [8]
(0 :;:: 90°,ifJ E (0,27r») and (0 = 39.23°,ifJ E (0,271'»,
respectively. The set with 00 :;:: 90° is completely
p((, valt + vbb) == F(vaa + vbb)
(4) measured, and the set with 00 :;:: 39.23° is partially
measured. 2) 2D FBP reconstruction: Use the set of
It means that the 2D Fourier transform of p((, S) is equal x~ray transforms with 0 ::::: 90° }o reconstruct a set of 2D
0
to the data of the 3D Fourier transform of f (x) on the images. The filter used is H(e, if) ::;: w(va)lval/2, where
pro jection plane P.
w(va ) ::;; (0.5 + 0.5 cos(7I'Ival/vc) is the Hamming apodizing
We express the unit vector by spherical coordinates, window with cutoff at Nyquist frequency, Vc. 3) Forward
(0, l/J), as
(sin 0 cos ifJ, sin 0 sin <p, cos 0).
The projection: Forward projection the reconstructed 2D
data acquired by the P+ SPECT system can be· images to fill the unmeasured portions of the set of x··ray
consolidated into two sets of 3D x-ray transforms with transforms with 0 :;:: 39.23°. 4) 3D FBP reconstruction:
0
(0
90°,ifJ E (0,21r)) and (0 = 39.23°,<p E (0,211)), Use the filters in Equations (6) and (7) to reconstruct
respectively. For a set of 3D x-ray transforms wIth a 3D image from the two sets of 3D x ray transforms.
(0 == 00 , l/J E (0,21f)), by using Equation (4), we obtain The same Hamming apodizing window as in 2D FBP was
Fourier data on a set of projection planes that have normal used.
unit vector ( ::;:: (sin 00 cos ifJ, sin 00 sin <p, cos ( 0 ), ifJ E (0,211').
When these planar Fourier data are distributed to and D. Reduction of image artifacts
summed over the 3D Fourier space, they nonuniformly
The straightforward implementation of the 3D FBP
fill the region 11' /2 - eo ~ 0 S 11' /2 + 00 in Fourier
reconstruction utilitying the filter in Equations (6) and
space, but leave the two conic spaces () < 11'/2 - 00 and
(7) cart produce image artifacts, as shown ift -Figure 2b,
o > 11'/2 + 00 empty. The Fourier space filter for the set
because the filter of Equation (7) changes abruptly from
of x-ray transforms is the function that makes the above
Ival/2 to Iv a l/4. To reduce the artifacts, we modified the
Fourier data have equal weights in the nonzero region of
filter so that it changes gradually. If we define a :;:: 107r /2 - 00 :s; 0 ::; 7r /2 + 00 • By calculating ~h~ de~sity of t.he
7r /21, the new filter can be written as
above Fourier data, we obtain the multlplIcatlve Founer
space filter (derivation is not presented because of page
IVai sin E>
a :s; 'l1
limits)
H((, v) = /41
(8)
{
sinE>(1 _ e-t('l-;;9)2) 'l1 < Q:S; E>
... ...
IVai. 0
(5)
H(e,v) = -sm o·
2
for 00 = 8' x-ray transforms, and
When 00 = 90°, the two empty conic spaces vanish; the
IVai
Q ::; \lI
Fourier space is completely filled. Equation (5) reduces
e-
=
8
v;
to H({, v) =Jvi + v~/2, the familiar ra~p filter of 2D
filtered-backprojection (FBP) reconstructlOn. Here? we
have two sets of 3D x-ray tra:nsforms with 00 :;:: 90° and
.oo =8 (8 = 39.23°). They share the region 1f /2 - 8 ::;
11997 International Meeting on Fully 3D .Image Reconstruction
4
H((, v)=
IVal(}
4
IVai
+ e-t{Q;9)2)
'l1 < a ::; 8
a> E>
2
1101
(9)
rI
n
I;
[J
[J
o
[J
[J
[]
n
U
rl
lJ
r
I
III.
COMPUTER SIMULATIONS
A Shepp phantom and a five-disk phantom were used
in the simulation. The parameters for the Shepp phantom
can be found in [9]. The five-disk phantom consists of five
20-cm diameter and 2-cm thick disks, and the spacing
between adjacent disks is 2 cm. SPECT projection data
were simulated by calculating line integrals through the
phantoms. The simulation does not consider physical
factors such as attenuation, scatter, and spatially variant
detector response. A SPECT having one 50 cm x 40
cm camera was simulated, and the camera collected
projection data on a 320 x 256 matrix with 1.56 mm pixel
size. The system used a parallel-beam (PB) collimater
and a P + collimator (described in Section II A) and
took 120 projection data equally spaced over 360°. The
simulation has the following three characteristics: 1) To
reduce the discretization error in the simulation, each
pixel is represented by a 8 x 8 equally-spaced point
array, and the average of their line integrals is assumed
to be the pixel value. 2) To assess the sensitivity more
realistically, the data in each portion of the P + collimator
are multiplied by cos <p to account for the fact' that the
packing densitity is reduced by cos <p for slanted holes,
where r.p is the angle between the hole direction and the
normal vector of the camera. 3) To account for the 2 cm
thickness of the collimator, the data in the shadow of
the P+ collimator are set to zero. After consolidation,
we obtain two sets of 3D x-ray transforms, each has 120
x-ray transforms in a 128 x 128 matrix with 1.56 mm
pLxel size. Images were reconstructed in 128 x 128 x 128
arrays with 1.56 mm voxel size.
IV.
RESULTS AND DISCUSSION
Sagittal slices y=-24.22 mm through the reconstructed
images of the Shepp phantom and their two profiles
through the point (x=0.78 mm, z=-60.16 mm) are shown
in Figure 2. The images were reconstructed from PB
data (2a) and from P+ data by the straightforward (2b)
and modified (2c) 3D FBP algorithms. Diamond-shaped
artifacts are visible in (2b) because of the abrupt change
in the Fourier space filter. These artifacts are reduced to
less than 0.5% of the background level by the modified
algorithm (2c). Sagittal slices y=0.78 mm through the
reconstructed images of the five-disk phantom and their
two profiles through the point (x=-80.47 mm, z=-80.47
mm) are shown in Figure 3. Here, the three images show
the same high quality, because the magnitude of artifacts
in straightforward 3D FBP is only a few percent of the
background, but the image contrast is 100%.
The relative sensitivities of PB and P + SPECT for the
two phantoms are shown in Table 1.
Table 1. Relative sensitivities
Phantom PB
P+
Shepp
1.0
3.97
Disk
1.0
3.51
On a SUN SPARC 20 workstation, the reconstruction
times for PB and P+ SPECT are listed in Table 2.
Table 2. Reconstruction times
Steps
PB
Consolidation
2D FBP
57
Forward projection
3D FBP
Total
57
V.
(seconds)
P+
59
57
228
237
581
CONCLUSION
A P + collimator was designed for brain SPECT studies
using large rectangular SPECT cameras. An exact 3D
FBP algorithm was developed for the P+ SPECT. The
results show that by using the P + collimator and the
3D reconstruction algorithm, we can achieve a sensitivity
more than three times higher than that of conventional
PB SPECT and keep artifacts to less than 0.5% of the
background level, with clinically acceptable reconstruction
times of less than 10 minutes.
VI.
REFERENCES
[1} C. Wu, D.L. Gunter: and C.-T. Chen "Parallel-plus
collimator for SPECT and its reconstruction," J. Nucl.
Med., vol. 35, pp. 33-34, 1994.
[2] P.E. Kinahan and J .G. Rogers "Analytic 3D image
reconstruction using all detected events," IEEE Trans.
Nucl. Sci., vol. NS-36, pp. 964-968, 1989.
[3] R.J. Jaszczak, C.E. Floyd, S.H. Manglos, K.L. Greer,
and R.E. Coleman "Cone beam collimation for SPECT:
analysis, simulation, and image reconstruction using filtered
backprojection," Med. Phys., vol. 13, pp. 484-489, 1986.
[4] G.T. Gullberg, G.L. Zeng, F.L. Datz, P.E. Christian,
C.H.Tung, and H.T. Morgan "Review of convergent
beam tomography in single photon emission computed
tomography," Phys. Med. Bioi., vol. 37, pp. 507-534, 1992.
[5] P. Grangeat "Mathematical framework of cone beam
3D reconstruction via the first derivative of the Radon
transform," in Mathematical Methods in Tomography,
(Lecture Notes in Mathematics), G.T. Herman, A.K. Louis,
F. Natterer, Eds. New York: Springer, 1991, pp. 66-97.
[6] C. Wu "Fully three-dimensional reconstruction in PET and
SPECT by the use of three-dimensional Radon transforms,"
PhD Dissertation, University of Chicago, Chicago, 1994.
[7] M. Defrise, R. Clack, and D. Townsend "Solution to the
three-dimensional image reconstruction problem from twodimensional parallel projections," J. Opt. Soc. A m. A, vol.
10, pp. 869-877, 1993.
[8] F. Natterer The Mathematics of Computerized Tomography,
New York: Wiley, 1986.
[9] M. Defrise and R. Clack "Filtered backprojection
reconstruction of combined parallel beam and· cone beam
SPECT data," Phys. Med. Bioi., vol. 40, pp. 1517-1537,
1995.
I
,
l_JI
I
for fJ o = 90° x-ray transforms. Here we chose IlJ = 0 and
a = 9.06°. Note that the filter is still an exact filter, only
the weights between the two set of x-ray transforms are
changed from equal weighting to variable weighting.
11997 International Meeting on Fully 3D Image Reconstruction
1111
I
J
-1W
- - -.-. - (c)
--
1--- _ _
---_____
xx
(d)
Fig. 1 An illustration of tho P+ collimator for brain SPE~C'I'. a) Front view, the collimator is divided into two upper portions
and three lower portions. b) Side view. In this view, the angle between the camera and the z-axis is 30°, the angle between the
beams of the lower portions and the z~axis is 90°, and the angle between the beams of the upper portions and the z-axis is 30 0 •
c) 'rop view of two upper portions. In this view, the angles between the beams of the two portions and the collimator arc ±45°.
d) Top view of three lower portiolls. In this view, the angle between the beams of the central portion and the collimator is 90° I
and the angles between the beams of the two side portions and the collimator are ±45°.
(n)
(b)
(c)
Fig. 2 Sagittal slices y=~24.22 mm through the reconstructed images of the Shepp phantom and their two profiles through the
point (x=0.78 mm, z=-60.16 mm). The images were reconstructed from PB data (a) and from P+ data by the straightforward
(b) and modified (c) 3D FBP algorithms. Note the grey scale is [1.005, 1.04] for (a) and (c), but it is [1.005, 1.10] for (b).
(a)
(b)
(c)
Fig. 3 Sagittal slices y=0.78 mm through the reconstructed images of the five-disk phantom and their two profiles through the
point (x=-80,47 mm, z=-80.47 mm). The images were reconstructed from PB data (a) and from P+ data by the straightforward
(b) and modified (c) 3D FBP algorithms. The grey scale is [0, 1.2] for all images.
11997 International Meeting on Fully 3D Image Reconstruction
1121
On Combination of Cone-Beam and Fan-Beam Projections in
Solving a Linear System of Equations
Grant T. Gullberg and G. Larry Zeng
Department of Radiology, University of Utah, Salt Lake City, UT 84132, USA
[]
[]
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[]
[J
[J
D
[j
[1
[J
[]
[1
Ij
Background
The idea of using cone-beam and parallel-beam collimators simultaneously on a multi-detector
SPECT scanner was first proposed by J aszczak et al. in 1991 at the first Fully 3D Image Reconstruction
Meeting [1]. Two years later, Gullberg and Zeng suggested using two cone-beam and one fan-beam
collimators on a three-detector system [2].
Combining cone-beam and fan-beam
collimation with simultaneous transmissionemission imaging on a three-detector SPECT
system offers an advantage over current threedetector SPECT systems with fan-beam
collimators and two-detector (orthogonally
oriented) SPECT systems with parallel-beam
collimators. The fan-beam collimator is secured
to the simultaneous transmission -emission
detector, and cone-beam collimators are secured
to the other two detectors, which acquire
emission data only (see Fig. 1). An important
aspect of this arrangement is that it solves the
data insufficiency problem of planar orbit, conebeam tomography. The emission data from the
Figure 1. A simultaneous transmissionfan-beam collimated detector can be used to fill
emiss~on three-detector SPECT system.
in the data missing from the cone-beam
projections. It has been suggested that an
iterative transmission ML-EM (fan-beam) algorithm be used to reconstruct the transmission data, and
an iterative emission ML-EM algorithm be used to reconstruct the emission data with both cone-beam
and fan-beam data [3]. If attenuation correction is not required, an analytical algorithm is also available
to reconstruct combined cone-beam and fanjbeam projections [3].
'I
Goals
When cone-beam projections and' fan-beam projections are combined, a proper weighting can
improve the combined linear system. This paper uses the condition number as a criterion to investigate
the optimal weighting between cone-beam and fan-beam projections. When imaging equations are
formed, the coefficients contain model mismatch errors. The condition number indicates how sensitive
the coefficients are to perturbations. Due to the large number of linear equations, the usual singular
value decomposition (SVD) technique cannot be used to find the singular values. An iterative Lanczos
method is employed to estimate the maximum and minimum singular values.
11997 International Meeting on Fully 3D Image Reconstruction
1131
Imaging Equations
Let the cone-beatn ancI fan-beatn iInaging equations be
Fc X ::::: Pc andFfX = Pf'
(1)
respectively. I-Iere X is a vector representation of the image array, and Pc and Pj are arrays of projection
data for the cone-beatn and fan-beatn geometries, respectively. COlnbining these two sets of equations
yields
(2)
A least-squares solution can be obtained frorn
(3)
If weighting is introduced between cone-beam and fan-beam equations, the least-squares problem (3)
becomes a weighted least-squares problem:
(4)
where [( 1 -
B)FlFe + BFJFf]
is a symmetric positive or semi-positive definite square matrix.
The Perturbation of Linear Systems
LetA::: (l-B)FJFc+BFJFf andB = (1-B)FJp c +BFJp j .ThenEq.(4)becomes
AX::::: B
(5)
'
where A E !l( nXn , BE !l( n ,and X E !l(n . A perturbed system can b
e wntten
as
(6)
~
where A
E !l(
nxn
,
~
J:j
E !l(
n
n.
,and Xe E !l( . It IS known that
I/XeII X"
- XII $; K(A) (IIA/I
IIBII)
2
cITAfi + cITBIT + O(c )
where the condition number K(A) is defined as K(A) =
(7)
IIAII/lA-1/1. If Lz-norm is used, then
(8)
where 0'1 (A) and O'n(A) are the maximum and minimuI? singular values of matrix A, respectively.
Equation (7) implies that in order to obtain a stable solution one needs (i) a small condition number,
11997 International Meeting on Fully 3D Image Reconstruction
1141
il
K(A) ; (ii) a large IIAII ; and (iii) a large
IIBII .
An Illustrative Example
The theory supporting this research is that the condition number of a linear combination of two
matrices can be smaller or larger than the condition numbers for either one of the matrices. To illustrate
this point, consider a 2x2 matrix (( 1 - P)FJFe + pFJFf) , where
[]
[]
11
tJ
[I
FJF c = [1 0.21 and FJF f = [2 o.02l
0.2 "; 9
0.02 0.2
J
(9)
J
The condition number for FJF c is 9.05022, and the condition number for FJF f is 10.01223. For the
equally weighted (i.e., P = 0.5) combination of these two matrices, (FJ F c + FJFf)' the condition
number is improved to 3.07726. However, if one chooses P = 0.82, the weighted combination of these
matrices, (0.18FJF c + 0.82FJFf)' gives an almost perfect condition number of 1.06344. The ideal
condition number is unity. The curve of condition number versus parameter Pis shown in Fig. 2.
11.0
9.0
~
I-
Q.)
E 7.0
:::J
C
c
a 5.0
+=>
:a
c
a
u
3.0
1.0 0 .0
0.2
0.4 parameter ~
Figure 2. Condition number versus parameter
defined in Eq. (9).
[J
[j
[J
[J
W
[J
u
P for
0.6
0.8
matrix
1.0
, as
This is convincing evidence that when combining two sets of equations, a proper weighting can
greatly reduce the overall system sensitivity to perturbations. Caution: For some other examples, the
F c and FJ F f .
condition number curves can reach above the values of the condition numbers of
FJ
Consideration of Total Projection Counts
Equation (7) suggests that a large norm of B = (1 - P)FJPc + pFJPf is desired. Here Pc and
Pf are cone-beam and fan-beam projection data, respectively. If the total projection counts for the conebeam data are more than those for the fan-beam data, the maximum IIBII implies p ~ 0 , that is, conebeam data should be heavily weighted.
A Realistic Study
An imaging system with cone-beam and fan-beam collimators was considered. The detector size
was 65 x 65, the focal length was 84 pixels, and the number of views was, 64 for both imaging
geometries. The image, voxel size was the same as the detector pixel size. The reconstruction region
11997 International Meeting on Fully 3D Image Reconstruction
1151
was an ellipsoid with sel11i-axes of 30, 30, and 22.5
84669 voxels. The axis of rotation was the z-axis.
The condition 11ll1nber for the cone-beam (~ ==
nUlnber for the fan-beanl (~ ::::: 1) system FJFf
(~ ::;: 5.0) of these two syste111S yielded a condition
number curve versus parmneter ~.
in x, y, and z directions, respectively, containing
0) systenl FJ Fe was 767407, and the condition
was 54258. The equally weighted cOlnbination
number of 615082. Figure 3 shows the condition
1000000.0
~
800000.0
bo
Q)
.0
E
::J
600000.0
C
c:
.Q
......
15
400000.0
c:::
0
0
200000.0
.2
0.4 parameter I-'A 0:6
0.8
.0
Figure 3. Condition number versus parameter B for matrix
«1 _ ~)FTF + ~FTF ) for a
.
b
f
e
e
f f
·
reaIIStlC cone- emn "an-beam system.
From this curve it can be observed that the combined system can have a condition number worse
(Le., larger) than those for both of the original systenls if the combination coefficients are not properly
chosen. The minimum point on this curve is at ~. = 1, suggesting that the fan-beam itself yields a
system with the trunlmum condition number. However, the requirement of a large
(1 - B)FJP c + ~FJpf suggests B = 0 if the cone-bearn data have more counts. Trading-off between
a smaller condition number K(A) and a larger norm IIBII yields a value of ~ between and 1.
°
Conclusions
Combining cone-bean and fan-beam collimators in a SPECT system produces high detection
sensitivity (due to cone-beam collimation) and complete measurements (due to fan-beam collimation).
One must be very careful when choosing the combination coefficients, otherwise the combined system
can be more unstable than either of the original systems, and will result in reconstruction artifacts.
Equation (7) shows a means to trade-off between a smaller condition number and a large norm
(measurement statistics). Previous iterative ML-EM reconstructio11s [3] suggest that a value of Bclose
to unity can be chosen.
References
[1] R. J. Jaszczak, J. Li, H. Wang, and R. E. Coleman, "Three-dimensional SPECT reconstruction of combined
cone beam and parallel beam data," Phys. Med. BioI., vol. 37, pp. 535-548,1992.
[2] G. T. Gullberg and G. L. Zeng, "Three-dimensional SPECT reconstruction of combined cone-beam and fanbeam data acquired using a three-detector SPECT system," in Proceedings of the 1995 International Meeting
on Fully Three-Dimensional Imaging Reconstruction in Radiology and Nuclear Medicine, Aix-Ies-Bains, pp.
329-331.
[3] G. L. Zeng and G. T. Gullberg, "Three-dimensional SPECT reconstruction of combined cone-beam and fanbeam data acquired using a three-detector SPECT system," submitted to Phys.Med. Biol. 1996.
11997 International Meeting on Fully 3D Image Reconstruction
1161
n
l
Circular and Circle-and-Line Orbits
for Conebeam X-ray Microtomography of Vascular Networks
!
Roger H. Johnson 1•3,4, Hui Hu 2 , Steven T. Haworth3 ,
Christopher A. Dawson 3,4, and John H. Linehan l ,3,4,
IMarquette University, 2GE Medical Systems, 3Zablocki VA Medical Center,
and 4Medical College of Wisconsin, M.ilwaukee, Wisconsin
[]
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INTRODUCTION:
Since vascular diseases constitute the most serious health problem in western society, there is an
intense interest in methods to image the vascular tree. Conventional angiographic studies provide
planar views of the contrast-enhanced vasculature, typically of the heart or brain. Recently,
methods for 3D cardiac imaging have been explored using either biplane angiograms (Wahle,
1996) or a larger number of views (Saint-Felix, 1994). Our goal is to develop volumetric imaging
methods for basic vascular research.
MATERIALS AND METHODS:
Dynamic 3D imaging with today's technology must be performed with a very small number of
projections. Such limited data sets generally contain information adequate for reconstruction of
vessel medial axes or, at best, synthesis of a binary representation of the vascular network. While
many studies require dynamic information, and can benefit from these binary images, in some
applications it is desireable to make reliable measurements of, for example, vessel diameters, or to
visualize intralumjnal manifestations of disease such as plaque. For some such studies, lowcontrast, high-resolution imaging is the most suitable of currently-available methods, even though
the temporal dimension must be sacrificed. We report on static volumetric imaging of the
pulmonary vascular tree.
We acquire transmitted x-ray projections from a dedicated microangiography system using a
microfocal x-ray source, four-axis specimen micromanipulator, and an image intensifier detector
equipped with a video CCD camera. The source, which is of the demountable (turbo-pumped),
solid-anode variety, is operable over the 5- to 100-kVp and 10- to several hundred micro amp
ranges and produces focal spots as small as three by four microns. The computer-controlled
specimen manipulator repeatability is one micron in translation and 0.0010 in rotation. The 9-, 7-,
5-inch image intensifier is dual-optically coupled to a· room-temperature CCD· camera which
outputs a standard RS-170 (640 by 480 pixels) video signal. Image data may be acquired to
SVHS tape for dynamic studies, but we utilize direct frame-averaging digitization to collect
projection data for conebeam reconstruction.
Because of its modest computational requirements and relative ease of implementation, Feldkamp's
algorithm (FDK) has been the most widely implemented method for 3D conebeam reconstruction
from transmitted x-ray projections (Feldkamp, 1984). An object fer) is reconstructed from its
projections, Pq,(Y,Z) , by first convolving the weighted projection data in the Y (horizontal)
direction with the Shepp-Logan or other filter kernel, h, of choice:
J
Pc, (Y,Z) = dY' .,fd2 +
~2 +Z2 P<t>(Y' ,Z)h(Y -
Y') ,
then backprojecting the filtered data from every angle:
_
feo (r)
lJ
1
d2
=-4
2 fd<I> (d
- '. i
1C
+r'x
Pc, (Y",Zo) _ df.j'
_ d,
Yo- d + r .x' 'ZO-d+r'x'
•
We have used the FDK algorithm to reconstruct image volumes from microangiograms of excised
dog, rat and ferret lung lobes. Although qualitatively pleasing and useful for many purposes, these
11997 International Meeting on Fully 3D Image Reconstruction
1171
iInages suffer fr0111 the well-known artefacts arising from the approximate nature of the algoritlu11.
These problelTIS (inability to fully recover object densities and spatial distortion at the periphery)
becol11e 1110re severe with the large cone angles required for high-magnification lnicrovascuhu'
iInaging.
The circle-and-line conebemn scanning orbit (Zeng, 1992) cOlnbined with a recently proposed
reconstruction algorithlTI (Bu, 1995) lm'gely overcomes the deficiencies of FDK reconstruction.
Figure 1 shows the source trajectories for the circular (left) and circle-anel-line (right) orbits:
rotation axis
rotation axis
linem'
scan
circle-and-line orbit
single circular orbit
Figure 1
The inaccuracies of the FDK algorithm arise from the fact that the data available from a single
circular scan forms an incomplete set since there are many nearly-horizontal' planes intersecting the
object which do not contain a source point (Tuy, 1983). Figure 2 depicts a cross-section through
Radon space and shows the object support and the data available from a single circular scan
(Grange at, 1991):
liIle integrals available from circular scan
outer boundary
of object support
shadow zone
Figure 2
In Figure 2, the small heavy circle at the center indicates the boundary of the object support and the
two large shaded circles contain the available data. Line integrals are available for the shaded
regions of the object, but the cross-hatched regions represent the "shadow zone", for which the
circular orbit provides no data.
In circle-and-line scanning, the shadow zone data is supplied by a linear scan. After rotating the
object through 360 degrees, the line integrals missing from the circular orbit geometry are acquired
by translating the specimen parallel to the rotation axis, supplementing the circular orbit with a
finite, orthogonal line, and satisfying the data sufficiency condition (Tuy, 1983). Figure. 3 shows
the geometry and parameters used in the derivation of the circle-and-line algorithm. The source S is
11997 International Meeting on Fully 3D Image Reconstruction
1181
ri
I
I
!
j
located a distance d from the vertical axis of rotation. The imaginary detector, with coordinates Y,
Z, is scaled to coincide with the axis of rotation.
axis of rotation
Z'
fl
[]
n
Figure 3
The reconstructed object function f(r) consists of three parts:
fer) = fc o (r) + fC I (r) + fL (f)
LJ
LI
The first term is equivalent to the FDK reconstruction given above. The second term is computed,
also from the circular-orbit data, using an analogous convolution-backprojection procedure:
() f dY ~ d
Pc (Z) = ~
P<t>(Y,Z)
I
aZ
d 2 + y2 + Z2
11
L
1
F
J CI
r-j
l
(r) - __l_ldc'P
4
-
7r
2
j
Z
(d + - . AI )2 PCI
r X
The third component of the reconstruction,
[I
1
(Z )
0 Z
=~
0
d+r'x'
IL 0=) , is obtained from the linear-scan data as:
7r
iLCf)= 4 2(d - A,)fdZofdE>H(Zo,E>,l)
7r
+r . x
0
df·j'.
d(z-z,)
,
1= d+r.x' S108+ d+r.x' cos8
[]
where
L (1,8) = ffdY dZ ..J
Zo
d
d2 + y2 + Z2
P, (Y,Z) D (YsinE> + ZcosE>-I) , and
rJ
II
[ J
11997 International Meeting on Fully 3D Image Reconstruction
0
w" (8,1)={
~
when 2lzo cose + z~ cos 2 e - d 2 sin 2 e > 0
otherwise
RESULTS:
We show iInages reconstructed using the FDK and circle-and-line conebemn algorithms to recover
512 3-pixel object volumes frOIn a number of 4802-pixel conebemn projections. We imaged
contrast-enhanced lungs at magnifications ranging from about 2X to 9X. We utilized x-ray
techniques of 40 to 70 kVp and 20 to 50 micro amps (tungsten anode; ntinhnal inherent Be
filtration). We averaged between two anel several hundred video fratnes per view (exposure tilnes
of <1 to about 10 seconds), producing a range of SNR's in the projection elata. We used between
45 and 720 views over 360 degrees to reconstruct each volume, supplelnented with linear-scan
data in the case of the circle-and-line orbit. We employed software corrections for any shift of the
rotation axis and for the spatial distortions of the hnage intensifier. Projection preprocessing
included dark-field subtraction (no x-ray bearn) and flood-field division (x-rayon; no object in
beam) to correct for fixed pattern camera noise and nOll"ulliform illulnination intensity.
Our results indicate that useful 3D images of the vasculature can be obtained using our
micro angiography inlaging system and conebeam reconstruction methods. The circle-and-line
algorithm produces superior and quantitatively more accurate results than the FDK algorithm,
particularly at high magnification (large cone angles). Image quality for the present system is
limited by the spatial resolution and dynamic range of the CCD video camera. The increase,
relative to fanbemn scanning, in detected scattered radiation due to the absence of any beam
collimation does not appear to present a significant problem for soft-tissue specimens less than ten
centimeters in diameter.
ACKNOWLEDGEMENT:
Supported in part by National Heart, Lung and Blood Institute Grants HL-19298 and HL-24349,
the Department of Veterans Affairs and the W.M. Keck Foundation.
REFERENCES:
Feldkmnp, LA, Davis, LC and Kress, JW, "Practical Cone-beam Algorithm", 1. Opt. Soc. Am.
1(6):612-619, 1984.
Grangeat, P, "Mathematical Framework of Cone Beam 3D Reconstruction via the First Derivative
of the Radon Transform", In: Mathematical Methodsin Tomography, Herman, GT,
Louis, AK and Natterer, F (eds.), Springer, Berlin, 1991, pp 66-97.
Hu, H, "A New Cone Beam Reconstruction Algorithm for the Circle-and-Line Orbit", Proc. 1995
International Meeting on Fully 3D Image Reconstruction in Radiology and Nuclear
Medicine, pp 303-310.
Saint-Felix, D, Tousset, Y, Picard, C, Ponchut, C, Romeas, R and Rougee, A, "In Vivo
Evaluation of a New System for 3D Computerized Angiography", Phys. Med. BioI.
39:583-595, 1994.
Tuy, HK, "An Inversion Formula for Cone-beam Reconstruction", SIAM 1. Applied Math
43(3):546-552, 1983.
Wahle, A, Oswald, H and Fleck, E, "3D Heart-vessel Reconstruction from Biplane Angiograms",
IEEE Computer Graphics and Applications 16(1):65-73, 1996.
Zeng, GL and Gullberg, GT, "A Cone-beam Tomography Algorithm for Orthogonal Circle-andline Orbit", Phys. Med. BioI. 37(3):563-577, 1992.
11997 International Meeting on Fully 3D Image Reconstruction
.1201
[]
Kinetic Parameter Estimation from
SPECT Cone-Beam Projection Measurements *
Ronald H. Huesmant, Bryan W. Reuttert, G. Larry Zeng t and Grant T. Gullberg t
t Center for Functional Imaging, Lawrence Berkeley National Laboratory,
University of California, Berkeley, CA 94720, USA
t Department of Radiology, University of Utah, Salt Lake City, UT 84132, USA
Introduction
[1
Kinetic parameters are commonly estimated from dynamically acquired nuclear medicine data by first
reconstructing a dynamic sequence of images and subsequently fitting the parameters to time activity
curves generated from regions of interest overlaid upon the reconstructed image sequence. Since SPECT
data acquisition involves movement of the detectors (fig 1) and the distribution of radiopharmaceutical
(fig 2) changes during the acquisition the image reconstruction step can produce erroneous res~lts that
lead to biases in the estimated kinetic parameters. If the SPECT data are acquired using cone-beam
collimators wherein the gantry rotates so that the focal point of the collimators always remains in a
plane, the additional problem of reconstructing dynamic images from insufficient projection samples
arises. The reconstructed intensities will also have errors due to insufficient acquisition of cone-beam
projection data, thus producing additional biases in the estimated kinetic parameters.
Figure 1: Cone-beam SPECT scanner.
(I
lJ
r
-~
; I
rJ
Figure 2: Compartmental model for
99mTc-teboroxime in the myocardium.
To overcome these problems we have investigated the estimation of the kinetic parameters directly
from the projection data by modeling the data acquisition process of a time-varying distribution of
radiopharmaceutical detected by a rotating SPECT system with cone-beam collimation. To accomplish
this it was necessary to parameterize the spatial and temporal distribution of the radiopharmaceutical
within the SPECT cone-beam field of view. We hypothesize that by estimating directly from conebeam projections instead of from reconstructed time-activity curves, the parameters which describe
the time-varying distribution of radiopharmaceutical can be estimated without bias.
*This work was supported by U.S. Department of Health and Human Services grant ROI-HL50663 and by U.S.
Department of Energy contract DE-AC03-76SF00098.
I I
~J
11997 International Meeting on Fully 3D Image Reconstruction
1211
The direct estimation of kinetic parameters from the projection measurements has bec01ue an active
area of research. However, to our knowledge no one has accomplished direct estimation fr01n full 3D
projection data sets. In the work at the University of Michigan, Chiao et al. (1, 2] perfonnecl estimates
of ROI kinetic parameters for a one-compartluent model and estimates of paralneters of the boundary
for the ROls fr01u shnulated transaxial PET measurements. They demonstrated that the biases in the
kinetic parmneter esthnators were reduced by allowing for estimators of the boundary of the ROls to
be included in the esthnation process. In other work at the University of British Columbia, Lilnber et
al. [3] fit the pal'ametCl.'s of a single exponential decay (to model fatty acid metabolism in the heart)
directly fronl simulated pl'ojections acquired with a single rotating SPECT detector system.
Estinlatioll of timeaactivity curves £i.·onl projections has been investigated by several groups. We
have described a Inethod to estimate the average activity in 'a 2D region of interest [4], and Defrise et
al. [5] extended these ideas to 3D. l'o compensate for physical factors such as attenuation and detector
resolution, Carson [6] described a Inethod to esthnate activity density assumed to be unifonn in a set
of regions of interest using 111aximum likelihood, and Formiconi (7] similarly used least squares.
The present research builds on the work of Carson and Formiconi as well as on our previous rea
search [8] wherein a one~compartment model fit dynamic sequences in a 3 x 3 array directly from projection Ineasurements. This work showed a bias in estimates from the reconstructed time activity curves,
which were eliminated in estimating the parameters directly from the projections. The estimation was
performed in a two step process: by first estimating the exponential factors using linear time-invariant
system theory, then estimating the multiplicative factors using a linear estimation technique.
The research presented here formulates the. problem as a minimization· of one non-linear estimati?n'
problem for a 3D time~varying distribution measured with planar orbit cone-beam tomography. A
one-compartment model is assumed for the simulated myocardium tissue with a known blood input
function, which would correspond to the kinetics of teboroxime in the heart [9, 10]. Parameters are
estimated by minimizing a weighted sum of square differences between the projection data and the
model predicted values for a rotating detector SPECT system with cone-beam collimators. The estimation of parameters directly from projections is compared with estimation of kinetic parameters from
tomographic determination of time-activity curves for four regions of interest.
Estimation of Kinetic Parameters Directly from Projections
The parameters are determined from a model of the projection data that assumes a one-compartment
kinetic model for each tissue type as shown in fig 2. The expression for uptake in tissue type m, is:
Qm(t)
= k2i
l
B(r)e-klW-r)dr = k2ivm(t) ,
where B(t) is the known blood input function, k2i is the uptake parameter, and
parameter. Total activity in the tissue is given by:
Qm(t) +
f:: B(t) = k2iVm(t) + f:: B(t) ,
(1)
k12
is the washout
(2)
where fJ: is the fraction of vasculature in the tissue.
This analysis starts with an image segmented into blood pool, M tissue types of interest, and
background as is schematically shown in fig 3. In order to obtain tissue boundaries, the object (patient) is assumed motionless during data acquisition, and a reconstructed image (for example, via the
projections at the time of strongest signal, or via the summed projections) is segmented to provide
anatomical structure. The image intensity at each segmented region is not used. From the segmented
image the lengths of the blood pool, tissue, and background regions along each projection ray for each .
projection angle are calculated. The number of projection rays per projection angle is denoted by N,
11997 Internati()nal Meeting on Fully 3D Image Reconstruction
.
1221
n
I.
f
800
700
600
500
400
300
200
100
100
Figure 3: Phantom: The outer
surface is the limit of the background activity, and the ellipsoid
enclosing the small ellipsoid and
three spheres represents the outer
surface of the left ventricle.
n
L)
200
300
400
500
seconds
600
700
800
900
Figure 4: Time-activity curves for blood, background, and four tissue regions of interest: The bulk of the left ventricular myocardium
is denoted by Tissue 2, and the spherical defects are denoted by
Tissue 1, 3, 4. The Blood time-activity curve corresponds to the
small ellipsoid indicating the inner wall of the left ventricle.
the number of projection angles per rotation by J, and the number of rotations by I. Thus, there are
a total of I J N projection rays distributed in time and space. For a typical projection ray at angle j
and position n, the length of the blood pool along the projection ray is denoted by Ujn, the length of
the background by Vjn, and the length of heart tissue m by wftt. The amplitude of the background
activity is denoted by g, and the background is assumed to be proportional to the blood activity. The
projection equations can be expressed as:
M
Pijn
= UjnB(tij) + VjngB(tij) + L
w~ [k2ivm(tij)
+ f::"B(tij)]
(3)
m=l
where the time tij is proportional to j + (i ~ l)J. The constants Ujn, Vjn, and w~ are pure geometrical
weighting factors for blood, background, and tissue m, respectively, and these equations are linear in
the unknowns g, k2i, and fJ:. The nonlinear parameters, ki2, are contained in vm(tij).
The criterion which is minimized by varying the model parameters is the weighted sum of squares
function
[]
I
J
N (
2 _ ' " ~ ~ Pijn ~
X - L..tL..tL..t
i=l j=l n=l
* )2
Pijn
2
a··
~Jn
'
(4)
where aijn are weighting factors, and Pijn are the measured data. Typically, aijn is either the statistical
uncertainty of the measured data or unity (for an unweighted least squares fit).
Computer Simulations
A simulation was performed to evaluate the ability to estimate the kinetic parameters directly from
cone-beam projection data. A simulated image, shown in fig '3, contained background, blood, and four
tis~ue regions of interest. The blood input function and simulated tissue activity curves are shown
in fig 4. The blood input function was assumed known, and simple one-compartment models were
11997 International Meeting on Fully 3D Image Reconstruction
1231
kf2
a
b
c
d
e
0.150
0.146
0.150
0.150
0.002
kt2
0.060
0.060
0.060
0.060
0.0002
kt2
0.900
0.932
0.937
0.900
0.056
kt2
0.600
0.610
0.630
0.600
0.021
9
0.200
0.211
0.200
0.200
0.0001
k~l
0.765
0.813
0.769
0.765
0.010
k~l
0.540
0.428
0.542
0.540
0.001
k~l
0.960
1.090
0.976
0.960
0.082
k~l
0.960
1.047
1.124
0.960
0.043
!;
!';
!;
!~
0.150
0.166
0.133
0.150
0.017
0.100
0.211
0.102
0.100
0.002
0.200
0.246
0.271
0.200
0.200
0.229
0.038
~0.003
0.200
0.029
Table 1: Results of parameter estimation: (a) simulated, (b) noiseless Feldkamp [11], (c) noiseless Formiconi [7], (d) noiseless direct; (e) direct ullcertainties for 10,000,000 events. Units for k21 and k12 are min~l.
used within four regions of interest of a simulated left ventricle of the Inyocardituu. Boundaries of
the luyocardial regions were assumed known, and background activity was proportional to the input
function. The pal'ameter 9 was the ratio of background to blood. There were 13 parameters to estimate:
the amplitudes, decay rates, and vascular fractions for the four myocardial regions, and the mnplitude
of the overall background. The 15 minute data acquisition protocol consisted of 10 revolutions of a
single-head SPECT system with cone-beam collimators, acquiring 120 angles per revolution and 48x30
lateral samples per angle. Neither attenuation nor scatter were included. Each projection had unit bin
width, and line length weighting was assumed.
Parameters were esthnated by minimizing a weighted sum of squared differences between the projection data and the model predicted values (eqn 4). The result of estimating the kinetic parameters
directly from the projection data for the simulation is given in Table 1. Parameter esthnates from
conventional analysis of noiseless simulated data had significant biases (up to about 20%). Estimation
of parameters directly from the noiseless projection data was unbiased as expected, because the model
used for fitting was faithful to the simulation. In addition, multiple local minima were not encountered,
regardless of noise levels simulated. Parameter uncertainties for 10,000,000 detected events ranged from
0.3% to 6% for wash-out parameters and from 0.2% to 9% for uptake parameters.
Summary
The combination of gantry motion and the time-varying nature of the radionuclide distribution being
imaged results in inconsistent projection data sets. Estimating kinetic parameters from time-activity
curves taken from reconstructed images [11] results in biases. Some of these biases are reduced and
some are increased if the time-activity curves are estimated from the projection data [7). Estimating the
kinetic parameters directly from cone-beam projections removes all bias for noiseless data as expected.
References
[1) Chiao PC, WL Rogers, NH Clinthorne, JA Fessler, and AO Hero. Model-based estimation for
dynamic cardiac studies using ECT. IEEE Trans Med Imag, 13(2):217-226, 1994.
[2) Chiao PC, WL Rogers, JA Fessler, NH Clint horne , and AO Hero. Model-based estimation with
boundary side information or boundary regularization. IEEE Trans Med'Imag, 13(2):227-234,
1994.
[3] Limber MA, MN Limber, A Cellar, JS Barney, and JM Borwein. Direct reconstruction of functional
parameters for dynamic SPECT. IEEE Trans Nucl Sci, 42:1249-1256, 1995.
[4] Huesman RH. A new fast algorithm for the evaluation of regions of interest and statistical uncertainty in computed tomography. Phys Med Bioi, 29(5):543-552, 1984.
11997 International Meeting on Fully 3D Image Reconstruction
1241
[5J Defrise M, D Townsend, and A Geissbuhler. Implementation of three-dimensional image reconstruction for multi-ring positron tomographs. Phys Med Biol, 35(10):1361-1372, 1990.
[6J Carson RE. A maximum likelihood method for region-of-intrest evaluation in emission tomography.
J Comput Assist Tomogr, 10(4):654-663, 1986.
.
[7J Formiconi AR. Least squares algorithm for region-of-interest evaluation in emission tomography.
IEEE Trans Med Imag, 12(1):90-100, 1993 .
r~
.
\
J.
)
[8J Zeng GL, GT Gullberg, and RH Huesman. Using linear time-invariant system theory to estimate
kinetic parameters directly from projection measurements. IEEE Trans Nucl Sci, NS-42(6):23392346, 1995.
[9] Smith AM, GT Gullberg, PE Christian, and FL Datz. Kinetic modeling of teboroxime using
[J
dy~amic SPECT imaging of a canine model. J Nucl Med, 35(3):984-995, 1994.
[10] Smith AM, GT Gullberg, and PE Christian. Experimental verification of 99mTc-teboroxime kinetic
parameters in the myocardium using dynamic SPECT: Reproducibility, correlations to flow, and
susceptibility to extravascular contamination. J Nucl Cardiol, 3:130-142, 1996.
[11] Feldkamp LA, LC Davis, and JW Kress. Practical cone-beam algorithm.
1:612-619, 1984.
J Opt Soc Am A,
r-;
jl ___ .\
[J
r'~
J
\
LJ
[J
ii
:.. J
11997 International Meeting on Fully 3D Image Reconstruction
1251
An Analytic Model of Pinhole Aperture Penetration for
3 .. D Pinhole SPECT Image Reconstruction
Mark F. Smith and Ronald J. Jaszczak
Departlnent of Radiology, Duke University Medical Center, Durham, NC, USA
I. Introduction
In single photon imaging with a pinhole collimator, photons penetrating the attenuating
luaterial close to the pinhole aperture broaden the tails of point spread functions (PSFs) and degrade
the resolution of planar and reconstructed SPECT itnages. Penetration is greater for radionuclides
with InedilUTI and high energy eluissions and it is a significant factor affecting our efforts to achieve
high resolution iInaging of 1.. 131 radiolabeled monoclonal antibodies adlninistered intratull10rally for
brain tumor therapy. In this paper we develop an analytic formula for pinhole aperture penetration.
Analytic predictions are cOlnpared with results fro1n photon transport simulations. The analytic
model of aperture penetration is used in the design of resolution recovery filters to cOlnpensate for
penetration blur. These filters are applied to simulated and experimental pinhole SPECT studies.
II. Theory
Recent studies by our research group have shown that experimentally acquired point spread
functions for pinhole imaging with a knife~edge aperture can be accurately modeled using photon
transport simulation programs (1, 2). For point sources on the central ray of a pinhole SPECT system
(Fig. 1) the tails of the PSFs due to penetration near the pinhole aperture are approximately linear on
log-scale plots and can be fit by decaying exponential functions of the form exp(-yx) (1).
The resolution of a pinhole collimator with a knife-edge aperture has been modeled by a simple
formula (3) but a model of the tails of the PSFs has not yet been developed. An expression for the
tails of PSFs for the simpler case of penetration through a shielding layer of constant thickness has
been given, however (4). In this section we develop an analytic formula for the roll-off coefficients
of the exponential tails of PSFs for pinhole imaging with a knife-edge aperture.
Consider a knife-edge pinhole aperture with a point source offset froIn the central ray (Fig. 2).
Let the acceptance angle of the aperture be a, the linear attenuation coefficient of the pinhole
material be /-l and the focal length of the pinhole collimator be f. Furthermore let the pinhole be
located at the origin of the coordinate system and denote the coordinates of the point source by
(x s' Ys' zs). For simplicity let Ys = 0 and let the projection P of the point source onto the gamma
camera be the origin of a local (r, fJ) coordinate system on the camera face. For a projection pixel at
position (l', e), the raypath length in the attenuating medium can be found by solving a quadratic
equation for the intersections of this raypath with the cone defining the air-matter boundary of the
pinhole aperture.
The amplitude of the PSF can be expanded for snlall offset r. After some algebraic
manipUlation we find that the tails have the form
e
(
e
r . cos 2 2· 2xs - r . cos
I (l', e) = 10 1+ 23 ...
[
(1 + f 1zs)( x s + Zs )
1+ f 1Zs
J]
exp( -y r)
(1)
where
r=
2
(12) (1 + [xsIZs}2 )112 (1- ([ xslzs }cot( aI2)sin() p )112
/-lcot a
x
1+ flzs
x
1-([xs lzs }cot(aI2))2
(2)
The exponential roll-off coefficient rsimplifies to
r = [2/-lcot(aI2)}1[1 + f Izs}
(3)
for a point source on the central ray above the pinhole (xs=O) and it reduces to the expected limit of
2/-lcot(aI2) when Zs is large and the rays incident upon the gamma camera are nearly vertical.
III. Comparison of Analytic Expressions with Photon Transport Simulations
Projection data for an 1-131 point source were modeled using a photon transport simulation
code (2). The knife-edge pinhole was located in the center of a 2.6 em thick tungsten slab. The
pinhole diameter was 1 mm and the focal length of the collimator was 14.5 cm. The 364 keV
emission was modeled with penetration through the pinhole aperture region and shielding; the
detection of photons scattered in the insert and shielding was not modeled. The projection pixel size
was 1.78 mm and decaying exponential functions were fit to the tails of the point spread functions.
11997 International Meeting on Fully 3D Image. Reconstruction
1261
Smith and Jaszczak
One set of simulations was performed with the point source on the central ray. For a pinhole
aperture with an acceptance angle a = 100°, log-scale plots of profiles through the PSFs are linear as
expected (Fig. 3), indicating that the tails can be fit by decaying exponential curves. The increase of
rwith position Zs above the pinhok was well-modeled by theory (Fig. 4). When the source position
above the pinhole was fixed at Zs = 12 cm, the variation of the roll-off coefficient r with collimator
acceptance angle a was also modeled quite well (Fig. 4).
The off-axis distance Xs of the point source was varied with a vertical distance Zs of 12 cm and
an acceptance angle a of 100°. The theoretical predictions matched the measured roll-off coefficient
for the azimuthal angle 8=90° (Fig. 5). The analytic prediction of ris the same for 8=0° and 8=180°
(equation 2), but the measured curves were slightly different from each other and this prediction
(Fig. 5). These differences are due to higher terms in the expansion for y. The variation of r with
azimuthal angle e is shown for an offset distance x s=6 cm (Fig. 6). The differences between the
theoretical and measured roll-off coefficients also are due to higher order expansion terms.
[;
IV. Application of Resolution Recovery Filters for Aperture Penetration in 3 D Reconstruction
a
A. Filter design
The analytic model of pinhole aperture penetration was used in designing resolution recovery
filters for application to pinhole SPECT projection data prior to 3-D image reconstruction. We
follow the approach of Wang et al. (1) and use the PSF for a point source at the axis of rotation as
the modulation transfer function (MTF) in a 2-D Metz filter (5). If b is the distance from center of
rotation to the pinhole, then the MTF is a circularly symmetric function of the form exp( -rr)where
[I
IJ
r =[2/1 cot(a/2)J /[1 + f
/ b J,
(4)
The 2-D Fourier transform of the MTF normalized by its zero-frequency value is
h(fx,fy) = 11[1 + (2nlr)2(f;
+ f; )J3/2
(5)
The resolution recovery filter for pinhole aperture penetration then has the 2-D Fourier transform
2 x
.
'.
M( fx,fy) = [1- (1- (h( fx,fy)) ) J/h( fx,fy)
-,
B. Application of the resolution recovery filter to compensate for penetration blur
[-.1
r·;
LJ
i !
{1
~J
r
(6)
1
tJ
r'
I I
L }
An 1-131 pinhole SPECT tumor study was simulated using our photon transport code. Projection data were generated at 120 equally spaced angles over 360° for a tumor model atthe center of a
20 cm diameter, 20 cm high water-filled cylinder. The 3.0 cm diameter tumor model consisted of a
2.2 cm diameter core surrounded by a 0.4 cm thick shell. A shell to core activity concentration ratio
of 5: 1 modeled a greater density of active tumor cells at tumor periphery. The axis of the cylinder
was aligned with the axis of rotation of the gamma camera. The distance from the pinhole to the axis
of rotation was 14 cm, the focal length was 14.5 cm and the aperture acceptance angle was 100°. The
projection array was 256 x 128 and the pixels were 1.78 mm. Only the primary 364 keY emission of
1-131 was modeled, without scatter in the cylinder or detector. No Poisson noise was added.
Images were reconstructed by a filtered backprojection method (6) in two ways, 1) without any
resolution recovery filter and 2) with a Metz filter of order x=100 applied to the projection data. A
multiplicative Chang attenuation correction (7) was applied as part of image reconstruction. With
application of the Metz filter the resolution of the tumor shell is improved and the contrast of the
shell with the core increases (Fig. 7).
A resolution recovery filter was also applied to projection data from an experimental phantom
study acquired on a clinical scanner. The tumor phantom was 6.2 cm in diameter with a 1.0 cm outer
shell. The phantom was filled with 3.3 mCi of an 1-131 solution with a shell to core activity concentration ratio of 4.1: 1. The tumor phantom was imaged in a 23 cm diameter water-filled cylinder for
60 min. The energy windows were 364 keY ± 9% and 304 keY ± 9%. The projection array was 128
x 64 with 3.56 mm pixels. A scatter subtraction factor k=1.0 was used and the projection data were
filtered either with 1) a 2-D Hann filter (cutoff frequency = 1.40 cycles/cm in projection space) or
2) a Metz filter of order x=10. A multiplicative Chang attenuation correction was applied as part of
the filtered backprojection reconstruction. With the Metz filter the shell of the turnor phantom is
better resolved and the contrast between the shell and core is increased (Fig. 8).
V. Conclusion
An analytic model has been developed for the exponential tails of point source response
11997 International Meeting on Fully 3D Image Reconstruction
1271
Smith and Jaszczak
functions for pinhole iInaging with a knife-edge aperture. There is good agreelnent between analytic
predictions ancllneasured exponential roll-off coefficients froln photon transport siInulations, though
higher order approxiInations are necessary for a more exact Inatch. The analytic fonnula for the 1'011off coefficient was used in designing resolution recovery filters for IB 131 iInaging. When these are
applied to the projection data prior to 3-D pinhole reconstruction, the resolution of the reconstructed
SPECT iInages for llulnerically siInulated and experimental phantom studies is iInproved.
References
1. Wang H, Jaszczak RJ, Coleman RE. Monte Carlo modeling of penetration effect for iodine-131 pinhole imaging.
IEEE Trans Nllcl Sci in press.
2. Smith MF, Jaszczak RJ, Wang H, Li J. Lead and tungsten pinhole inserts for 1-131 SPECT tumor imaging:
experhnental measurements and photon transport simulations. IEEE Trans Nllcl Sci in press.
3. Mortimer RK, Anger HO, Tobias CA. The gamma ray pinhole camera with image amplifier. Cony Record IRE, Pt 9
1954; 2-5.
4. Barrett HH, Swindell W. Radiological Imaging.' The The01Y of Image Formation, Detection and Processing. 1981,
New York: Academic.
5. Metz eE, Beck RN. Quantitative effects of stationary linear image processing on noise and resolution of structure in
radiolluclide images. J Nucl Med 1974; 15:164-170.
6. Li J, laszczak RJ, Greer KL, Coleman RE. A filtered backprojection algorithm for pinhole SPECT with a displaced
centre of rotation. Phys MedBioll994; 39:165-176.
7. Chang L8T. A method for attenuation correction in radionuclide computed tomography. IEEE Trans Nucl Sci 1978;
NSM25:638 643.
8. Smith MF, laszczak RJ, Wang H. Pinhole aperture design for 131 I tumor imaging. IEEE Trans Nucl Sci in review.
M
point source
,
'~,~a',~~~
Zs
,,
,
pinhole
aperture
scintillation camera
scintillation camera
Fig. 1. Diagram for a point source on the central ray, which is
the ray perpendicular to the gamma camera passing through
the pinhole. The focal length is f and the source is located a
distance ~ above the pinhole. (Not to scale.)
Fig. 2. Diagram for a point source offset from the central ray (not
to scale.) The solid ray indicates the projection of the point source
through the pinhole onto the gamma camera at point P. The dashed
ray is the path through the pinhole aperture material to the position
(r, (J) with respect to the coordinate system with origin at P.
Profile, Point Source 8 cm Above Pinhole
1 0° ~""""""""""''''''''''''''''''''''''''''r"'""''"''TT"''''"rr"'-i;I
1 0. 5
96
112
126
144
Pixel, Index
Profile, Point
Sour~e
16 em Above Pinhole
10o~~~~~~~~~=
1 o· 5 '--'--,--",~-""",-",,--,--,--,~.&.-......"""""'-'
96
112
128
144
160
Pixel Index
Fig. 3. Profiles through the PSFs for point sources 8 and 16 em above the pinhole on the central ray. The projection pixel size is 1.78
mm and each profile is nonnalized to its peak value. The approximate linear slope of the curves on these log-scale plots indicates that
an exponential fit can be made to the profile tails~ The slopes are different for the two source positions.
11997 International· Meeting on Fully 3D Image Reconstruction
1281
{--1
I
\.
t
}
Smith and J aszczak
Gamma vs. Height above Pinhole
'"
E
.~
5'--'---'-~-r-"T"-,---r--r-'T---r---r--,---,,---,
Gamma VS. Acceptance Angie (Alpha)
12~~~~~~~~~~~
4
10
3
2
4
2
o
o
OL.-..--L-'---t-o..-L-L---l-"'---.I-,-L.-L-<---.J
12 16
z (em)
20
24
28
O.
20
Fig. 4. For a point source on the central ray, variation of the roll-off coefficient
variation of rwith acceptance angle a of the pinhole aperture (right).
Gamma vs. Offset (Theta=900)
6
a
4
E
E
[]
16
above the pinhole (left) and
-gamma(meas,O deg)
- 'gamma(meas,180 de g)
··'··gamma(theor)
E
E
jl8
0
/
V/
12
111
,.theor
0
/,/
a
8
2
o
o
Zs
20~~~==~~~~~1
8
111
r with height
Gamma vs. Offset (Theta=O, 180°)
10~---r~~~~~~~-r---r--
[]
40 60 80 100120 140
alpha (degrees)
/.
~ . :-:::.~~.'::::':~.'
4
.../,.'
OL.-..-L~~~~--l-~~-L-<---.J
2
4
6
8
10
offset (em)
12
14
0
2
4
6
8
10
offset (em)
12
14
Fig. 5. For a point source offset from the central ray, variation of the roll-off coefficient r in the 8=90° direction with offset distance
(left) and in the 8=0° and 8=180° directions (right). The first order analytic prediction is the same for 8=0° and 8:;::180°. The
difference between the predicted and measured values increases with offset.
Xs
(-_l)
Gamma vs. Azimuthal Angle (Theta)
5n-~~---r---r-~~~~~~
'"
E
E
a
3
2
OL..-<........--L--'-~-'---'--'---'---'-..........,
o
90
180
270
360
theta (degrees)
fJ
Fig. 6. For a point source offset 6 cm from the central ray, variation of the roll-off coefficient rwith azimuthal angle 8. Differences
between the measured and first order theoretical curve are du~ to unmodeled higher order tenns in the expansion for r.
Profiles, TUmor with 4 mm Shell
1.0
~ 0.8
~
0.6
E
«
~
0.4
a. 0.2
0.0 I--_ _ _- J .
<48
No Resolution Recovery Filter
[
128
168
208
Pixel Index
Fig. 7. Transverse slices and profiles of a 3.0 cm diameter tumor from a simulated SPECT study (pixel size 1.78 mm). Resolution and
contrast are improved with application of a Metz filter to compensate for blurring due to pinhole aperture penetration.
Profiles, Tumor near Center
[1
J
88
Metz Filter
I
1.0
LJ
Q)
'0
0.8
:::I
~0.6
E
« 0.4
~0.2
0.0
~~~~~~--'-~~~~~~
24
40
56
72
Pixel Index
88
104
Metz Filter
Hann Filter (no resolution recovery)
Fig. 8. Transverse slices and profiles of a 6.2 cm diameter tumor phantom from a SPECT study acquired on a clinical gamma camera
(pixel size 3.56 mm). The Metz filter improves resolution and contrast. The shell intensity varies due to spatial-dependent resolution.
11997 International Meeting on Fully 3D Image Reconstruction
1291
Comparison ofFrequency...Distance--Relationship and Gaussian...Diffusion Based Methods
of Compensation for Nonstationary Spatial Resolution in SPECT Imaging.
Vandana Kohli, MSEE1,2, Michael King, PhD l , Stephen Glick, PhD!, and Tin.. Su Pan, PhD3
IDepartment of,Nuclear Medicine, The University of Massachusetts Medical Center, 55 Lake Ave
North, Worcester,~.L\. 01655, 2Depa..rtment of Electrical Engineering, The University of
Massachusetts Lowell, 1 University Ave, Lowell, MA, 01854 ,and ~The Applied Science
Laboratory, General Electric Company, Milwaukee, WI, 53201.
One method which can be employed to compensate for spatial resolution in single photon emission computed
tomographic (SPECT) imaging is restoration filtering. Most restoration filters used in nuclear medicine have
assumed a statihnary point spread function (PSF) (not varying with location of the activity relative to the camera);
however, the camera response is distance..<Jependent. One way to fonnulate a distance-dependent restoration filter
is to use the frequency-distance ..relationship (FDR) (Lewitt et a/1989, and Glick et a/1994). The FDR states that
the distance of the source from the center--of..rotation (COR) producing the signal is concentrated along planes in
Ltu'ee..rumensions (3D) given by the negative of the anguiar frequency divided by the transaxial spatial frequency in
the 3D Fourier transfonn of the set of sinograms. The advantages ofFDR restoration filtering include: 1) it is fast
compared to inclusion of the distance..<fependent resolution (DDR) in iterative reconstruction; 2) since linear
filtering is employed, the result is not data..<Jependent so long as the·same filter is applied (however, it is datadependent if an image-dependent filter is applied); 3) it can achieve good stationarity and isotropy of response
(Glick et a/1994); and 4) it can yield improved quantitative accuracy over low pass filters (pretorius et a/ 1996).
The disadvantages ofFDR restoration filtering include: 1) it is limited as to how much resolution recovery can be
obtained without amplifying noise; 2) if the noise regularization is adapted to match the local signal to noise
content of the Fourier transfonn of the sinograms, a stationary and anisotropic response is no longer obtained; 3) if
opposing acquisition views are combined, or filtering is symmetric between near and far fields, circular noise
correlations will result in transverse slices (Soares et a/1996); 4) FDR is only an approximation (albeit, a good
one); 5) at low transaxial spatial frequencies and, in particular, when the transaxial spatial frequency is zero, FDR
becomes more of an approximation (all distance pass through the origin making the selection of the PSF in the
axial frequency direction here arbitrary); 6) FDR does not account for the alteration in the sinograms by
attenuation; and 7) FDR restoration filtering correlates the noise in the projections which will be input to the
reconstruction algorithm.
Another method to correct for the non..stationary camera response is the incorporation of a blurring model in an
iterative reconstruction algorithm (Tsui et a/1988, Floyd et a/1988, Fonniconi et aJ 1989, Penny et a/1990,-Zeng
and Gullberg 1991, McCarthy and Miller 1991, Zeng and Gullberg 1992, Beekman et aI1993). The problem with
this method has been the immense increase in computational burden imposed when'maximum..likelihood
expectation-maximization (MLEM) reconstruction includes 3D modeling ofDDR The use of ordered..subsets with
MLEM (OSEM) has been shown. to dramatically reduce the number of iterations required for MLEM to reconstruct
slices (Hudson and Larkin 1994, Kamphuis et aI1996), and thus decrease the computational burden. Besides
reducing the number of iterations required to reconstruct images, one can decrease the computational burden by: 1)
including the modeling ofDDR only in the projection step (Zengand Gullberg 1992); and 2) using Gaussiandiffusion to model DDR so that only small convolutional masks are applied (McCarthy and Miller 1991). The
advantages of modeling DDR in iterative reconstruction include: 1) there appears to be a better signal/noise tradeoff with resolution recovery allowing improved accuracy of quantitation over FDR restoration filtering (pretorius et
al submitted); 2) the noise characteristics of the data are not altered prior to reconstruction; and 3) the
approximations inherent in application ofFDR are avoided. The disadvantages include: I) processing time (FOR is
faster since the restoration filtering needs only to be applied once); 2) degree to which a stationary and isotropic
response is reached depends on the location of the source in the slices, the camera orbit (circular versus bodycontouring), the number of iteratio~s used, the algorithm, and the source distribution (pan et al submitted)~ and 3)
11997 International Meeting on Fully 3D Image Reconstruction
1301
aliasing alters the discreet response significantly from the desired response of the convolutional filters when
Gaussian diffusion is employed (King et al submitted).
r--~,
i, I!
\
'
r\
!J
I
I
[I
fJ
[j
[J
rI
H
r
The objective of this investigation was to provide a systematic comparison between these two approaches to
compensation of DDR The studies were designed to illustrate and expand upon the above points of comparison
with the goal of further clarifying the relative merits of the two approaches.
The mathematical cardiac-torso (MCAT) phantom (Tsui et a/1994) was used to simulate the non-uniform
attenuation of SPECT imaging in the thorax region. The phantom had an elliptical outline and had a width, or
major axis, of36 cm. Five identical, 3D, Gaussian point sources ofFWHM = 1:248 cm (standard deviation = 0,53
cm) were placed at least 3 FWIllv.t's apart to serve as the source distribution. The size of the point sources was
about the thickness of the myocardium in the human thorax. The five locations were selected to allow investigation
of resolution recovery at a variety of locations varying in attenuation and distance from the COR
We simulated attenuation and detector response during projection as previously described (pan et a/1995).The
collimator simulated was a low energy high resolution (LEHR) parallel hole collimator on our Picker, International
Prism 3000 SPECT system. To obtain the DDR model for the system, point sources of Tc-99m were imaged in air
at 5, 10, 15, 20, 25, and 30 cm from the face of this collimator. Gaussian functions were fit in the horizontal and
vertical directions to the resulting point spread functions and the fitted sigma's averaged in the x and y direction.
Each projection data set consisted of 120 angles of 128 by 128 pixels of pixel size = 0.317 cm. The projection 'data
were rebined from 256 by 256 projections of pixel size = 0.1585 cm to reduce the problems of sampling and
aliasing. The circular camera orbit simulated had a radius of 28 cm(from the detection plane). In addition to using
the activity map with only five point sources, we also added a background of 10% of the maximum activity0f·the~;
point sources. This additional activity allowed us to investigate the influence of the background activity in the
resolution recovery of the point sources. Poisson noise was not included in the simulations.
All reconstructions were via 1 to 10 iterations of OSEM. The 120 projection angles were divided into 15 subsets in
the OSEM reconstruction. The reconstruction time for 1 iteration of OSEM with both attenuation and DDR
correction for 128 slices and 120 angles was about 1.5 hours on a DEC alpha workstation Model 600. The
reconstruction strategies compared were: 1) projections which solely included the effects ofDDR which were FDR
filtered prior to OSEM; 2) projections which included the effects of nonuniform attenuation and DDR which were
FDR filtered prior to OSEM reconstruction which accounted for the nonuniform attenuation; 3) projections which
included the effects of nonuniform attenuation and DDR which were corrected for nonuniform attenuation (Bellini
et a11979, Glick et a/ 1995), and FDR filtered prior to OSEM reconstruction which did not account for the
nonuniform attenuation; 4) projections which included the effects of nonuniform attenuation and DDR
reconstructed by OSEM which accounted for the nonuniform attenuation and DDR; and 5) projections which
included the effects of nonuniform attenuation and DDR, and reconstructed by OSEM which accounted for the
nonuniform attenuation but not DDR Reconstruction strategy 1 was investigated to determine the limitations of
FDR in the absence of attenuation. This was of interest to determine the "best case" performance of FDR filtering
since FDR does not account for attenuation. Comparison of reconstruction strategies 1 and 2 allowed assessment of
the impact of attenuation on FDR restoration when attenuation is not compensated for prior to FDR filtering.
Comparison of reconstruction strategies 1 and 3 allowed assessment of the degree to which pre-correction for
attenuation could alleviate the degradation in FDR performance in the presence of attenuation. By comparing
strategies 2 and 3, the relative merits of analytical pre-filtering attenuation correction, and correction for
attenuation after FDR filtering within OSEM could be detennined. Comparison of strategies 3 ,4, and 5 allowed
assessment of the degree to which DDR can be compensated for by a combined pre-reconstruction approach for
attenuation and DDR, and by the inclusion of them in OSEM. Comparison of the same strategies applied to the
acquisition sets with and without the presence of the 10% background allowed assessment of the impact of
surrounding activity on the point sources.
Once the slices were reconstructed, 3D Gaussian distributions were fit and the FWHM in the directions of X (along
the major axis of a transverse slice of the phantom), Y (perpendicular to X in the slice), and Z (axial direction of
the phantom) were calculated. We defined the normalized FWHM (nFWHM) as a measure of resolution recovery
with the nFWHM equal to the measured FWHM divided 1.248 cm (the source FWHM). The closer nFWHM: is to
1
L
11997 International Meeting on Fully 3D Image Reconstruction
131\
1; the better the resolution recovety was. The nFWHM was also calculated in the X and Z directions for the each
source at each projection angle with no resolution compensation, and with FDR restoration to help determine the
impact ofFDR restoration on the data input to reconstruction.
Currently, not all of the reconstructions have been completed or analyzed. The following results have been
obtained thus far.
First~
even in the absence of Poisson noise and attenuation, and with regularization of the FDR filter to produce a
constant Gaussian response equivalent to that of the camera at a distance equal to the radius ..of..rotation (ROR), the
nFWHM's become extremely erratic in the filtered projections as the distance increases beyond that of the ROR.
The average nFWHM with this degreo of regularization was 1.28. Even under these ideal conditions, increased
regularization for distances greater than the ROR (the far field) is required to avoid the projections becoming
exceedingly noisy. When such regularization is applied (for example, by clipping the FDR filter so that it does not
have coefficients·larger than 10.0), the far field response is no longer erratic, but larger nFWHM's are obtained for
far field locations. This indicated that there will be a trade--offbetween an isotropic and stationary response, and
the regularization of noise.
With the addition of uncorrected attenuation, low frequency streaking is seen in the sinograms after FDR
restoration filtering. The nature of these streaks is dependent on the coefficients of the FDR filter in the region of
the 3D Fourier Transfonn of the sinograms where FDR is unable to specify a distance (the plane for the transaxial
spatial frequency equal to zero). In the absence of attenuation, all projections have the same total count and only
noise contributes to terms in this plane when the angular frequency is not 0, In the presence of attenuation there is
variation in the number of counts within the projections as a function ofangle, and signal as well as noise
contributes to the off..axis terms. The result is a. degradation in the ability ofFDR to compensate for DDR. With the
exception of a point source located in the lungs, degradation in FDR performance caused by attenuation appears to
be corrected by application of analytical attenuation correction
Finally, with both attenuation and DDR modeled in the projector and backprojector pair, only 2·3 iterations are
required to obtain near stationary and isotropic nFWHMts of approximately 1.0 when solely the point sources are
present in the projections. The presence of the 10% background has a profound impact on the convergence of
OSEM. Even at 10 iterations of OSEM, the nFWHMt s are not stational)' or isotropic, and are not near the ideal
value of 1.0.
Acknowledgments:
This research was supported by National Heart, Lung, and Blood Institute under Grant HlI'·50349. Its contents are
solely the responsibility of the authors and do not necessarily represent the official views of the National Heart,
Lung, and Blood Institute.
References:
Beekman FJ, Eijkman EGJ, Viergever MA, Borm GF and Slijpen ETP 1993 Object shape dependent PSF model
for SPECT imagingIEEE Trans. Nucl. Sci. 4031-39
Bellini S, Piacentini M, Cafforio C and Rocca F 1979 Compensation of tissue absorption in emission tomography
IEEE Trans. Acou.s Speech. Sig. Proc. 27 213-18
Floyd CE, Jaszczak RJ, Manglos SH and Coleman RE 1988 Compensation for collimator divergence in SPECT
using inverse Monte Carlo reconstruction IEEE Trans. Nucl. Sci. 35 784-87
Formiconi AR, Pupi A and Passeri A 1989 Compensation of spatial response in SPECT with conjugate gradient
reconstruction technique Phys. Med. BioI. 34 69-84
11997 International Meeting on Fully 3D Image Reconstruction
1321
Glick 8J, Penney BC, King MA and Byrne CL 1994 Noniterative compensation for the distance-dependent detector
response and photon attenuation in SPECT imaging IEEE Trans Med Imag 13363-74
Glick SJ, King MA, Pan T-S, and Soares EJ 1995 An analytical approach for compensation of non-uniform
attenuation in cardiac SPECT imaging. Phys Med Bioi 40 1677-1693
Hudson HM and Larkin RS 1994 Accelerated image reconstruction using ordered subsets projection data IEEE
Trans. Med.Imag. 13601-9.
fl
tj
Kamphuis C, Beekman FJ and Viergever MA 1996 Evaluation of OS-EM vs. ML-EM for ID, 2D and fully 3D
8PECT reconstruction. IEEE Trans. Nuc/. Sci. 43 2018-24
[01
King MA, Pan T-S, and Luo D-S Fast spatial-domain convolution which accounts for system spatial resolution.
IEEE Trans Nucl Sci, submitted.
I
I
Lewitt RM, Edholm PR and Xia W 1989 Fourier method for correction of depth dependent collimator blurring.
SPIE Proc. 1092 232-43
[]
fl
McCarthy AW and Miller MI 1991 Maximum likelihood SPECT in clinical computation times using meshconnected parallel computers IEEE Trans. Med Imag. 10 426-36
.1
[]
Pan T-S, Luo D-S and King MA 1995 Design of an efficient 3-D proJector and backprojector pair for SPECT::'1h:
Proceedings of Fully 3D Image Reconstruction in Radiology and Nuclear Medicine ( Aix-Ies~Bhlns,
Savoie,France) pp. 181-5
- Influence of OSEM, elliptical orbits and background. activity on SPECT 3D resolution recovery Phys. Med.
Bioi. submitted.
Penney BC, King MA and Knesaurek K 1990 A projector, back-projector pair which account for the twodimensional depth and distance dependent blurring in SPECT IEEE Trans. Nucl. Sci. 37681-6
r- i
LJ
Pretorius PH, King MA, Glick SJ, Pan T-S and Luo D-S 1996 Reducing the effect of non-stationary resolution on
activity quantitation with the frequency distance relationship in 8PECT. IEEE Trans. Nucl. Sci,. In press.
[J
Pretorius PH, King MA, Pan T-S, de Vries DJ, Glick SJ, and Byrne CL Reducing the influence of the partial
volume effect on 8PECT activity quantitation with 3D modeling of spatial resolution in iterative reconstruction.
Phys. Med. Bioi. submitted.
[j
Soares EJ, Glick 8J, and King, MA Noise characteristics of frequency-distance principle (FDP) restoration
filtering. IEEE Trans. Nucl. Sci,. In press.
[.I
Tsui BMW, Hu HE, Gilland DR and Gullberg GT 1988 Implementation of simultaneous attenuation and detector
response correction in 8PECT IEEE Trans. Nucl. Sci. 35 778-83
r- j
lJ
[J
r:
I
I
Tsui BMW, Zhao XD, Gregoriou GK, Lalush DS, Frey EC, Johnston RE, and McCartney WH 1994 Quantitative
vardiac SPECT reconstruction with reduced image degradation due to patient anatomy. IEEE Trans. Nucl. Sci. 41
2838-2844.
Zeng GL and Gullberg GT 1991 Three-dimensional iterative reconstruction algorithms with attenuation and
geometric point response correction IEEE Trans. Nucl. Sci. 38 693-702
-1992 Frequency domain implementation of the three-dimensional geometric point response correction in SPECT
imaging IEEE Trans. Nucl. Sci. 39 1444-53
:
l
l_
11997 International Meeting on Fully 3D Image Reconstruction
1331
Comparison of Scatter Compensation Methods in Fully 3D
Iterative SPECT Reconstruction: A Simulation Study.
Freek J. Beekman!, Chris Kamphuisl, Eric C. Frey2,
Imaging Science Institute, University Hospital Utrecht.
Department of Biomedical Engineering, The University of North Carolina at Chapel Hill.
1
2
Abstract
Effects of different scatter compensation methods incorporated in fully 3D iterative reconstruction
were investigated. The methods were: (i) The inclusion of an noisy "ideal scatter estimate" (ISE)i (ii)
Like (i) but with a noiseless scatter estimate (ISE-NF)i (iii) Incorporation of scatter in the point spread
function during iterative reconstruction ("ideal scatter model", ISIvI)i (iv) No scatter compensation, (v)
Ideal scatter rejection, as was simulated by assuming a perfect energy l~esolution and 110 loss of sensitivity
for primary photons. A cylinder containing cold spheres was used to calculate cOlltrastftto-lloise curves.
For a brain study, global errors between reconstruction and "true" distributions were calculated. Results
show that ideal scatter rejection is superior to all other methods. In all cases considered, ISM is superior
to ISE and performs approximately as well as (brain study) or better than (cylinder data) ISE-NF.
Both ISM and ISE improve contrast-to-noise curves and reduce global errors, compared to no scatter
compensation. In case of ISE, blurring of the scatter estimate with a Gaussian kernel results in slightly
reduced errors in brain studies, especially at low count levels. The optimal kernel size strongly depends
on the noise level.
INTRODUCTION Detection of photons which have undergone scatter events in the patient is one
of the main factors of image distortion in SPECT. Over the last two decades, many methods have been
proposed to compensate for scatter in order to reduce the resulting degradation of image contrast and loss
of quantitation.
One class of scatter compensation methods are subtraction-based. In these methods, an estimate of
the scatter contribution in the projection data is subtracted from the measured projection data. Many of
these methods use data from multiple energy windows to obtain the scatter estimate (e.g., [1,2]). These
different methods have different accuracies [3J and, to further complicate matters, the accuracies depend
on the exact calibration procedure used to determine various parameters and calibration curves.
A second class of scatter compensation methods is the incorporation of an accurate model of the image
formation process into an iterative reconstruction algorithm (e.g. [4-6]). This has the advantage that the
accuracy of the scatter compensation should be limited only by the ability of the model to describe the
image formation process. In addition, there are potential advantages in terms of noise properties [6J.
The focus of the present work is to compare subtraction-based and reconstruction-based scatter compensation methods. Rather than choosing a subset of these methods, two limiting cases for subtraction
based methods are investigated. The first method is to assume that we are able to estimate the true-noise
free (mean) scatter component of the projection data. This represents an ideal scatter subtraction method
that has the best possible properties in terms of accuracy and noise. This method will be referred to as
ideal scatter estimation-noise free (ISE-NF). The second limiting method is to assume that we can estimate
the true scatter component of the projection data, but that this component is corrupted by Poisson noise.
The effects of the noise in the projection data can then be modulated by the use of low pass filtering. This
method, ideal scatter estimation (ISE) represents a scatter subtraction that has ideal behavior in terms
of accuracy, but less than ideal, and more realistic, noise properties. These two limiting-case subtraction
methods are compared to the limiting case of ideal scatter modeling (ISM), i.e., where the reconstruction
algorithm incorporates the exact model of the image formation process. In addition to these three compensation methods, comparisons were made to the case of no scatter compensation (NSC), and ideal scatter
rejection (ISR) (the case of a detector that can perfectly reject scattered photons). All methods considered
compensate for attenuation and the full distance depe~dent camera response. These various methods are
compared using two phantoms,· a simple phantom containing cold rods in a hot background, and the 3D
Hoffman brain phantom. A quantitative comparison is made using measures of image contrast and noise
(for the simple phantom) and global accuracy (for the brain phantom).
METHODS Noise free projections of a cylinder and a digital 3D Hoffman brain phantom [7J containing
99m-Technetium were simulated on a 128 x 128 pixel grid with 2 x 2 mm pixels. The digital phantoms are
128 3 grey value images with 2 x 2 x 2 mm voxels. The small voxel size for simulation (compared to 4 mm
11997 International Meeting on Fully 3D Image Reconstruction
1341
voxels and projection pixels for reconstruction) was used to partly simulate the continuous character of
real activity distributions. Before reconstruction, the projection images were collapsed to 64 x 64 images
with a pixel size of 4 x 4 mm. The SPECT simulator [8] projects each voxel with a distance and object
shape dependent PSF [9]. Primary and scatter data can be stored in separate files. The cylinder (diameter
222 mm, height 300 mm) contained 20 spherical cold spots with a diameter of 20 mm. The rotation radius
of the detector was 130 mm for both the cylinder and the brain study. The number of projection angles
was 120.
The noise free simulated projection data were scaled to the desired number of total counts, and Poisson
noise was generated using these projection values as the means. For the cylindrical phantom calculations
were made at noise levels corresponding to 20, 40 and 80 million counts in the projection data. In the case
of the brain phantom, the total number of counts summed over all projections were 6, 12, and 24 million
counts.
For the ISE method, the amount of noise in the scatter estimate was calculated by considering the
number of counts in the lower energy window of the Dual Window data (scatter window 92-125 keV) [1].
To obtain statistically significant differences between the contrasts, the average contrast of 80 spheres was
determined.
An adaptation of the ML-EM algorithm [10] was used. Details of our implementation of the used PSF
model in ISM combined with a rotating projector back-projector can be found in [6]. In case of NSC, ISR,
ISE and ISE-NF only the primary part of the model was included in the PSF. The scatter background in ISE
and ISE-NF was included in the denominator of the algorithm. To avoid prohibitive long reconstruction
times all reconstructions were accelerated by using 15 subsets of projections (the Ordered Subset EM
algorithm [11]).
Several image metrics were used to quantitatively evaluate the scatter compensation methods. Cold
region contrast (C), used for evaluation of cylinder data, is defined by G = I~~~I where 1 is, the. average
activity in a cold sphere and b is the average activity in a region with constant background activity. The
noise. in a background region was expressed in the normalized standard deviation (NSD). For a global
measure of accuracy of the brain reconstruction the sum of squares of differences (SSD) wa.s calculated.
The SSD is defined as the sum over all voxels of the squared pixel intensity differences between the
reconstructions of the brain and the digital brain phantom. Before the SSD is calculated, the total number
of counts in the reconstruction is normalized to the total number of counts in the brain phantom. For
further quantitative assessment, image profiles through reconstructed images are shown. The profiles used
for this investigation were taken through slices with a width and thickness of 4 mm (Le. one pixel).
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RESULTS I. Cylindrical phantom studies. For comparison of contrast at equal noise level rather than
at equal iteration number, contrast to noise (CTN) curves (Figure 1, left) were generated. This enables
comparison of average contrast of 80 spheres at equal back-ground noise level in terms of NSD. In the
right frame of figure 1, contrast for each method is plotted as a function of total counts in the projections
for an NSD of 0.2. The curves show that ISR is only slightly superior to ISM, and ISM is superior to
ISE even when a noise free scatter estimate is used (ISE-NF). All scatter compensation methods improve
CTN-curves.
Images and profiles at equal NSD are shown in Figure 2 for the 40 million counts data set. Note the
extremely small difference in shape between ISE-NF and ISM. Although shapes of the profiles of ISE-NF
and ISM are very siII,lilar, the contrast is slightly higher in ISM which explains the better CTN curves of
ISM. In general, the differences between the scatter compensation methods in terms of CTN-curves are
significant, but the differences in the presented images are not spectacular.
II. Brain phantom studies. The sum of squares of differences (SSD's) between pixels in the phantom
and the reconstructed images were calculated as a function of iteration number. Figure 3 shows that at
lower iteration numbers the SSD decreases. After more iterations, the SSD starts to increase, mainly
because noise is dominating the image.
The right frame in Figure 3 shows the effect of Gaussian pre-reconstruction filtering of the scatter
estimate on, the SSD. The curves for blurred scatter estimates are for a kernel width resulting in the
smallest SSD. Only the curve for the optimal kernel (the one which reaches the lowest SSD) was selected
for the graph. The lowest SSD is obtained by ideal scatter rejection, at all count levels investigated. ISM
always results in a lower SSD than ISE, and in an equal or lower SDD than ISE-NF.
11997 International Meeting on Fully 3D Image Reconstruction
135\
40 Million Counts
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Total Counts in Projections
Figure 1: Comparison of NSC, ISE, ISE-NF, ISM, and ISH.. Left) Contrast to noise curves of cold spots in the
cylinder data for 40 million counts. Right) Comparison of contrast, reached at equal back-ground noiRe level (NSD
0.2) for 20, 40 and 80 million count data sets.
=
NSC
C = 0.55
ISE
Figure 2: Contrast (C), images and profiles of the cold sphere phantoms, displayed at equal Normalized Standard
Deviation (NSD = 0.2) in a background region, for 40 million counts projection data. From left to right: NSC, ISE,
ISE-NF, ISM, and ISR.
CONCL US IONS and DISCUSSION The complete rejection of scatter (ISR), a$ can be approximated by systems with an extremely high energy resolution and a narrow energy window, is superior to
all compensation methods considered in this paper. The theoretical assumption was made that exactly the
same amount of primary photons were acquired by ISR, as in a 20% wide energy window of a conventional
gamma camera system with an energy resolution of 11%. In practice however, a semiconductor detector
has a much lower count efficiency than a NaI gamma-camera. ISR is better than removing a noise free
scatter estimate (ISE-NF) because subtracting the noise free scatter estimate will not remove the noise due
to the scattered photons and the compensated data will be noisier than data that was acquired without
scatter.
The use of an accurate scatter model during reconstruction (ISM) is superior to the use of an accurate
scatter estimate, even in the case that the scatter estimate is noise free (ISE-NF). One of the practical
drawbacks of ISM are long reconstruction times. Over the last couple of years, several methods have been
developed to reduce these reconstruction times. When some of these methods are combined speed up
factors of about two orders of magnitude (e.g.[12]) can be reached.
The accuracy of reconstruction in ISM is determined by the accuracy of scatter models. Investigations
to further improve scatter models are still going on. In addition, the accuracy of both attenuation and
scatter modeling during reconstruction is influenced by the accuracy of attenuation maps. This influence
11997 International Meeting on Fully 3D Image Reconstruction
1361
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Figure 3: Left frame: Comparison of the SSD as a function of iteration OS-EM with 15 subsets, for NSC, ISE,
ISE-NF, ISM, and ISR (12M count level). Right frame: Effect of Gaussian filtering of the scatter estimate for ISE
on MSE measured from the Hoffman phantom. ISEblurO.7 refers to a blurring kernel with a standard deviation of
0.7 pixels.
[J
is currently being investigated.
Subtraction based methods (ISE and its, in practice impossible, noise free limit ISE-NF) gave the worst
results of all compensation methods considered. Important practical advantages of ISE compared to ISM
is that ISE is very fast and inherently compensates for scattered photons originating from sQurcesout of
the field of view. The disadvantages are already considered in the introduction. In practiceth&':'~!nount of
noise in scatter estimates are often higher than for the ISE method presented here.
In this paper, scatter was included in the denominator of the OS-EM algorithm. This has already
been performed by others for the ML-EM algorithm. Inclusion of a noise free scatter estimate, results in
convergence to an ML solution at high iteration numbers. We do not expect that a "correct" algorithm for
including a noisy scatter estimate as an alternative for ISE like implemented herein will result in different
conclusions. This is because even in the mathematically correct case of ISE-NF, ISM results in equal or
lower errors than ISE-NF.
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References
-1
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II
L)
[1] R. J. Jaszczak, K. L. Greer, and C. E. Floyd, "Improved SPECT quantification using compensation for scattered photons,"
J. Nucl. Med., vol. 25, pp. 893-900, 1984.
[2] M. A. King, G. J. Hademenos, and S. J. Glick, "A dual photopeak window method for scatter correction," J. Nucl. Med.,
.
vol. 33, pp. 605-612, 1992.
[3] M. Ljungberg, M. A. King, G. J. Hademenos, and S. E. Strand, "Comparison of four scatter correction methods using
Monte Carlo simulated source distributions," J.Nucl.Med., vol. 35, pp. 143-151, 1994.
[4] C. E. Floyd, R. J. Jaszczak, K. L. Greer, and R. E. Coleman, "Inverse Monte Carlo as an unified reconstruction algorithm
for EOT," J.Nucl.Med., vol. 27, pp. 1577-1585, 1986.
[5] E. O. Frey and B. M. W. Tsui, "A practical method for incorporating scatter in a projector-backprojector for accurate
scatter compensation in SPECT," IEEE Trans. Nucl.Sci., vol. 40, pp. 1107-1116, 1993.
[6] F. J. Beekman, C. Kamphuis, and M. A. Viergever, "Improved quantitation in SPECT imaging using fully 3D iterative
spatially variant scatter compensation," IEEE Trans. Med. 1m., vol. 15, pp. 491-499, 1996.
[7] E. J. Hoffman, P. D. Cutler, W. M. Digby, and J. O. Maziotta, "3D-phantom to simulate cerebral bloodflow and metabolic
images for PET," IEEE Trans.Nucl.Sci., vol. 37, pp. 616-620, 1990.
[8] F. J. Beekman and M. A. Viergever, "Fast SPECT simulation including object shape dependent scatter," IEEE
Trans.Med.Im., vol. 14, pp. 271-282, 1995.
[9] F. J. Beekman, E. G. J. Eijkman, M. A. Viergever, G. F. Borm, and E. T. P. Slijpen, "Object shape dependent PSF
model for SPECT imaging," IEEE Trans.Nucl.Sci., vol. 40, pp. 31-39, 1993.
[10] K. Lange and R. Carson, "E.M. reconstruction algorithms for emission and transmission tomography,"
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[11] H. M. Hudson and R. S. Larkin, "Accelerated image reconstruction using ordered subsets of projection data," IEEE
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[12] C. Kamphuis, F. J. Beekman, M. A. Viergever, and P. P. van Rijk, "Accelerated fully 3D SPECT reconstruction using
Dual Matrix Ordered Subsets (abstract)," J. Nucl. Med., vol. 37-5, p. 62P, 1996.
11997 International Meeting on Fully 3D Image Reconstruction
1371
Inversion of the Radon transform in two and three dimensions using
orthogonal wavelet channels
Eric Clarkson, Department of Radiology, University of Arizona, Tucson, AZ
85724
Abstract
Using very general properties of the continuous wavelet transform and its
adjoint it is possible to derive a class of inversion formulas for the Radon transform
in two and three dimensions. To implement the procedure a set of orthonormal
wavelet channels which span the object space is required. Other than these two
conditions, there are no other special properties that the wavelet functions need
to have. In each dimension there are both real space and Fourier space versions
of the inversion method. In what follows upper case letters will indicate Fourier
transforms of the corresponding functions in lower case.
For the two-dimensional case suppose that f(x, y) can be expanded in the
following form:
00
f(x, y) =
L fn(x)h~(y).
(1)
n=l
With the hn being fixed wavelet functions and the in variable functions depending
on f. Let l(c/J,p) be the line passing through (pcosc/J,psinc/J) and perpendicular to
(cosc/J)i + (sinc/J)]. The Radon transform of f is:
A(¢,p)
=J
fdl.
(2)
Jz(¢IP)
If the equation for this line is written as y = x~t, let AO(S,t) = A(¢(S),p(s,t)).
The Radon transform now looks like this:
.110(8,
t) =
~ ill.r fn(x)h~ (x-t)
V(;2+1
82 ~
-s- dx.
(3)
We may write this as
(4)
where Wn = W {fn, hn } ,the wavelet transform of in with respect to hn. Let
W(8, t)
=
lJ11 .110(8, t).
11997 International Meeting on Fully 3D Image Reconstruction
1381
The wavelet inner product is given by
,-)
lj
(6)
We will say that the wavelets hn form orthonormal wavelet channels if (hn, hm)w =
onm, in which case the functions in can be recovered from w via the adjoint of the
wavelet transform: in = W T {w, hn }. Explicitly, we have
fn(X) =
[]
11W(S,t){£~ (x~t) ~:dt.
(7)
This follows from the fact that W T {W {In, hn}, hm} = (hn' hm)w in. In terms of
Ao, in is given by
[]
in(x)
[]
=
11
lit
lit
(
x-t)
Ao(s,t)hn -.s
~dS
- 2 - 2 dt .
Simpler expressions result if write the equation for the line as y
Al(m, b) = Ao(m- 1 , -bm- 1 ). Then
Al(m, b) = yfl + m21 f
[]
(8)
s +ls
= mx + b.
(x, mx + b) dx
Let
(9)
is the Radon transform in these parameters. To recover in we now have
---I
[ _1
(10)
Therefore we may write an inversion formula for the Radon transform:
[J
I(x,y)
L]
=
r r Al(m,b) [fhn(mx+b)h~(Y)] V~
~dmdb.
Jilt Jilt
(11)
n=l
vfsT.
In the Fourier domain we have Wn(s, k) = Fn(k)H~(sk)
For the different
forms Aq of the Radon transform of I, the function Aq is the Fourier transform of
Aq with respect to the second coordinate. The recovery of Fn from Ao is given by
(13)
1.
H1
I
I
LI
11997 International Meeting on Fully 3D Image Reconstruction
1391
while for Al we have
I' A \m,
(
_~)
1-7" (~) • ~ dm
m .un
m V~'m"
P. (k) -
(14)
Jll'(~~l
-n,"",-
For this procedure to work we must be assured that, for the chosen set of orthonormal wavelet channels, an expansion of the form in equation (1) is always
possible for an arbitrary object function f. We have constructed a set of orthonormal wavelet channels that are Haar-like in Fourier space and which satisfy this
condition when the object functions are square integrable. An explicit inversion
formula exists in this case for the Fourier domain.
For the three-dimensional Radon transform a similar procedure is possible.
We write
00
L lij(z)hi (x)hj (y).
(16)
VI .+a
. ab +b2 L Uij(a, b, c).
(17)
I(x, y, z)
=
The integral of I on the plane z =
i,j=l
ax + by + c is
2
00
A3(a, b, c) =
i,j=l
=
and wj =
If Wij = W {/ij, hi}, wij(t) = Wij(a, t) and ufj(b, c) = Uij(a, b, c), then ufj
W {wij, hj } . To retrieve lij from A3, let ua(b, c)
W T {u a , hj }. Then
=
1_,_:2b+b2A3(a, b, c)
00
wj(t)
= L wfj(t)
(18)
i=l
(19)
with
fij(Z) =
1
R3
da~
(20)
kij(z, a, b, C).,3(a, b, C)2 b2 dc,
a
If we Fourier transform A3 with respect to c the corresponding formula is
Fij(k) =
1
R2
Hi (ak)Hj (bk)A3 (a, b, k) ~
~~
1
1+a
2
+
- ,,-,b,'
b2 a
(21)
This method can be generalized to higher dimensions in a similar fashion.
11997 International Meeting on Fully 3D Image Reconstruction
1401
-- -------- ---
r-l
\
I
'
I
Submitted to 1997 International Meeting 'on Fully Three-dimensional Image Reconstruction in Radiology and
Nuclear Medicine. June 25-28, 1997, Pittsburgh. PA, USA.
r-l
I
j
Towards Exact 3D-reconstruction for Helical Cone-Beam
scanning of Long Objects. A New Detector Arrangement and a
New Completeness Condition.
II
[-j
Per-Erik Danielsson, Paul Edholm, Jan Eriksson, Maria Magnusson Seger
f]
Image Processing Laboratory, Dept. of Electrical Engineering
Link6ping University, S-581 83 Linkoping, Sweden
[--I
_J
State-of-the-art
[]
Exact 3D-reconstruction from cone-beam projections is now a well developed art with several efficient
algorithms [1], [2], [3]. However, most if not all of these have a severe drawback. They assume that the
whole object fits in a window limited by the available 2D-detector. This assumption is not valid in most
practical applications including medical full body X-ray tomography. While future 2D-detectors'are likely to cover the width of the patient Gust as the ID-detectors presently in use) it is not forseeablethatthey
can be extended to cover the length of the patient from head to foot.
If
cJ
[]
This is the background for the long object problem. While allowing the object to extend (indefinitely)
below and above the detector window, how should cone-beam projection data be aquired and processed
to obtain an exact or near-exact result with greatest efficiency?
[]
r-i
The helical scanning mode is an obvious candidate for data acquisition. Several authors have proposed
such systems including [4], [5] using reconstruction techniques borrowed from Feldkamp et.a!' [6].
LJ
U
fl
L_!
Exact reconstruction techniques are usually founded in completeness criteria for Radon data. For the long
object problem with no obvious origin of the object space we have found it advantageous to use the following alternative completeness criterion. Sufficient projection data for exact reconstruction of the long
object is acquired (complete data capture) if each point in the object is exposed over a rotation angle of
at least 180 degrees.
[]
[- \
I)
L_J
Q
r
Orlov's theorem
i
LJ
The proof of this sufficiency condition is based on Orlov's theorem [7] which states that if an object is
projected with a parallel beam and the set of projection angles mapped onto the unit sphere describes a
certain curve, then these projections provide complete information for synthesis of all projections taken
from orientations with projection angles corresponding to points inside the convex hull of the curve. Figure 1 provides an example. All projections represented by the shaded area are retrievable, given projections taken from the curve. The proof of the theorem follows from observations in the Fourier domain
and is omitted here for the sake of brevity.
An interesting effect is achieved when the source path is extended so that it covers two opposite points
on a diameter. See Figure 2. The convex hull contains everything inside and on great circle arcs connecting A and B. In Figure 2, two such arcs are shown denoted 1 and 2, respectively. But since A and B are
diametrically opposite, we can fill the whole sphere with such arcs and the convex hull of the curve S
is then the whole unit sphere of projections. Hence, all projections are retrievable from data acquired
along a curve with diametrically opposed endpoints and therefore this is a complete data capture.
I
I
\
_oj
11997 International Meeting on Fully 3D Image Reconstruction
1411
Figure 1
Figure 2
v
s
Detector
Figure 3
An optimal detector arrangement.
Figure 3 shows a fixed source-detector system inside which a long cylindrical object rotates with angular
velo~ity (t) and translates upwards with velocity v. The source S resides on the outer rim of the object space
and the detector is "glued" to its 'outer surface. Sideways; the detector is extended to cover up to one full
turn of the cylinder, less if the object does not extend all the way out to the cylinder rim. The height of
the detector is 2:rc ~. and it is limited in the vertical direction by two slanted lines having a slope which
exactly fits the pitch of the helical movement. Unwrapped and portrayed in the plane of the sheet, the
detector area attains the shape shown in Figure 4.
11997 International Meeting on Fully 3D Image Reconstruction
1421
2n~
[-
co
l
-n12
[I
[J
[]
Figure 4
[I
[]
s
[J
[]
[]
~
Figure 5
The fan-beam angle y is defined in Figure 5 which shows the system from above. A line Ql - Q2 is traversing the object space and is shown in two specific positions. Both positions are such that one end of the
line coincides with a ray from the source. Such a line is called a 1t-line.lts projection enters and exits the
detector at two points In and Out indicated in Figure 4 and 5. In between, all points on the line have rotated
an angle 1t with respect to the source position. Hence, all points including Ql, Q and Q2 have been fully
exposed, and since this line is chosen arbitrarily all points in the object space have been exposed during
an angular interval 1t. The detector arrangement of Figures 3 and 4 provides us with complete and non-redundant projection data for the whole object of unlimited length.
rl
I
I
LJ
11997 International Meeting on Fully 3D Image Reconstruction
1431
Conclusions
The above detector arrangement may be implemented in various ways. Thus, it is not mandatory to place
the detector physically on the cy linder in Figure 3. It may be arranged in any manner, for instance as pieceways fiat panels at any distance from the rotation axis, as long as the detector data are limited to the window described by Figure 3 and 4. It should also be noted that the full width from y :::::
to r = is
not necessary for objects which have a diameter less than 2R. Fot instance, if the object has the width
R the detector area of Figure 4 may be limited to the section from y == -1C / 6 to y == 1t/6 .
-I
I
The exploitation of the above detector arrangements for exact or near-exact reconstl1lction will require
appropriate preweighting and filtering followed by backprojection. Several options for reb inning is also
available. The results of this ongoing effort is omitted in this abstract for the sake of brevity but will be
reported in the full length paper.
Acknowledgement
The support for this work from Swedish Council for Engineering Sciences, grant No. 95-470 is gratefully
acknowledged.
References
[1]
P. Grangeat. "Mathematical framework of cone beam 3D reconstruction via the first derivative
of the Radon transform", Mathematical Methods in Tomography, G.T. Herman, A.K. Louis, F.
Natterer (eds.), Lecture notes in Mathematics, Springer Veriag 1991.
[2]
M. Defrise, R. Clack, "A cone-beam reconstruction algorithm using shift-invariant filtering and
cone-beam backprojection", IEEE Transactions on Medical Imaging, Vol. 13, No.1, pp.
186-195, 1994.
[3]
C. (Axelsson) Jacobson, P.E. Danielsson, "3D Reconstruction from Cone-Beam Data in
O(N310gN) time", Physics in Medicine and Biology, Vol. 39, pp. 477-491, 1994.
[4]
O. Wang, T.H. Lin, P.C. Cheng, D.M. Shinozaki, "A general cone-beam reconstruction algorithm", IEEE Transaction on Medical Imaging, Vol. 12, pp. 486-496, 1993.
[5]
S. Schaller, T. Flohr, P. Steffen, "A new approximate algorithm for image reconstruction in conebeam spiral CT at small cone-angles", IEEE 1996 Medical Imaging Conference, Anaheim, Nov.
3-9,1996.
[6]
L.A. Feldkamp, L.C. David, J.W. Kress, "Practical cone-beam algorithms", Journal o/Optical
Soc. Am., Vol. A6, pp. 612-619, 1984.
[7]
S.S. Orlov, "Theory of three-dimensional reconstruction. I. C()nditions for a complete set of projections", Sov. Phys. Crystallogr. pp. 511-515, 1975.
[8]
P.E. Danielsson, "Invention regarding optimal detector for tomographic 3D-reconstruction of
long objects", (in Swedish), Swedish patent application, January 1997.
11997 International Meeting on Fully 3D Image Reconstruction
1441
3D efficient parallel sampling perturbation in tomograp11Y
Laurent Desbat
TIMC-IMAG UMR CNRS 5525,
lAB, faculte de Medecine,
38706 La Tranche France
e-mail: [email protected]
1
Introduction
in 3D tomography based on the perturbation
of the Interlaced Hexagonal sampling scheme.
In [3], efficient sampling conditions have been
obtained for the parallel 3D X-ray transform
when the 2D detector trajectory is a circle
around the measured object (see figure 1):
g(</>, s, t)
=
1:
f(sO +te3 + u()du,
[]
[]
[J
[]
E [-1,1], t E [-1,1].
Efficient
sampling
condi-
tions
(1) Efficient sampling lattice schemes are based
with ¢ E [0, 27r], ( = (- sin ¢, cos</>, O)t, e =
(cos¢,sin</>,O)t, e3 = (0,0, l)t, Y = sf) +
te3, s
2
on sam pIing conditions due to Petersen and
Middleton [7]. We consider the n dimensional sampling on sets {Wk, k E LZn} gen-·
erated by the non-singular matrix W. vVe say
that the set I( contains the essential support
of j if fertK Ij(~)ld~ is negligible compare to
feEJRn Ij(~) Id~. The Petersen and Middleton
sampling condition for the lattice {Wk, k E
LZn} is that the sets I( + 27r W- t k, k E LZ n
have no overlapping (sampling errors can be
driven by fertK Ij(~) Id~, see [6]).
The essential support of the Fourier transform of the 2D Radon transform has been given
--~
JS in [8, 6]. This result is extended to 3D tomography for (1) when f is supposed to be
essentially b band limited [3,2]. Let fJv(O", T) =
(27r)-1 f;7r fJ(</>, 0", T)e-ivrPd¢, where 9(</>,0", T) is
the 2D Fourier transform of (1) according to
the variables sand t. The essential support of
Figure 1: Definition of the parameters </>, s, t of 9v(0", T) is shown to be contained in
the 3D measurement parallel geometry.
1(3 = { (v, 0", T) E LZ X IR X IR; 10"1 < b,
A given efficient scheme is not always easIvl < max (10"1/'!9, (1/'!9 - 1) b) ,
ily feasible with an existing measurement tool.
ITI < c(b,O")},
This is the reason why, the study of perturbations of the interlaced scheme have been pro- where < '!9 < 1 arbitrary close to 1 and
posed in 2D and applied to a process tomograb if lal ~ O"1J,b = max (1., (1 - '!9) b)
phy problem in oil industry [1]. The purpose of c(b,O") =
Vb 2 - 0"2 if O"1J,b < 10"1 < b
{
this work is to study new efficient geometries
otherwise.
°
°
r
1
lJ
11997 International Meeting on Fully 3D Image Reconstruction
1451
The sampling scheme generated by the matrix
WIll
= ~
b
2fJ'
[
0]
0
-fJ'
1
0
0
-1/J3" 2/J3" '
with 1)1 = fJ / (2 - fJ) < fJ < 1 (chosen close to
1 such that b/fJl E IN), satisJies the sampling
condition (it can be shown that the sets 1(3 +
21fWjj}k, k E ZZ3 are mutually disjoint, see [2])
for the X-ray transform of an essentially bband-limited function for
1)
>
VI - (-13 - 1)2
and b suJIiciently large. This scheme is ~~
more e·fficient than the best 3D rectangular
grid generated by:
Ws
1)
=?:
0 0]
3.1
Faridani's generalized san1pling
theorelTI
Because efficient sampling schemes are not always easily exactly feasible, see [1], we are interested by the study of its perturbations. An
essential tool for this purpose is the theorem
proposed by A. Faridani [4, 5J. We present here
only the simplest version of the theorem. This
theorem allows to consider sampling on unions
of shifted (too coarse) lattices U~=O A,. with
A,. = {a,. -I- Wk, k E LZ'n}j we suppose in the
following that ao = (0, ... , 0) t • The overlapping of]( by the sets [( +21f(W- 1 )tk, k E 2Z n
yields a decomposition of ]( == Ur=:l ](1, where
\:Il E {1, ... ,L}, 3{kl,O, ... ,kl,MI-l} C 7Z n ,
1 :; MI < 00 such that Ve E [(I, k E
zzn :
(~-21r(W-lrk)
(2)
0 1 0
[
bOO
1
¢:?
EJ(
(4)
k E {kl,O, ... , kl,MI-1}
Moreover, it can be easily shown that the IH It turns out that with IvI == max~1 M/
scheme is the union of two standard schemes (the maximum number of overlapping of ](),
generated by the diagonal matrix:
and a r E lR n be chosen so that \:Il, l =
1, ... , L, the following linear system (with
D
= ~ [20gl
~ ~
o 2/-13
kl,o == (0, ... , O)t):
]
~M-1
L.w=O
f31r ==
1
for j == 1, ... , Ml - 1 :
1
2i1r
L...,r=O f31r e- {(W- Q'r),kl J)lR n
more precisely
~M-l
I
{WIHk, k E LZ
{ D k, k
E
3
(3)
}
yz3} U{x + D k, k E ZZ3} ,
8
(5)
-
0
- ,
have a solution for (3~ E <C, r == 0, ... , M - 1,
then for f E S(1Rn) we have a Fourier interpolation form ula
with the shift
M-l
Swf(x) ==
L 2:
f(ar+Wk)gr(x-ar-Wk)
(6)
where
L
3
Efficient sampling perturbations
The purpose of this section is to show that
the IH scheme can be generated with coarse
grids, see eq. (8), and then to provide a way to
study perturbations of this scheme based on a
generalized sampling theorem presented in section 3.1 and on proposition 1 of section 3.2.
gr (y) == (21f)-n~21 det WI Lf3~XKI (-y).
1=1
The interpolation error can be driven by
~el~K IJ(e)lde. This theorem can be generalized for function periodic in a first set of variables (see [4]) for W feasible, Le., such that
Ted E W2Zn, where T is the period of f in
its dth variable. A general theorem for Locally
Compact Abelian groups is given in [5]
11997 International Meeting on Fully 3D Image Reconstruction
1461
3.2
IH scheme perturbation
thus
As in 2D tomography [1] the study of the
perturbation of the 3D IH scheme is based
Let us conon the following remarks.
sider the non-singular diagonal matrix Dp =
diag(Pl,P2, ... ,Pn) with Pd E IN,Pd =f=. O. Then
Vk E LZ n ,3!(q,r) E LZnxyzn,k = Dpq+rwith
Vd = 1, ... , n, 0 :::; rd < Pd. Thus if we denote
II = {(r1, 1'2, ... , rn)t, 0 :::; rd < Pd}, then
](e3
+ 21f D- 1D;l yz3
(10)
U (](e3 + 21fVVn;yz3 + 21fD-ID;lr)
U U(](e3 + 21fWjJ}(LZ3 + 1/2 (0,1, O)t)
rED
We can conclude from the last equality that the
sets
](e3
21f n- l D;lyz3 can be reorganized
yzn =
Dpyzn l'
rED
in a 2PlP2P3 tiling of the LZ X ]R 2 space. Thus,
the maximum number of overlapping M of the
Thus for example, the IH sampling scheme
set ](e3 (in fact of the whole LZ X ]R2 space)
(n = 3) can be written from eq. (3)
is 2plP2P3. As det DDp = 2PlP2P3 det WIH,
3
(8) as efficient schemes as the IH scheme can be
{WIHk, k E LZ }
produced with DDp.
N ow the choice of the shift a r in the case of
3
{Dr+DDpk,k E LZ })
=
perturbation of the IH scheme can be reduce
rED
to the study of perturbations of unitary matrix
{xs+Dr+DDpk, k E LZ3}) , by the next proposition.
U
+
+
(7)
(u
[J
U(U
rED
[J
~l
and thus can be produced by 2PlP2P3 shifted
coarse grids generated by the matrix D D p ,
with the 2P1P2P3 shifts a~, E E {+, - }, l' E II
given by :
[]
a~
Dr,
a; =
Xs
(9)
+ Dr = D (r + 1/2 (1, 1, l)t) .
The Faridani's theorem allows us to consider
sam piing on coarse grids D Dp and thus perturbations of the sampling IH schemes. As
the sets ](3
27rWn;k, k E LZ3 are mutually disjoint and in order to simplify the discussion, let us consider a set ]<e3 ~ ](3 such
that ](e3 21f W.u}k, k E yz3 tiles the yz X 1R 2
space. We have from (7) (n = 3), D;l LZ3 =
UrED LZ3
D;lr, and thus
+
+
[]
+
[]
21f D- 1D;1 LZ3
U21f n- LZ3 +27r n- n;lr.
1
II
1
rED
We can also directly verify that
21f D- 1LZ3
=
wci (yz3 + 1/2 (0, 1, O)t)
U27r Wn; yz3
21f
Proposition 1 Let us denote by Ul, 1
1, ... ,L the matrices of the linear systems (5),
i.e., Ul-J,r e = e-2i7r((W-IQrE),kl,i)IR2. In the case
of 2PlP2P3 shifted coarse grids generating the
interlaced scheme, with a r given by (9), Ul =
U do not depend on land 1/ yf2P1P2P3 U is unitary.
3.3
Application
In medical imaging, each projections contain
hundred thousands of data so that only very
regular sampling schemes are to be considered.
However, in industrial tomography, the measurement tools can be very simple and limited [1]. If we suppose that we have only
2 detectors (and one source of 'Y ray for example) then we can consider schemes generated by the disposition of the 2 detectors
shown in Fig. 2. Let us denote h = 1f /b and
7r / P = 1J'h where P is the number of projection on [0,1f]. For the projection angles
4jl1f / P, it E yz the system is translated on
the positions (2hj2, 2h/V3j3)t, (j2, j3) E yz2 in
the plane (s, t). Note that the projection angle
(4j1 2)7r / P has been also scanned but with
a step h' in the direction s. The next angle of
+
II
I
I
(!
11997 International Meeting on Fully 3D Image Reconstruction
1471
projection is (4jl + 1)7r I P, the same translation are done in the plane (s, t) plus a shift of
(h, h/V3)t. The projection angle (4jl + 3)7r / P
is at the same time scanned so that the next
considered angle takes the form 4h 7r / P. Because hi h' is generally not rational, the periodicity of the scheme in the direction s is only
possible if we consider that the next sampling
point in this direction is outside the support
of the measured function (here the unit cylinder). This yields coarse grids generated by the
matrix
7r/b
4'0!9' 2P20
[ o
0]
0
= DpD,
2/-/3
=
(2, P2, 1)t and P2 E .7.Z such that
2P27r /b > 2. As seen in the previous secwith p
2 detectors
h'=h cos 21liP
angle 2n1P "
;
1 source
Figure 2: 3D sclleme generated witll 2 Detectors for measuring a cylinder. Only 3 positions
of tlle measurement system are visualized in a
cross section (translations in the direction s of
step 2h). The system is also translated in tlw
orthogonal direction to this cross section.
tion 4P2 grids generated D(2,P2,l)D should
be sufficient to sample g( ¢, s, t). The 4P2
shifts corresponding to the perturbation of
the IH scheme presented before are given [4] A. Faridani. An application of a multidimensional sampling theorem to computed
by 7r/b(0,2j2,0)t, 7r/b('!9',2j2 + 1,1/v'a)t,
tomography.
In AMS-IMS-SIAM Confer7r/b(2'!9',2j2COS(rJ'7r/b),0)t, 7r/b(3'!9', (2j2 +
ence on Integral Geometry and Tomogra1) cos({)'lI-jb), 1/V3)t,j2 ::::: 0, ... ,P2 - 1. The
volume 113, pages 65-80. Comtempophy,
proposition 1 allows us to reduce the study the
rary Mathematics, 1990.
correctness of such a scheme to the inversibility of perturbated unitary matrices.
The
[5] A. Faridani. A generalized sampling thelarger is P, the smaller is the perturbation in
orem for locally compact abelian groups.
the sampling scheme. As in 2D, from relative
Math. Comp., 63(207):307-327,1994.
low values of P this new efficient scheme can
be shown to be correct, see [1].
[6] F. Natterer. The Mathematics of Comput-
erized Tomography. Wiley, 1986.
References
[1] L. Desbat. Efficient sampling on coarse
grids in tomography. Inverse Problems,
9:251-269, 1993.
[2] L. Desbat. Echantillonnage parallele efficace en tomographie 3D. CRAS serie 1,
1996. accepte pour publication.
[7] D.P. Petersen and D. Middleton 1962.
Sampling and reconstruction of wavenumber-limited functions in N-dimensional euclidean space. Inf. Control, 5:279-323,
1962.
[8] P.A. Rattey and A.G. Lindgren. Sampling
the 2-D Radon transform. IEEE Trans.
ASSP, 29:994-1002, 1981.
[3] L. Desbat.
Efficient sampling in 3D
tomography:
parallel schemes.
In
P. Grangeat and J .L. Amans, editors,
Three-Dimensional Image Reconstruction
in Radiology and Nuclear Medicine, pages
87-100. Kluver Academic, 1996.
11997 International Meeting on Fully 3D Image Reconstruction
1481
[:
A
Proof of proposition 1
Thus
= e-2i7r(r,D;lml,i) and Ui
Ui.
},r+
},r
_ =
e-2i7r(r,ql,j+D;lml,i)-i7r(E~=1 (QI,j)d+(D;l ml ,j)d).
Proof: We first establish the form of the kl,j
of (5). We first write 3! (ql,j, ml,j) E ZZ3 X ZZ3 thus the UI-},r do only depend on 1ni ,J' and
i7r
such that kl,j = Dpql,j + ml,j with 0 S; ml,jd < e- E~=1 (ql,j)d. As the kl,j can only take the
Pd, d = 1, ... ,3. Thus we have
2PIP2P3 forms DpQi,j + 17~I,j, mi,j E IT (PIP2P3
possibilities) and E~=l ql,jd odd or even (2
27r D- 1D;l kl,j
possibilities), we can conclude that, apart
1
from
a simple permutation of the rows (except
27r D-1ql,j + 27r D- D;lml,j
the
first
row), the matrices UI do not depend
b (ql,j1 j{J!, ql,j2' ql,jJ t + 27r D- 1D;lmld on t. Moreover,
'
f
11t .'
[]
[]
o
[]
Now we can consider the two following alternatives ql,j2 - ql,j1 - qld3 is even (or equivalently E~=l ql,jd is even) or is odd. In the first
case (E~=l ql,jd even) we can make the change
of variable ql,j2 - ql,jl - ql,j3 = 201,j2' Ql,jl
Ol,jl' ql,j3 = 01,j3 and we get
1
27rD- Qz,j
2. if j
i- j'
• if
m/,j
i- mi,j'
then we have
UI}' ,rIO
VI.,} ,r = 0
10
rEI1,tE{ +,-}
= 27rWnJ- oZ,j
because 2: rEI1 e- 2i7r (r,D;l m /,i)
• if m/,j = mi,j' then 2:~=1 QZ,jd and:/'
2:~=1 Ql,j'd have different parities,
thus
In the second case (E~=l QZ,jd odd) we can
make the change of variable Ql,j2 - QI,j1 - ql,i 3 =
20l,J2 + 1, Ql,jl = 0U 1 , Ql,j3 = 0l,j3 and we get
[]
[]
o
[j
[]
Thus we have
Thus as each of the 27rD-ID;lkld,j
1, ... ,2PIP2P3 belongs to only one of the
2PIP2P3 tiling (10),we can conclude that for
the PIP2P3 ml,j E II, kl,j can be written in two
forms kl,j = Dpql,j ml,j with with E~=l Q/,jd
odd Of even.
Now we can conclude by the following remark:
L
E
E
== PIP2P3-PIP2P3 = O.
o
+
if
if
Ulj,r eUlp,re
rEI1,tE{ +,-}
+
=-
=
[]
[]
D
[
I I'
, I
11997 International Meeting on Fully 3D Image Reconstruction
1491
Estimation of Geometric Parameters for cone beam Geometry
Yu-Lung I-Isieh, G. Larry Zeng and Grant T. Gullberg
Departlllent of Radiology, University of Utah, Salt Lake City, UT 84132, USA
Abstract
Mechanical and electrical shift of the gantry system can result in bm:ring and donut-like artifacts in the
reconstructed image (without any compensation). A robust estimation method has been reported for fan beam
geometry by using a phantom with five point sources arranged as a cross. The strong constraints among the
colineal' and co-orthogonal relationships for the five point sources guarantee the convergence of a nonlinear
estimation algoritlun to the true geometric parameters, regardless of the initial guess of the parameters. For cone
beam geometry, each point source can have sinograms in both transaxial and axial directions instead of only in
the transaxial direction, as is the case for fan beam geometry. This makes it more difficult to estimate the
geometric information for cone beam geometry. This study used both a two-dimensional, five-point-source
phantom and a three~dimensional, seven-point-source phantom to estimate the geometric parameters. The threedimensional; seven-point-source phantom has the point sources arranged in the shape of a diamond shape, which
establishes strong colinear and co-orthogonal constraints. For the five-point source calibration algorithm, the
initial value of the radius of rotation has to be chosen carefully to estimate the parameters of a cone beam
geometry. The stronger constraints in the seven-point-source phantom are expected to define the solution more
uniquely.
Background
A mechanical or electronic shift of the camera system can produce artifacts in the reconstructed image. For
example, a shift of the rotation axis in the transaxial direction call result in blurring and donut-like artifacts. For
parallel-beam geometry, this shift of the rotation axis can be easily estimated by halving the sum of the distance
between a point source measured in two opposite projection views. The geometric parameters are more difficult
to estimate for convergent collimators. Gullberg et. al [1, 2] proposed methods that used a single point source and
the Marquardt algorithm to estimate the geometric parameters for fan beam and cone beam geometries.
Parameters were estimated by fitting the analytical function for the sinograms of the point source to the measured
sinogram. For fan beam geometry only the difference between the analytical relation and the measured sinogram
in the trans axial direction needs to be minimized, and for cone beam geometry those in the trans axial and axial
directions should be minimized. Recently, Rizo et. al [3] applied this method to the calibration of a multiple-head
cone beam SPECT system. However, the initial guess of the geometric parameters can bias the estimates of the
geometric parameters if only a single point source is used. Hsieh [4] showed with computer simulations that the
objective function can converge to zero when using a single, two, or three point sources, but with non-unique
results that depend upon the initial guess of the solution. To solve for a unique solution to this nonlinear problem,
Hsieh et al. [4, 5] proposed a more robust algorithm to estimated the geometric parameters of a fan beam
geometry that used a phantom consisting of five point sources arranged in a cross. The focal length, the rotation
radius, the displacement of the rotation axis, and the point source locations can be accurately estimated regardless
of the guess as to the initial solution. The five transaxial sinograms with colinear and co-orthogonal relationships
between the five point sources make for a unique solution.
This study extends the previous approaches to develop a new robust estimation algorithm for cone beam
geometry. The goal is to estimate the true geometric information and the locations of the point sources uniquely
and accurately.
Theory
Two phantoms are proposed: One with five point sources arranged in a cross and another with seven point
sources arranged in a diamond as shown in Fig. 1.
11997 International Meeting on Fully 3D Image Reconstruction
1501
[1
r-'
The five-point-source calibration algorithm minimizes the objective function:
I
!
I
I )
5 5
,,22
~
~~
2
2
,,22
2
~2
X = AsL.,.L/~i,m-~i,m) crSi.m+A~L.,.L(Si,m-Si,m) cr~l.m+/vLLL +/v0L.,.0 ,
i=1 m
i=1 m
1=1
where AS' /vs' AV AO are the weighting coeff~cients. The first summations are the total sums of least squares
differences between the measured sinograms ~i, m and the calculated sinograms ~i, m in the transaxial direction,
and the second summations are the total sums of those least squares differences in the axial directions. The last
two terms are the sums of the least squares differences for the colinear (or midpoint) and the co-orthogonal
relationships for the point sources. To simplify the estimation algorithm, we only estimate the focal length of the
collimator, F, the circular rotation radius, R, the displacement of the rotation axis in the trans axial direction, 't,
and the pr~jected locations of the focal point on the detector plane, C and C~, as shown in Fig. 2. The symbols
~i, m and Si, m are the measured projection locations of the ith point source at the mth projection view. The
relationships
s
~.
I,m
[I
o
o
[]
= F.(-xsin(ro)+ycos(ro)-'t)+C and
xcos(ro)+ysin(ro)+F-R
S
S.
I,m
=
F·z
+c
xcos(ro)+ysin(ro)+F-R
~
are the calculated projection locations of the ith point source in the transaxial and axial directions as a function
of the geometric parameters.
The seven-point-source calibration algorithm minimizes the objective function:
7
2
X
~~
7
,,22
= AsL-L-(~i,m-~i,m)
i=1 m
3
~~
,,22
~2
crSI.m+A~L-L-(Si,m-Si,m) cr~l.m+ALL.,.L
;=1 m
1=1
3
-y2
+/V O L.,. 0
0=1
,
where the parameters are the same as described above.
The difference between the two point source phantoms, in addition to the number of points, is that one is
two-dimensional and the other is three-dimensional. A two-dimensional phantom with five point sources was
developed because a one-dimensional phantom (for example a single point source or three point sources on a
line) does not give a unique estimation for fan beam geometry. For fan beam geometry the strong constrains of
this two-dimensional phantom with five point sources on a cross can produce an unique solution. However, the
problem becomes more complicated for cone beam geometry. Thus, a three-dimensional phantom is being
investigated to determine if unique solutions can be estimated for cone beam geometry.
[I
[J
[I
[]
(a)
[I
Figure 1. (a) The two-dimensional phantom with five
point sources arranged in a cross. (b) The threedimensional phantom of seven point sources arranged in
a diamond. For the three-dimensional phantom, two point
sources use isotopes different from that of the other point
sources in order to distinguish the sino gram of each point
source.
detector plane
s'
Figure 2. The geometric parameters 't, R, C
estimated for cone beam geometry.
C~,
F to be
'1
...
~.
11997 International Meeting on Fully 3D Image Reconstruction
•
1
.~
1511
Cone Beam Calibration Using the Five ..Point"Sollrce Phantom (Methods and Results)
Five point sources filled with Tc-99m were ananged in a cross. This arrangement was positioned off the
center of rotation. The distance between two farthest point sources was set to 12 cm and that of two nearest point
sources was set to 10 cm. Another point source was positioned at the center of the intersection of these two
orthogonal lines. One hundred twenty projections with 64x64 arrays were obtained using the Picker PRISM
2000. During the first fit, the radius of rotation (R) was fixed (which could be read from the gantry system) and
all other parameters were estimated. Then using stronger the orthogonal and the linear relationships among five
point sources, all the parameters, including the actual radius of rotation were simultaneously estimated.
"'0' in the second estimation step were set to
.
Initial values are shown in Table 1. The weightings, "'~, "'~,
1, 1, 87, 42, respectively. Table 2, shows that the estimated results are close ·to the manufacturer's specifications.
The systeluatic errors (see Fig. 3) of two collimators in the axial and transaxial directions between the estimated
and lueasured sinogl'ams were within 1.0 pixel (4.67 mm). The average projection bin width was also estimated
by using the estimated distances between the sources, which were in units of bin width, and relating this to the
known point source distances, which were in units of cm.
"'v
Discussion
Although all the parameters, including the point source positions, should vary simultaneously in the
calibration process, it is difficult to generate a unique solution by using this five-point-source phantom for cone
beatn geometry. In the computer simulation study, this approach converged only to a "good" estimate when the
initial guess of the rotation radius was close to the actual radius. Although the information of the rotation radius
could be obtained from the gantry system for a useful initial guess, this procedure leaves room for further
Table 1. The initial values in cone beam calibration.
't
R
CI;
-10 b
38.5 b
50.0b
C~
F
80.0b
200b
~-
---_ ..
position
point 1
point 2
point 3
point 4
point 5
11
11.0
22.0
8.0
12.0
-15.0
~
1.0
5.0
-2.0
-6.0
2.0
l;
-10.0
25.0
31.0
9.0
13.0
Table 2. The calibrated results for two cone beam collimators.
parameters
(bin width)
bin width
(b)
't
R
C~
C~
F
manufacture
0.0 b
(0.0 mm)
38.507 b
(173 mm)
64.0b
(299 m,m)
64.0b
(299 mm)
150 b
(700 mm)
4.6693 mm
collimator 1
1.3359 b
38.7680 b
65.9503 b
64.6934 b
143.596 b
4.647mm
collimator 2
1.2095 b
39.2445 b
65.6627 b
64.1997 b
142.109 b
4.675 mm
11997 International Meeting on Fully 3D Image Reconstruction
1521
investigation.
We propose to use the seven-point-source phantom in a later study. Dual isotopes will be used to distinguish
the sinograms for each point source. Compared to the objective function of the five-point-source calibration
algorithm, two sinograms, one linear, and two orthogonal least squares functions are added. These strong
constrains may help to solve this nonlinear problem for uniquely determining the true parameters.
r~
lJ
[i
REFERENCES
II
[1]
l'
[2]
1-.1
!
Lj
[3]
fl
lJ
[4]
o
[5]
G. T. Gullberg, B. M. W. Tsui, C. R. Crawford and E. R. Edgerton, "Estimation of geometrical parameters
for fan beam tomography," Phys. Med. Biol., vol. 32, no. 12, pp. 1581-1594, 1987.
G. T. Gullberg, B. M. W. Tsui, C. R. Crawford, 1. G. Ballard and J. T. Hagius, "Estimation of geometrical
parameters and collimator evaluation for cone beam tomography," Med. Phys., vol. 17, no. 2, pp. 264-272,
1990.
P. Rizo, P. Grangeat and R. Guillemaud; "Geometric calibration method for multiple-head cone beam
SPECT system," IEEE Trans. on Nucl. Sci., vol. 41, pp. 2758-2764,1994.
Y-L. Hsieh, "Calibration of fan beam geometry for single photon emission computed tomography," M.S.
Thesis, University of Utah, 1992.
Y-L. Hsieh, G.L. Zeng, G.T. Gullberg, H.T. Morgan, "A method for estimating the parameters of fan beam
and cone beam SPECT system using five point sources", J.Nucl. Med., abstract book, vol. 34, no. 5, pp.
191, May 1992.
o
o
0.5
cU
[J
1
'j
[
point 1
&-epoint2
+......+point 3
v- _J{ point 4
.E
~
-point 1
2
+-+point 3
or --v point 4
---. point 5
G---e point
,........".'"\ . - - - point 5
~
Qj
...
"'--..,
EI;Q. .~~'-.~.~Ij~r'r~~m~
.~
o.o~~~·~'
';
j
'a
~-0.5
b
Li
-1.0
0.0
-1.0 L.......o...........-....................~................................~.......................J
0,0 60.0 120.0 180.0 240.0 300.0 360.0
60.0 120,0 180.0 240,0 300.0 360.0
angle
angle
1.0 ..............--,....................,..................................,................"T""""'.........,
-
point 1
~point2
0.5
. 0.5
[1
cU
cU
=
1
43 0.0
i
43 0.0
.~
.~
'a
'a
.~
t:l..
t:l..
[]
[-I
+--+point 3
V- -II point 4
,;., ..... ~~-:..-- point 5
,
"',
LlW"~
·a-0.5
~
-1.0 L.......o......................................................................................................J
0.0 60,0 120,0 180,0 240,0 300,0 360.0
-0.5
-1.0 L......o............................................................................................................J
0.0 60.0 120.0 180.0 240,0 300,0 360.0
angle
(a) Collimator 1
angle
(b) Collimator 2
Figure 3. The differences between the estimated and the measured sinograms in the transaxial (s) and the axial
directions (s) for cone beam projection data. The systematic errors of Collimator 1 and 2 are within 1.0 pixels.
IlJI
11997 International Meeting on Fully 3D Image Reconstruction
1531
Simulation studies of 3D whole-body PET imaging
C Comtat l , PE Kinahan!, T Beyer!, DW Townsend l , M Defrise 2, and C MicheI3
lUniversity of Pittsburgh, USA
2Catholic University of Louvain, Belgiuln
3Free University of Brussels, Belgium
INTRODUCTION
The utility of PET ilnaging is often lilnited by low data collection. rates and short imaging titnes,
resulting in images with high levels of statistical noise. WholeNbody oncology iInaging [1], in
particular, is constrained to short itnaging times at each bed position in order to maintain a total
scan duration that is acceptable to patients suffering from serious disease. The short imaging times
lead to increased statistical noise and further degradation in image quality and diagnostic utility as
cOlllpared to other PET inlaging protocols. Patients are typically scanned at multiple contiguous
bed positions over an axial length of 75-100 cm. Ideally, for oncology patients, the total imaging
time should be no longer than one hour or so, and therefore only 5-10 minutes of imaging can be
performed at each bed position. To limit the total scan duration, the transmission scan is often
omitted and/or the amount of overlap between adjacent bed positions is reduced. If bed overlap is
reduced too far, however, there will be an increase in image noise region of overlap [2]. In
addition, whole-body scans reconstructed without attenuation correction are non-quantitative, and
can lead to incorrect diagnoses, particularly for tumors located deep within the body.
For all PET imaging protocols, two different approaches to reducing statistical noise that have
been developed are volume, or 3D, imaging to increase sensitivity [2-4], and statistical
reconstruction methods to reduce noise propagation [5-10]. To achieve reductions in statistical
noise in clinically feasible times, 3D imaging and statistical 2D image reconstruction methods can
be combined by using the Fourier rebinning technique (FORE) that accurately rebins 3D PET data
into 2D data sets [11]. With the 2D data sets (sinograms) we can then use any of the accelerated
2D statistical itnage reconstnlction methods that have been developed in recent years, such as the
2D ordered-subset EM (2DOSEM) statistical reconstruction method [6].
Even with the reduction in image noise offered by the combination of FORE+2DOSEM, the
diagnostic utility of an attenuation corrected whole-body scan will be affected by the number of
emission and transmission counts collected during the study. The numbers of collected counts, in
turn, depends on the relative partition of the scan time between the emission (E) and transmission
(T) scans and the time spent scanning each bed position [2]. For a given choice of reconstruction
algorithm the fixed constraints are the total scanning time and the axial extent of the field-of-view,
while the parameters to be varied are the Err partition and the amount of bed overlap. The goal of
varying the parameters is to minimize image noise.
We are performing simulation studies of whole~body images in order to determine near-optimal
choices for these parameters in terms of minimizing image noise. For the simulation studies, we
include the major acquisition effects (for whole-body PET) of emission, transmission, and random
coincidence statistics, as well as data correction effects of attenuation correction, and on-line
randoms subtraction. We do not include the effect of scattered coincidences or scatter correction as
this would substantially increase the complexity of simulation.
In this work we illustrate the differences between simulations that include only emission
statistics and those that also include the effect of transmission and random coincidence statistics, as
well as attenuation correction, and on-line randoms subtraction. Using this methodology we
compare the FORE-rebinned data reconstructed by analytic and statistical methods, and investigate
how the Elf partition and the amount of bed overlap affect image noise.
This work is supported by grants from the Swiss National Scientific Foundation and the Whitaker Foundation
11997 International Meeting on Fully 3D Image Reconstruction
1541
[]
IJ
[I
SIMULATION METHOD
For the simulation studies, we include the effects of emission, transmission, and random
coincidence statistics, attenuation correction, and on-line randoms subtraction. We did not include
the effects of scatter, detector normalization, and deadtime correction. The simulation method is
similar to that used by the EVAL3DPET package [12] and is based on adding noise to numericallycalculated line-integrals. In our implementation we also include the effect of transmission and
random coincidence statistics. For the whole-body phantom, both the emission and attenuating
medium are described in terms of cylinders and ellipsoids as shown in figure 1. The effect of
multiple bed positions is generated by applying a fixed z-shift to the centers of the cylinders and
ellipsoids. This is illustrated in figure 2.
.
[]
[I
[I
[I
Figure 1. Frontal and transverse views of the whole-body phantom object used in the simulation studies'.:The
phantom is described by a collection of attenuation objects (left) and emission objects (right), Representative brain,
lung, heart, and bladder objects are visible, all of which have volumes similar to true physiological volumes.
Bed position 1
Bed position 2
D
[]
z~
sphere center at
zo
sphere center at
zo + z-shift of one bed position
Figure 2. Illustration of how multiple bed positions are simulated by applying a z-shift to the centers of object
descriptions.
lJ
In 3D PET imaging the collected 'prompt' events consist of random coincidences and true
coincidences that have been scaled down by attenuation. Both types of events are Poisson
distributed. In this case a model of the noisy emission data ei for line-of response (LOR) i is
generated by:
[J
ei =p{NEei}+p{NR},
-1
[
J
lJ
o
[J
cai
Ns
where p{.u} is a Poisson random process for a given mean .u, ei is the noiseless unattenuated
emission data, ai is the corresponding attenuation correction factor (ACF), NE is the total number
of emission counts, NR is the total number of random counts, and Ns is the total number of LORs
s(ej /aj) scales the total emission counts to account for the
or sinogram bins. The constant c =
effect of attenuation. In practice, the true emission counts are estimated by an on-line subtraction
of events from a delayed-coincidence time window, where all events are presumed to be random
coincidences. The effect of attenuation is then corrected for by multiplying the net true emission
'L7
155
counts by the corresponding ACF,
data becolues,
ai'
With these corrections, the model of the noisy enlission
e, == cal (p{NE.~'}+ piNIJ} - p{NR}J.
NE
Ns
cal
Ns
Finally, if we take into account the statistics of the transmission scan, it is possible to show that
distribution of the ACFs is approxirr1ately Gaussian with mean 11, =: at and variance
== (a? b)/(WT2 tTlvT ), where IvT is the total number of transnlission counts and b is the ratio of
average counts in the blank scan to the total nUluber of transmission counts as given by
b =L7~1 ajl. The factors WT 2 and IT account for the width of the sluoothing filter typically
applied to the transmission data and the fraction of transmission counts that contribute to the image,
respecti vely.
at
Taking ail of these effects into account, the noisy corrected emission data
e,-
e, is generated by:
cG{a'tO"f}(P{NEe;}
{NR}J
--+p{NR}
NE
cal
Ns -p -Ns
(1)
where G{l1tU2} is a Gaussian random process for a given mean and variance. Note that to simulate
noiseless randoms subtraction, the last term can be changed from
-P{NR/Ns}
to
-(NR/Ns).
This approach allows for rapid generation of data, which in turn allows for the multiple
realizations necessary for the estimation of the variance in the fmal images. The noiseless emission
data are over-sampled by a factor of 4 and then averaged to simulate partial volume effects.
Starting from a single noiseless data set {e, a, I i = 1... Ns}, multiple realizations of ej can be
generated. The effect of changing the Err time partition is simulated by changing the ratio of
promts to trnsmission counts (NE+NR VS. NT), while the effect of changing the bed overlap is
simulated by changing the z.. shift applied to the object descriptions and adjusting NE, NT, and NR
accordingly. In other words, for a fixed total scan time and axial extent, increasing the bed overlap
leads to a shorter scan time per bed position, and thus a decrease in the number of emission,
random, and transmission counts collected.
t
RESULTS
Figure 3 shows reconstructions of two central bed positions of the phantom shown in figure 1.
The sinograms were simulated in 3D, then rebinned with the FORE method, and then
reconstructed in 2D with either filtered-backprojection (2DFBP) or ordered-subsets EM
(2DOSEM). In the cases shown here, the 2DFBP images were reconstructed with a Hamming
window cut. .off at 70% of the Nyquist frequency, while the 2DOSEM images were reconstructed
using one iteration of 8 subsets.
Figure 4 shows images of the whole-body phantom (8 bed positions) reconstructed by
FORE+2DFBP and FORE+2DOSEM using the same parameters as used in figure 3. The effect of
the large attenuation factors across the torso and arms is clearly visible, as well as a reduction in
image noise outside the phantom for the FORE+2DOSEM image.
DISCUSSION
The results shown above clearly demonstrate the importance of accurately modeling the
different components that contribute to image noise in whole-body PET imaging. The goal of this
project is to perform accurate simulations of whole-body scans to investigate choices for the
reconstruction method, the Err time partition, and the amount of bed overlap that reduce image
noise for a fixed total scan time. The simulation method described here will be validated by
comparisons with patient scans, and more detailed analyses of the choice of reconstruction
11997 International Meeting on F~IIY 3D Image Reconstruction
1561
D
[J
it
algorithm and imaging parameters will be presented. This simulation method is a useful tool to
assist in selecting the best reconstruction algorithm and in specifying imaging parameters before
embarking on the more rigorous, but time-consuming, SNR and ROC analyses of lesion
detectability.
L_ )
r ._;
lJ
[1
D
[1
Figure 3. Reconstructions of two central bed positions of the phantom in figure 1 simulated in 3D. Top left:
FORE+2DFBP reconstruction with only emission (Poisson) noise. Top right: FORE+2DFBP reconstruction with
emission noise and noiseless attenuation correction. Bottom left: FORE+2DFBP reconstruction with emission,
transmission, and randoms noise, as well as effects of attenuation correction and on-line randoms subtraction,
according to equation (1). Bottom right: FORE+2DOSEM reconstruction of same data used in bottom.left
reconstruction.
o
Figure 4. Frontal and transverse views of reconstructions of the whole-body phantom object. Transverse views are at
the same level as those shown in figure 3. Left: FORE+2DFBP reconstruction. Right: FORE+2DOSEM.
REFERENCES
[1] Dahlbom M, et al. J Nuc Med 33(6):1191-1199, 1992
lJ
[J
[2] Cutler PD and Xu M. Phys Med BioI 41:1453-67, 1996
[3] Townsend DW, et al.lEEE Trans Med Imag 10: 505-512,1991.
[4] Cherry SR, Dahlbom M, and Hoffman EJ.IEEE Trans Nuc Sci 39:1088-92, 1992
[5] Shepp LA and Vardi Y.IEEE Trans Med Imag 2:113-119, 1982
[6] Hudson HM and Larkin RS. IEEE Trans Med Imag. 13:601-609,1994
[7] Meikle SR, et al. Phys Med BioI 39:1689-1704, 1994
[8] Fessler JA. IEEE Trans Med Img 13:290-300, 1994
[9] Mumcuoglu EU, et al. IEEE Trans Med Imag 13(4):687-701, 1994
[10] Lalush DS and Tsui BMW. Med Phys 22(8):1273-1284, 1995
[11] Defrise M. Inverse Problems 11:983-994, 1995
[12] Furuie SS, Herman OT, et al. Phys Med BioI 39(5):341-354, 1994.
lJ
11997 International Meeting on Fully 3D Image Reconstruction
1571
Advantage of algebl'aic regularized algorithms over
Feldkamp mcthod in 3D cOllc .. beam reconstruction
The projection data p is the set of projections, for a focalpoint trajectory (I> and a set of vectors 8:
1. Laurette l ,2, J. DarcoUl't', L. Blanc M
Feraud2 , P.-M.
2
Koulibalyl, M. Barlaud •
, Laboratoil'e de Biophysique et Traitement de l'Image.
Faculte de medecine. Universite de Nice Sophia-Antipolis.
2 Laboratoire 13S. URA 1376 du CNRS. Universite de
Nice-Sophia Antipolis.
3 Laboratoire J. A. Dieudonne. URA 168 ell! CNRS.
Universite de Nice-Sophia Antipolis.
Introduction
It is now well-known that cone-beam collimation increases
sensitivity without loss of resolution of emission imaging.
Besides these gains, fully 3D approach provides a more
precise description of matter-radiation interactions.
Therefore lesion detection is improved when compared to
parallel and fan-beam collimation [Li94].
Analytical reconstruction methods have the advantage of
being fast but are very sensitive to the data sufficiency
conditions defined by Tuy [Tuy83]. The main advantage
of algebraic methods is the possibility to invert the
physical projection operator which accounts for
attenuation, Compton scatter and detector response.
However they are cumbersome and since this problem is
known to be illMposed, they require regularization, i.e. to a
priori select a subset among all the possible solutions.
Several constraints can be used as regularizing
information: positivity, smoothness assumption, edgepreservation, support, etc.
The present paper compares the performances of the
classical analytical Feldkamp's method with those of a
regularized algebraic method under the hypothesis that the
latter can better cope with the missing data inherent to the
use of a single-circle acquisition orbit.
1 Reconstruction algorithms
p
(2)
A condition ensuring that the 3D image can be
reconstructed in a stable way from a set of cone-beam
projections has been derived by Tuy [Tuy83], then by
Smith [Smi85] and Grallgeat [Gra91].
The single circular trajectory does not fulfil Tuy's
condition and no analytical method is able to perform an
exact inverse process. However they can be applied to
obtain an approached inverse. The most frequently used
method with a single circular orbit data is the one
developed by Feldkamp [FeI84]. Even though the
reconstruction formula is obtained after several
approximations, it can be shown that it is equivalent to
exact methods applied to this trajectory [Smi85].
2.2 Algebraic methods
In an algebraic context, the observed data p are linked to
the unknown image J through a discrete linear model of
the form:
p= XJ+rb
(3)
where X is a discrete linear operator depending on the
acquisition geometry. 11 represents the noise contribution.
To solve the problem which consists in finding the
unknown J from the data p, we adopt a Bayesian strategy.
In this context, we have chosen to estimate the maximum
a posteriori (MAP), i.e. to maximize the posterior
distribution p(f/ p) given by:
p(J/p) oc p(p/J) p(J)
(4)
Although the likelihood distribution p(p/f) theoretically
corresponds to a Poissonian process, assuming a Gaussian
distribution is a correct approximation, given the count
rates used in medical imaging, and has the advantage to
lead to faster algorithms [Kou95].
When the a priori distribution P(f) is uniform, no a
priori on the image is assumed (no regularization): a
Poissonian
1.1 Analytical methods
In an analytical context, both projection data p and
unknown image f are considered to be continuous. The
acquisition model is supposed to be linear and depends on
the acquisition geometry (parallel, fan-beam, conebeam, ... ). For the cone-beam geometry, the operator is
called the X-ray transform, which corresponds to the conebeam projection of the density functionj(x):
-too
XJ(F,e)= f J{F+te)dt
={XJ(F,e)} Fe(I),Oe8
.
(1)
where the focal point is denoted by F and e is a threedimensional vector indicating the direction of projection.
11997 International Meeting on Fully3D Image Reconstruction
p(p/J)
leads to classical ML-EM [Lan84]
while a Gaussian one results in minimizing a least-squares
criterion, what can be done by using ART method or
Conjugate Gradient (CG).
However trying to solve equation (3) is known to be an illposed problem, so regularization is necessary, that is to
introduce a p(J) non uniform. When p(J) is a Gibbs
prior, the criterion to minimize, because it contains a nonquadratic-except for a smoothness assumption-term
Jr(J) (see table 1), is much more difficult to solve: the
MAP-EM-OSL developed by Green [Gre90] or the MAPEM-SQ [Kou96] could be used to solve the system
induced by the use of a Poissonian likelihood distribution;
1581
-----
when choosing a Gaussian p(p/f) , the criterion can be
seen as a constrained least-squares which could be
minimized
by the MAP-GC-SQ developed by
Charbonnier et ai., [Cha96]. This classification is
summarized in table 3. Most of these methods have not yet
been implemented for fully 3D reconstruction: classical
EM is mostly used. Standard least-squares minimization is
used by Trousset et al. [Tr090] and a constrained leastsquares (MAP-GC-SQ) have been implemented by our
, group [Lau96].
[]
(a)
(b)
Figure 1. Computer simulation: (a) phantom dimensions; (b)
the ideal sagittal cut of the phantom,
I b rruc
' reconstructlOn meth0 ds.
T a hI e 1: CI aSSl'filcatlOn 0 falge
\
LJ
p(p/J) p(J)
~
[j
ML-EMI
I [pln(Xf)-Xf ]+A? Jr(f)
Ga..Issian.
1
Ga..Issian.
Gilb;
IIp- Xfl1
IIp- Xfl1
MAP-EM-OSL
MAP-EM-SQ
CG,AIIT
2
2
+
f.? Jr(f)
MAP-OC-SQI
Existing 3D versions.
The model we have chosen uses a Gaussian likelihood
distribution and a Gibbs a priori distribution which
determines positive images with uniform areas separated
by sharp edges. This can be expressed through the
following criterion:
[I
~J
[L-J
,
Algorithm
I[pln(Xf)- xi]
Pcis<rnian GiI:OOan.
I
[]
1
J(f)
J(/) =lip - Xfl1 +A2 El (f) +Jl2 E2 (I)
2
I
(5)
IIp- xfl1 2
n
w
is the data consistency term; E 1(/)
corresponds to the energy of a Markov Random Field
(MRF) (see [Lau96] for more details); E 2 (/) expresses
[j
the positivity of the image [Mum94].
Solving the problem defined in equation (5) leads to the
Euler equations:
where
(Xl X -A2i\(/)-1l 2rr(f))f = Xl p
(6)
where ~(/) is a discrete approximation of the weighted
Laplacian operator, allowing the reconstruction of sharpedged images, and rr(/) is a binary matri~ describing the
discs of 25 mm radius and 3 mm height. The distance
between two consecutive discs is 11 mm. The central disc
was positioned at the centre of the circular orbit plane.
Figure Ib shows a noise-free sagittal cut of the phantom
generated by the computer.
Pinhole projections were generated by computing the
matrix product XI The focal length was 200 mm. We have
taken three values for the radius of the circular orbit (200,
150 and 100 mm) in order to observe the variation of the
sagittal artefacts, From a 256x256x256 object, 64 64x64
projections were generated over 2n. No noise was added
on the projections because we only want to' study the
artefacts induced by each method. No physical
interference was simulated.
Five methods were used to reconstruct 128x128x128
objects with always the same voxel size (0.8 mm):
Feldkamp'S method with a ramp filter, conjugate gradient
(CO 1), conjugate gradient with ROS (C02), conjugate
gradient with region of support (ROS) and positivity
constraint (C03) and finally conjugate gradient with ROS,
positivity constraint and edge-preserving regularization
(C04), that is to say the MAP-OC-SQ. The projection
model uses Dirac model for the voxels.
The reconstructions were performed on a Digital Alpha
Station 200 4/100. The reconstruction times are reported
in table 2. For the non-regularized algorithms, GC1 and
GC2, the process is stopped after 10 iterations. For GC3
and GC4, the indicated number of iterations corresponds
to full convergence, considered to be reached when the
NMSE between two successive images is inferior to 10-5 :
positivity of each image voxel. Equation (6) is non-linear
and is solved by half-quadratic minimization [Cha96,
Oem92].
NMSE(fkJ'k+l)
II
I
k+I
k 112
k 2
II/ l1
2 Method and results
Figure 2 presents the reconstructed sagittal cuts described
in figure 1b.
2.1 Computer simulation
Table 2. Reconstructions times.
A simulated Defrise phantom, shown in figure 1, was used
to compare the algorithms. This phantom contains seven
Nurlta'ofitmfuls
IIEmioodmiioo (s)
Rro:n<ml:ti<n~ (rrn)
Mirix amnx:ticn (rrn)
Tdal(rrn)
CDl
em
CG3
10
10
184
35
10
45
35
7
54
35
50
6
6
13
56
CD4
77
40
75
6
81
RiIkarr.p
15
15
r~'
; I
lJ
r
,
lJ
11997 International Meeting on Fully 3D Image Reconstruction
1591
f':::200 mill
f=150 mm
f':::lOO mm
Feldkamp
GCI
GC2
GC3
GC4
Figure 2. Sagittal cuts of the reconstructions. From left to right: the radius of the circular orbit takes 200, 150 and 100 mm values. From
top to bottom: Feldkamp, GC 1, GC2, GC3 and GC4 reconstruction methods.
11997 Internatioh'al Meeting on Fully 3D Image Reconstruction
1601
[FeI84] Feldkamp L A, Davis L C and Kress 1984 J W
"Practical cone-beam algorithm" J. Opt. Soc. Am. 1 612These results show that analytical and non regularized
619.
[Gem92] Geman S and Reynolds G 1992 "Constrained
algebraic methods suffer from insufficient data collected
restoration and the recovery of discontinuities" IEEE
by single-circular orbit acquisition. In the case of
analytical algorithms, the explanation is given by Tuy's
Trans. Pattern Anal. 14367-383.
[Gra91] Grangeat P 1991 "Mathematical framework of
condition. All these methods include an integration steps
cone-beam reconstruction via the first derivative of the
where all the planes passing through a point are necessary
Radon transform" Mathematical Methods in Tomography
for its exact reconstruction. A circular orbit does not give
(Lecture Notes in Mathematics 1497) ed G T Herman et
access to all the planes. However, Grangeat's method,
al. (Berlin: Springer) pp 66-97.
known to be superior to Feldkamp's one, includes an
[Gre90] Green P J 1990 "Bayesian reconstructions from
interpolation scheme in Radon space to complete the
emission
tomography data using a modified EM
missing data.
algorithm" IEEE Trans. Med. 1m. 9 84-93.
In the case of algebraic methods, we propose the
following hypothesis. The projectionibackprojection
[Kou95] Koulibaly P M, Darcourt J, Migneco 0,
r:
Barlaud M and Blanc-Feraud L 1995 "Comparaison du
operator defines a "shadow zone" around the objects
MAP-EM-OSL et d'ARTUR
deux algorithmes
which becomes wider as the concerned object is far from
deterministes
de
reconstruction en
tomographie
the focal plane. When solving the system, the iterative
d'emission" Innovation et Technologie en Biologie et
algorithm cannot see if a point of the shadow zone really
belongs to the true object. So all the points of this area
M edecine 39 643-65.
will be set to non-null values.
[Kou96] Koulibaly P M, Charbonnier P, Blanc-Feraud L,
When using the object RQS, it can be noticed that the
Laurette I, Darcourt J and Barlaud M 1996 "Poisson
discs are better defined. However large negative values
statistic and half-quadratic regularization for emission
appear in the extremities of the shadow zone. This could
tomography reconstruction algorithm" Proc;c Int. Conf. on
be explained by the fact that these negative values are
Image Processing (Lausanne).'
introduced in order to preserve a constant energy and
[Lan84] Lange K and Carson R 1984 "EM
balance the higher energy in the better defined areas.
reconstruction algorithms for emission and transmission
This problem is solved by incorporating a positivity
tomography" J. Comput. Assist. Tomogr. 8306-316.
constraint. The gain in contrast is very significant.
[Lau96] Laurette I, Koulibaly P M, Blanc-Feraud L,
Furthermore it can be noticed that the reconstructed object
Charbonnier P, Nosmas J-C, Barlaud M and Darcourt J
is even closer to the real one. This constraint really leads
1996 "Cone-beam algebraic reconstruction using edgeto a better geometric definition of the discs. As it has been
preserving regularization" Three-Dimensional Image
shown in [GuI90], EM algorithm which includes a
Reconstruction in Radiology and Nuclear Medicine ed P
positivity constraint, is less sensible to the missing data
Grangeat and J-L Amans (Dordrecht: Kluwer AcadewJc
than Feldkamp'S method.
Publishers) pp 59-73.
Finally, edge-preserving regularization further improves
[Li94]
Li J, Jaszczak R J, Turkington T G, Metz CE,
the final image by smoothing the uniform areas separated
Gilland D R, Greer K L and Coleman R E 1994 "An
by sharp edges. The reconstructions obtained by the
evaluation of lesion detectability with cone-beam, fanalgorithm including all the constraints is very close to the
beam and parallel-beam collimation in SPECT by
original phantom.
continuous ROC study" J. Nucl. Med. 35 135-140.
[Mum94] Mumcuoglu U, Leahy R, Cherry S Rand Zhou
f------'fhese~results~are~in~favour-of~th-~use_andllevelopmentof'-----~Z·--T9V~FaSf~giadTent-based methods for Bayesian
regularized algebraic methods for 3D cone-beam
reconstruction of transmission and emission PET images"
reconstruction to promote its clinical use.
IEEE Trans. Med. 1m. 13687-701.
_I
[Smi85] Smith B D 1985 "Image reconstruction from
Acknowledgements
cone-beam projections: necessary and sufficient
conditions and reconstruction methods" IEEE Trans. Med.
This work was made possible by a grant from the Region
Imaging MI·4 14-25.
Provence-ALpes-Cote d'Azur and the financial support of
[Tuy83] Tuy H K 1983 "An inversion formula for conethe University of Nice-Sophia Antipolis.
beam reconstruction" SIAM J. Appl. Math. 43546-552.
[Zen90] Zeng G. L. and Gullberg G. T. 1990 "A study
References
of reconstruction artifacts in cone beam tomography using
filtered backprojection and iterative EM algorithms "
[Cha96] Charbonnier P, Blanc-Feraud L, Aubert G and
IEEE Trans. Nuc. Science 37 759-767.
Barlaud M 1996 "Deterministic edge-preserving
regularization in computed imaging" IEEE Image
Processing. In press.
[J
3 Discussion
LJ
l]
[J
[]
l-i
[J
11997 International Meeting on Fully 3D Image Reconstruction
161\
A new symmetrical vertex path for exact reconstruction
in cone-beam C.T.
F. Noo
if'
R. Clack t
ABSTRACT
A new vertex path is proposed for cone-beam medical CT
that can be implemented by uniform rotation of the x-ray
source and simple translations of the patient bed. The new
vertex path has convenient symmetry properties which allow the derivation of a new reconstruction algorithm to
handle noise in a nearly optimal way. The algorithm is
also able to correctly handle axially-truncated projection
data when this path is used. Reconstruction results are
shown with simulated data and real data.
1. INTRODUCTION
Cone-beam (CB) tomography remains a challenge in x-ray
medical imaging for several reasons. Firstly, we observe
that new technologies have to be developed to build high
quality large 2D detectors. Current CB systems often use
an image intensifier coupled with a CCD camera, and the
resulting quality of the data is relatively poor. Secondly,
we note that no satisfactory vertex path (x-ray source trajectory) and algorithm combination has yet been proposed
that can yield an exn,ct. reconstruction in the presence of
truncated projection data. An attractive scanning mode
is the helix path as it can be realized by smoothly rotating the x-ray source and translating the patient bed.
Unfortunately, no algorithm exists yet that can exactly
reconstruct from axially truncated projection data along
the helix path. Axial truncation is inherent to most applications in cone-beam CT.
Building on existing CB methods [~, 2, 3], Kudo and
Saito devised a hybrid filtered backprojection (HFBP) algorithm [4] that is appropriate for exactly reconstructing
axially truncated data. However, the Extended Completeness Condition to be satisfied by the vertex path for this
algorithm to work is fairly restrictive. The vertex paths
satisfying this condition which have been published so far
all contain a circular subpath: the cir~le and one line [5],
the circle and multiple lines [4], and the circle and helix [6].
"'F. Noo is a Research Assistant supported by the Belgian National
fund for Scientific Research, and with the Institute of Electricity
Montefiore, B28, University of Liege, B-4000 Liege (Belgium). Email: [email protected]
tR. Clack is with the Departement of Radiology, University of
Utah, Salt Lake City, UT 84132
tT. J. Roney and T. A. White are with the Idaho National Engineering Laboratory, Idaho Falls, ID 83415
T. J. Roney+
T. A. White +
The idea behind the HFBP algorithm is to process a
subset of the measured data (namely the circle data) using
the algorithm of Feldkamp et al [7] so as to get a good estimate of the image. Supplementary data are only used to
correct artefacts in the approximated image, and thereby
achieve an exact reconstruction.
The HFBP algorithm has two undesirable characteristics. Firstly, the supplementary subpath data only make
a small contribution to the reconstruction since the artifacts in the approximate image are usually minor. This
is a serious drawback from the statistical point of view.
Considering for example a circle plus one line path, it is
not· obvious how to define the relative numbers of vertex
points to be sampled on the circle and on the line so as
to properly handle noise. Secondly, it was shown in [6]
that singularities exist in some filtered projections. A new
composite algorithm was proposed in [6] that avoids singularities, but requires processing part of the data with a
2D filter instead of simply applying the ID ramp filter of
Feldkamp'S algorithm.
In this paper, we present a new scanning mode, called
the "cross path" , that can be realized by uniformly rotating the x-ray source and smoothly moving the patient bed.
This new vertex path satisfies the Extended Completeness
Condition of [4] and is therefore appropriate for processing axially truncated data. Furthermore, it has symmetry
properties that allow the derivation of a new reconstruction algorithm which handles noise in a nearly optimal
way, and which avoids singularities of the HFBP method
without involving any 2D filter.
The organization of the paper is as follows. In section 2, we define the cross vertex path and verify that it
satisfies the Extended Completeness Condition. In section 3, we present the new reconstruction algorithm. Section 4 is then devoted to simulations. We illustrate the
ability of the new algorithm to exactly reconstruct the
classical 3D Shepp phantom in the presence of truncation,
and we present results from real data. Conclusions are
given in section 5.
2. THE CROSS VERTEX PATH
2.1. Data acquisition
We consider imaging a patient lying on a bed oriented
along some axis that we denote as the x 3-axis. The effective linear attenuation coefficient in the patient is repre-
11997 International Meeting on Fully 3D Image ReConstruction
1621
IiLJ
°r~t·,·Hi:·H. . ~.
o·
-0.5
..
.. :
..•
.,;
2
4
;
-10
:. . : ........ ':..
,.. "
.. ,
8
tl
, .. ,' ,
. ,. :. .,'
;:
: '
6
,,'
10
12
14
16
Figure 1: Axial motion of the patient bed (in units of h) as
a function of the angular position of the source (in radians).
X-rays are emitted while the bed moves cosinusoidally.
sented by the function f(;J2) where ;J2 = (Xl' X 2 , X3)'
The x-ray source is always a distance R from the X 3 axis and rotates about the patient with a uniform, angular
speed w. The patient bed can be translated in the X3
direction following some motion law g(wt) where t is the
real time. Typical examples are the collection of circle
data for g(wt) == 0, or g(wt) proportional to wt for helix
data. In this paper, we choose a harmonic motion for the
patient bed:
g+(wt)
[]
= hcoswt
and g-(wt)
g+(wt) = (Rcoswt,Rsinwt,g+(wt))
= (R cos wt, R sin wt, g- (wt))
u
r
.
I II
I
L---J
lJ
VI -
1. any plane which is affected by axial truncation (i.e
such that Grangeat's formula [1] can not be applied)
must intersect the subpath a constant number of
times,
2. the vector tangent to the subpath at any point must
be such that axial truncation does not affect the result of applying a ID ramp filter along its direction
in the detector
= -hcoswt
The data acquisition we propose consists of two measurement periods. The first measurement period begins when
the source, which is considered to rotate uniformly about
the patient, arrives in the plane X 2 = O. The patient bed is
then moved cosinusoidally along the x 3-axis with the same
frequency was the source. X-rays are emitted and CB projections are collected while the source performs a complete
turn. Next, the patient bed is stopped and the source is
rotated a 180 degrees without emitting any rays. The second measurement period begins just after the source has
completed that half-turn rotation. The cosinusoidal motion of the patient bed is repeated, and CB projections are
again measured on a complete turn.
The motion of the patient bed during the data acquisition is shown in figure 1. This motion is very smooth. The
speed of the bed always tends to zero when the bed must
be stopped, or when the direction of the motion must be
reversed. The maximum acceleration that must be given
to the bed depends on the angular speed of the source and
on the amplitude h of the harmonic motion of the bed,
and can be adapted to best convenience of the patient.
Mathematically, the cross vertex path consists of two
subpaths g+(.) and g-(.) defined by equation~
g- (wt)
provided that any plane intersecting that region also intersects the vertex path. The geometry of the cross vertex
path is such that Tuy's condition is satisfied at least for
a cylindrical region-of-interest O(r, d) centered on the X 3 axis, of radius r < R, and of half-height d = h
r2 / R2.
A complementary sufficiency condition on the vertex
path for handling projection data which are truncated
along the axial direction is the Extended Completeness
Condition of [4]: the vertex path must contain some subpath satisfying the following two properties:
wt E [0',271")
wt E [371",571")
The cross vertex path is appropriate for handling axial
truncation. Planes for which Grangeat '8 formula cannot
be applied all intersect twice each of the subpaths g+ (.)
and g - (.). Furthermore, the ratio h / R can be chosen sufficiently small that both subpaths admit at any point a
tangent vector nearly parallel to the (Xl' X 2 ) plane.
The dimensions of the region that can be exactly reconstructed depend on the size of the 2D detector. If the 2D
detector is sufficiently large to enclose, given any source
position along the cross path, the CB projection of the
cylindrical region-of-interest O(r, d), then most of that region can be exactly reconstructed [4, 6].
3. THE RECONSTRUCTION'ALGORITHM
3.1. Detector geometry
In this section, we let A = wt, so the g(A) denotes the
position of the x-ray source. The detector is considered as
a planar arrayal ways oriented so as to be parallel to the
x3-axis and to the vector tangent to the cross path at g(A).
The distance between the vertex point and the detector is
equal to D and is the same for any source position along
the path.
Given a fixed A, the CB projection p(u, v, A) represents
the set of line integrals diverging from the vertex point and
in the image coordinate system. Both subpaths are ellipses
(see figure 2) defined by the intersection of two oblique
planes with the imaginary cylinder of radius R. The subpath g+(.) lies in the plane hXI = Rx 3 , and g-(.) lies in
the plane hXI = -RX3'
2.2. Completeness condition
According to Thy's condition [8], a vertex path is sufficient for exact reconstruction of some region-of-interest
Figure 2: The cross vertex path consists of two ellipses which
are symmetric about the plane X3 = 0
11997 International Meeting on Fully 3D Image Reconstruction
1631
FBP method that uniformly processes each CD projection,
thereby achieving a nearly optimal handling of noise.
The expression for lvI+(s, /1" A) for vertex points along
«+ (.) is given below (symmetry arguments can be used to
find M-(s,tt,A)):
M+ (s, 11" A) :::: M;with
+ M;ts(s, tt, A)
M: = 1/4 and
M;!"s(s, tL, A) ::::
1(1-1/)(tt))
ifII(s,tt,A)ng-(')=0
{ ('l/J(I1,i) + 'l/J(tt2) - 21/)(tt)) otherwise
t
where tti and /12 define the orientation of the lines of intersection of II(s, /1, A) with the detector of vertices along
a-(') and contained in II(s,/1, A), and
Figure 3: Geometry of data acquisition. The source is a dis~
tance D from the detector. Each pixel (u, v) defines with the
vertex point a line along which the xMray attenuation p(u, v,..\)
is measured.
0
touching the detector at some point (u, v). The coordinates (u, v) are defined according to unit orthogonal axes
Qu and Qv respectively parallel and perpendicular to the
vector tangent to the cross path at A (see figure 3). The
projection of the vertex point onto the detector defines the
origin (u, v) ::: (0,0).
'l/J(tL) ==
{ exp ( cos2 ~-sin2 ~o )
if / cos /1/ 2 sin 11'0
otherwise
where tto is some small angle chosen here as rr /20.
The component M;; of M+(s, /1, A) results in applying
a iD ramp filter along the u-axis in the detector. The
component M;ts is continuously differentiable and results
in a nonstationary filter.
3.2. FBP reconstruction
The reconstruction algorithm we propose is an adapted
version of the nonstationary FBP algorithm of [2] and [3] in
which the CB projections are successively scaled, filtered
and then backprojected in 3D to get the reconstructed
image. The exactness of the method relies on an appropriate handling of the redundancy in the CB data using a
weighting function M(s, tt, A) in the filtering step.
The significance of M(s, p" A) is as follows. The variables sand tt specify the position of some line V(s, tt) in
the A-detector, which associated with the vertex point g(A)
define a plane II (s , p" A) (see figure 3). In the absence of
noise, the plane II(s, tt, A) receives identical contributions
from all vertices that it contains. The role of M(s, tt, A) is
to ensure that the total contribution is unity.
Given a vertex path, several choices are possible for
M(s, tt, A), among which continuously differentiable forms
are preferred for numerical stability [2]. Vertex paths that
contain a subpath presenting properties 1 and 2 of section 2.2 are special as they admit along that subpath a
weight M(S,p"A) independent of s which reduces the nonstationary filter to a stationary filter [6].
As was mentioned in section 2.2, the cross vertex path
satisfies the Extended Completeness Condition of [4]. Data
along the subpath g+ (.) can be applied a stationary filter
while data along g- (.) are applied a nonstationary filter.
Or conversely, data along g+ (.) can be applied a nonstationary filter while data along g-(.) are applied a stationary filter. Invoking the linearity of the CB reconstruction
problem, exact reconstruction can be achieved by applying
to each CB projection both filters, stationary and nonstationary, and averaging the result. .The consequence is a
cos 2 /1:
4. APPLICATIONS
4.1. Simulated data
Reconstructions of the classical 3D Shepp phantom were
used to verify the capability of the new algorithm to give
exact reconstruction of low contrast objects in the presence
of truncation. Results are shown in figure 4.
The cross vertex path was sampled with 256 points
(128 points on each subpath). An interlaced sampling
scheme was used between points along Q+ (.) and points
along g - (.). The source was a distance 350mm from the
rotation axis and a distance 700mm from· the detector.
The amplitude h of the harmonic motion of the "patient
bed" was taken as 100mm.
The detector consisted of 128 x 128 square pixels of side
4mm for simulation without truncation, and of 128 x 86
pixels for simulation with truncation. Axial translation of
the detector was applied so as to keep the projection of
the object centered in the middle of the detector array.
Data were reconstructed on a grid of 100 x 100 x 100
cubic voxels of side 2mm. Images were displayed using the
compressed greyscale [1.0,1.04] so as to illustrate the low
contrast features inside the phantom.
4.2. Real data
Real data were collected from a drum (of radius 285mm
and of height 800mm) containing many small objects such
as bottles, pipes, and broken glass; and two tall vertical cylinders holding small spheres of high density material. The drum was placed on a rotating platform and the
11997 International Meeting on Fully 3D Image Reconstruction
1641
i'
ii
j
A potential drawback of the new scanning mode is the
need to translate the detector as the source rotates so as
to keep the region-of-interest centered in the projection.
[!
6. REFERENCES
[1] P. Grangeat, Analyse d'un systeme d'imagerie 3D par
reconstruction d partir de radiographies X en geometrie
conique, Ph.D. Thesis, Ecole Nationale Superieure des
Telecommunications, France, 1987.
[2] M. Defrise, R. Clack, "A cone-beam reconstruction algorithm using shift variant filtering and cone-beam back projection," IEEE Trans. Med. Imag., 13, 186-195, 1994.
[3] H. Kudo, T. Saito, "Derivation and implementation of
a cone-beam reconstruction algorithm for nonplanar orbits," IEEE Trans. Med. Imag., 13, 196-211, 1994.
r~r
LJ
[4] H. Kudo, T. Saito, "An extended completeness condition
for exact cone-beam reconstruction and its application,"
IEEE Conf. Record of the 1994 Nuclear Science Symposium and Medical Imaging Conference, Norfolk, VA.,
1995.
[5] G. L. Zeng, G. T. Gullberg, "A cone-beam algorithm for
orthogonal circle-and-Iine orbit," Phys. Med. Bio!., 37,
563-578, 1992.
fl
L!
[6] F. Noo, M. Defrise, R. Clack, "FBP reconstruction of
cone-beam data acquired with a vertex path containing
a circle", IEEE Conf. Record of the 1996 Nuclear Science
Symposium and Medical Imaging Conference, Anaheim,
CA, (to appear) 1997.
[J
[J
Figure 4: Reconstruction of the 3D Shepp phantom: (a-b) without truncation, (c-d) with truncation (e-f) differences between
images of the two first rows. Axial truncation can be handled
to give an exact reconstruction in a region determined by the
size of the detector.
[]
[j
[J
source was moved harmonically to collect cross data of parameters R = 1740mm, D = 2337mm, and h = 400mm [9].
The number of vertices was 256. The detector consisted
of a large immobile scintillation screen optically coupled
to a CCD camera, and provided projections discretized
into 185 x 256 square pixels of side 4.32mm. The reconstruction, shown in figure 5, was performed on a grid of
128 x 128 x 160 cubic voxels of side 5mm. The poor quality
of the image is presumed to be due to some misalignment
or inaccurately determined parameters of the scanner, and
work is proceeding to verify this assumption.
[7] L. A. Feldkamp, L. C. Davis, J. W. Kress, "Practical conebeam algorithm," J. Opt. Soc. Am., A6, 612-619, 1984.
[8] H. Tuy, "An inversion formula for cone-beam reconstruction", SIAM J. App!. Math., 43, 546-552, 1983.
[9] T. J. Roney, S. G. Galbraith, T. A. White, M. O'Reilly,
R. Clack, M. Defrise, F. Noo, "Feasibility and Applications of Cone Beam X-Ray Imaging for Containerized
Waste," Proceedings for the 4th Nondestructive Assay
and Nondestructive Examination Waste Characterization
Conference, Salt Lake City, pp. 295-324, 1995.
5. CONCLUSION AND DISCUSSION
u
r~
I
!
-,
I
I
L.J
A vertex path has been proposed for cone-beam CT that
• can be realized using uniform rotation for the source
and harmonic motion for the patient bed
Figure 5: Reconstruction from real data of a drum containing
small objects like bottles, broken glass, pipes and tall cylinders:
• admits a reconstruction algorithm which handles noise (a) slice X3 = -180mm (b) slice X2 = 40mm, the reference
in a nearly optimal way and which is capable of exact
frame is centered in the drum
reconstruction in the presence of ruda,l truncatiQn,_
11997 International Meeting on Fully 3D Image Reconstruction
1651
Fast Accurate Iterative Reconstruction for Low-Statistics
Positron Volume Imaging
A. J. Reader, K. Erlalldsson, M. A. Flower, R. J. Ott
Joint Depnrtment of Physics, Inslillltc of Cnncer Research, Royal Marsden NHS Trust, Downs Rond, Sulton, Surrey SM2 5PT
UK
Abstract
A fast accurate method of implementing threeHdimensional iterative reconstruction
techniques is presented. The method is ideally suited to low"statistics reconstructions 01' to any
imaging situation or system where the number of lines of response (LORs) exceeds the typical number
of events acquired. The new fast accurate iterative reconstruction (FAIR) method has been applied to
Expectatiol1 Maximisation Maximum~Likelihood (EMRML [1]) and has been compared with the
conventional implementation, which requires rebinning and storage of the data in projections.
a
Introduction
Positron Volume Imaging (PVI) situations where the number of LORs exceeds the number of
events acquired occur in two main areas: in dynamic imaging and with large area rotating planar
detector cameras. Such imaging situations normally call for list-mode data storage as opposed to full
sparse projection data storage. For large area detector (LAD) cameras list-mode storage is even more
preferable, not just because of storage requirements but also to retain accurate positional information.
The requirement of iterative image reconstruction techniques is to model the measurement
process of the camera, which can be approximated by an X-fay transform. The standard
implementation of most iterative methods requires the data to be rebinned and stored in projection
format; a procedure involving large data storage requirements and loss of data accuracy for some
systems.
The method proposed here, FAIR, obviates these two drawbacks by direct use of an exact,
compact and unordered projection data list as the basis for iterative reconstruction.
Theory
The coincidence data from a typical LAD-based system are ill the form of seven element
events: (XJ,YJ,ZJ,X2,Y2,Z2,ex) (figure 1) where (XJ,YJ,Zl) is the detection position on the first
detector, (X2, Y2,Z2) is the position on the second detector, and ex is the gantry angle of rotation at the
moment of coincidence' detection. Each of these events is used to determine a position on a 2 D
paraUel projection (figure 2) according to the following transform of variables:
(Xl, Yl,Zl,X2, Y2,Z2,cx) -7 (y' ,z' ,<1>,9)
Rather than lose the accurate values of (y' ,z' ,<I>, e) by histogramming the event into a comparatively
coarse bin, the exact projection coordinates of the event are stored sequentially in a new listwmode
data file.
8
V2
Z1
L
D1
~
AXIS OF
ROTATION
%
D2
y
.... . . _.. ........:::.::::::.»-//
Xl
s
Figure 1: LAD geometry. The list-mode coordinate
frames are shown on the detectors, and the reconstruction
frame is shown in the centre.
Figure 2: The reconstruction coordinate frame showing a 2-D
parallel projection. Note the orientation change from figure 1.
11997 International Meeting on Fully 3D Image Reconstruction
1661
n
Most iterative reconstruction methods can be regarded as a repetitive sequence of forward
and back projection along the measured LORs. The FAIR method consists of forward projecting
through the current estimate, along each of the accurately defined LaRs in the list-mode data file:
q: = FP{fk}
(1)
where FP is the forward projection linear operator, or x-ray transform, fk is the kth estimate of the
image and q/ is the forward projected value for LOR i through the kth estimate. The corrective update
image / (consisting of voxelsj=l ... J) is obtained by
c; = BP{ :k }
(2)
where BP is the backprojection linear operator and the value of 1 is simply the single event measured
by a single LOR.
Equations (1) and (2) can be implemented for each event in the projection data list to obtain
the complete corrective update image ck for use in one iteration of the EM-ML method
ik k
i,k+l
=fl
J
[:
(3)
where Wj is a weighting matrix to account for the varying geometric acceptance which compensates for
LORs with no measured value.
The standard EM-ML method is obtained if equation (2) is replaced by
-"'
[.J
cJ = BP{ ~ }
[J-)
(4)
where mi are the histogrammed measured projection data.
Implementation
The computational implementation of equations (1) to (3) requires minimal computer
memory, requiring just 3 image volume matrices, one for the current estimate f k, one for the
corrective update image ck and one for the backprojection weights w.
Each detected list-mode event is transformed into a 2-D parallel projection coordinate which
is then stored in a new list-mode data file. Each event is then read from the file in turn to determine
the forward projection LOR through the current estimate f k: the reciprocal of the forward projected
value is taken for backprojection into the corrective image volume ck (equation (2». Attenuation
correction is carried out by attenuating the forward projected value. The attenuation factors are found
by forward projection through a linear attenuation map along the LORs specified in the projection
data file list, and the values stored sequentially in another file or in memory.
For the EM-ML implementation, the forward projection through the estimate followed by
backprojection into the correction image (equations (1) and (2) ) is repeated for each individual event
in the file. The final corrective image is divided by the weights matrix (which is analytically derived
to allow for LORs with no measured value and only needs to be calculated once) before being used to
update the estimate image volume (equation (3».
Both the backprojection and forward projection operations were implemented by calculating
contributions at discrete steps along each LOR using tri-linear interpolation.
[j
[J
[J
[]
[l
o
I
w.J
Comparison
Using measured data from the LAD camera at the Royal Marsden Hospital [2], a FAIR
implementation of the EM-ML method (FAIR-EM) was compared with a conventional
implementation whereby the data were rebinned into complete 2-D parallel projections. Typical
sampling intervals of 128x128 (3mm side) with 96 azimuthal angles (<I» (interval 1.875°) and 9 copolar angles (e) (interval 3.389°) were chosen. Images from both approaches were reconstructed into
image volumes of diameter 38.4cm within matrices of 128x128x128 voxels, 3mm side.
The two methods were compared using the point spread function (PSF) and a uniform
cylinder. The methods were also compared for speed of reconstruction. For the PSF comparison a
22Na point source was located at 6 different positions in the field of view (FOV): (0,0,0), (7,0,0),
(14,0,0), (0,0,7), (7,0,7), (14,0,7) «x,y,z) in cm). In each position 3.5xl05 events were acquired and
the resulting list-mode data were added together to create one file of 2.1xl06 events. The mean axial
'
i
11997 International Meeting on Fully 3D Image Reconstruction
1671
and tangential full widths at half maximum (FWHMs) and tenth maximum (FWTMs) were calculated
for each iteration of each method (cubic spline interpolation of the profiles was used). For the uniform
cylinder comparison a phantom of 680a (12cm diameter, 7.2cm long) was located in the centre of the
FOV and 2xl06 events were acquired. The signal to noise ratio (SNR) was found for each iteration of
each method by M /0'; where M is the mean count in an annular region (from radial position 3.3cm to
4.2cm) in each of the central 16 slices of the uniform phantom and (J is the standard deviation.
Provisionally, attenuation and scatter were compensated for using a post-reconstruction correction
image which was analytically derived using a linear attenuation coefficient of O.08cm- l .
Results
Axial FWHM
10 -'&----------~----,
9.5 -.
9·· - -
AXial FWTM
20~-r---------------------~
19 -- .
1a -- •
E 17 --
Ea.5-- .
.s. a-··
•
.
~ 15 -. . .
5! 14--
e 16 --
:: 7.5 . - .
~ 6.~ -.
EM
: .
6 _.
5.5 -. .
u:
FAIA~EM
234
5
6
7
2
5
6
Iteration
Tangential FWHM
Tangential FWTM
EM
FAI A-EM
7
8
29
_27
~ 25
-23-
~
~
~ 10 - .
~-+----4EM
3
5
4
6
7
21
19
17
•• - EM
15 +--+-+----t--f--t---f-~ FAIR-EM
7 .t--t---+-~I----t--+---t---t FAIR·EM
2
4
n,,",UUIVIl
..•
1
3
..............1__
14 -- .
13 -- . .
12· . .
:: 11
-=-.- : - : - ,
10+-~--~--~----~-+--~~
8
15~~-----------------------------------~
g
-:-.
~~:t_;::.~.
11 - - •
5-F--+=-~--+-~---+--~~
e
13·- .
12
8
1
2
3
Iteration
4
5
6
7
8
Iteration
Figure 3: Comparison of PSF widths (each datum point represents mean of 6 measurements for 6 positions)
65r-------~----~~------~
55
a:
45
Z 35
en
25
15
5 +I_-+__~~--~~==~~EM
FAIR-EM
2
3
4
5
6
7
8
Iteration
Figure 4: SNR results taken from annular region within uniform cylinder_
11997 International Meeting on Fully 3D Image Reconstruction
1681
[I
[l
Figure 5: Central slice of the unifonn phantom after 8 iterations of each method (2xl events). Left hand side: the FAIR-EM
method. Right hand side: the standard EM method. The FAIR-EM image has negligible edge enhancement and is a cleaner image.
(Images displayed using the ANAL YZE 7M biomedical imaging software [3])
[J
Gl
40
~
35
~
30
.E 25
I
[)
I/)
...
t!
~
20
15
10
5
0+---~-+--~--~--+---~-4
250
500
750 1000 1250 1500 1750 2000
Events (k)
Figure 6: Speed increase of FAIR-EM when compared with ~tandard EM
Ll
[l
[J
I]
[]
[J
rl
U
Discussion
It has been demonstrated that, for sparse data PYI situations, iterative reconstruction can be
made more computationally efficient and additionally for some systems can improve spatial
resolution. The results for the camera at this hospital show that the FAIR method achieves improved
resolution at less computational expense when compared to the typical implementation of the EM-ML
method (for 2xl06 events FAIR-EM was nearly 5 times quicker). The lower SNR values of the FAIREM method compared to the standard EM method could be due to accelerated convergence: this will
be investigated.
FAIR methods need only 3 image matrices and one list-mode data file, compared with 2 or 3
image matrices and 2 sets of complete projection data for standard implementations. Not only are
computer memory requirements and disk storage reduced but also FAIR requires significantly less
processing time, depending on the number of events.
The new approach to iterative reconstruction described here is readily transferable to other
iterative methods such as the iterative Image Space Reconstruction Algorithm (lSRA [4]) and the
Simultaneous Iterative Reconstruction Technique (SIRT [5]).
Possible areas of development include use of subsets [6] of the compact projection data list to
accelerate convergence. In the current implementation, which leaves the projection data unsorted, the
subsets would inevitably be 'random' as opposed to 'ordered': the significance of this effect will be
investigated.
References
1) Shepp L A, Vardi Y (1982) "Maximum Likelihood Reconstruction for Emission Tomography" IEEE Trans Med Imag 2:113-122
2) Cherry S R, Marsden P K, Ott R J, Flower M A, Webb S, Babich J W (1989) "Image quantification with a large area multiwire
proportional chamber positron camera (MUP-PET)" Eur J Nucl Med 15:694-700
3) Robb R A, BariUot C (1989) "Interactive Display and Analysis of3-D Medical Images" IEEE Trans Med Imag 8:217-226
4) Daube-Witherspoon M E, Mueh1lehner G (1986) "An Iterative Image Space Reconstruction Algorithm Suitable for Volume ECT"
IEEE Trans Med Imag 2:61-66
5) Landweber L (1951) "An Iterative Fonnula for Fredholm Integral Equations of the First Kind" Arner J Math 73:615-624
6) Hudson H M, Larkin R S (1994) "Accelerated Image Reconstruction Using Ordered Subsets of Projection Data" IEEE Trans
Med Imag 4:601-609
11997 International Meeting on Fully 3D Image Reconstruction
1691
Design and Implelnentation Aspects of a 3D Reconstruction Algorithm for the JUlich
TierPET System
A. Terstegge, S. Weber 2, H. I-Ierzog2, H.W. Mtiller~Gartner2, H.Halling
Zentrallabor fUr Elektronik, ZEL
2Institut fUr Medizin, 1ME
Forschungszcntrum JUlich GmbH, Germany
Email: A.Terstegge@kfa"juelich.de
I. INTRODUCTION
The research center in JUlich is currently evaluating the
TierPET system, a high resolution animal PET scanner [1].
During the development of this system, two main aspects were
studied: The detector system has been optimized for good
sensitivity and high resolution. This goal was achieved by
careful simulation studies of the detectors as well as various
preliminary measurements of different detector characteristics
[2]. The intrinsic resolution of the system is ~ 1 mm in the
center of the field-of-view (FOY).
The second important aspect is including corrections of the
measured data and the efficient implementation of a suitable
reconstruction algorithm, which should consider the high
intrinsic resolution of the system and minimize the introduction
of artifacts.
This abstract focuses on the choice and implementation of
the reconstruction algorithm, and demonstrates the quality of
the resulting system by presenting first reconstruction results
from real phantom measurements.
.
II. THE RECONSTRUCTION ALGORITHM
Transfonn methods like the filtered back-projection
algorithm (FBP) usually introduce some reconstruction
artifacts due to two circumstances: The statistics of the
annihilation process is neglected. Secondly, they use analytical
formulas, which a valid for a continuous data and image space.
Only after solving the reconstruction problem, the discrete
nature of these properties is introduced.
Algebraic methods like the ML-EM algorithm [3] do not
have these disadvantages. But they can be painfully slow due
to their iterative nature. The basic ML-EM algorithm, which is
shown in fig. 1, was implemented as a starting point.
set Xo
for i =0,1,2, ... (until convergence)
u= P Xi
u=b0u
Xi+ 1
==
Xi
® pT
U
endfor
Figure 1: The ML-EM algorithm
The operators ® and 0 denote a component-wise
multiplication and division. The vectors Xi and b denote
the n-dimensional image vector at iteration i and the
11997 International Meeting on Fully 3D Image Reconstruction
rn-dimensionnl. measurement vector, respectively. P is the
(m, 11.) system transfer matrix.
To speed up the convergence rate of this algorithm, the
following acceleration scheme was investigated. The ML-EM
can be interpreted as a scaled gradient method of the form
(1)
where the scaling is performed by a diagonal matrix
diag( Xi (n)), which is updated for every iteration.
The objective function I(x) is the likelihood-function, which is
minimized during the reconstruction process. There are several
other objective functions, which can be used in this context.
If, for example, I(x) is replaced by the "Least Squares" (LS)
criterion
Mi 1 =
mJnf(x)
= ~llb - P xll~,
(2)
then an analogous algorithm can be derived, as it was shown
in [4]. Since this algorithm uses the same scaling matrix as
the ML-EM algorithm, it is called "EM-like". To accelerate
the gradient algorithm using the LS criterion, one can introduce
conjugate gradient (CO) techniques. The LSQR algorithm [5] is
an numerical stable implementation of the CO method to solve
the unsymmetric problem Px = b.
This basic algorithm is extended in two ways: In contrast
to the ML-EM algorithm, the LS solution is not ensuring
non-negativity of x. Therefore the step length of the correction
vector is checked during every iteration, and the process
is restarted as soon as the constraint is hit. Secondly,
the procedure includes the scaling matrix (also called a
preconditioning matrix) Mil. Unlike theMLsEM algorithm,
this scaling matrix is fixed during the application of the CG
method. If a restart occurs, the diagonal matrix Mi 1 is
updated to contain the elements of the current solution vector
x. Because the resulting method is a preconditioned LSQR
algorithm, we call it PLSQR.
Except some minor corrections, the algorithm is similar to
the PCG algorithm proposed in [4]. It is shown in fig. 2.
If
there are only few restarts of the algorithm, the CG method in
the inner loop can highly increase the convergence rate of the
reconstruction. Compared to the ML-EM method, the PLSQR
algorithm decreases the number of iterations roughly by a factor
of three. A restart occurred about every fifth iteration.
Table 1 summarizes the storage requirements and the
work per iteration for various reconstruction methods. The
work involved in the matrix-vector products is excluded,
1701
rl
III. IMPLEMENTATION ASPECTS
=
: j
set Xo, i
0
(*) set M-1 = diag(xi(n))
d b - P Xo, {3 Ildll 2, UI (1/{3) d
VI
pT Ub
/V! M-I Vb VI
(1/,) V1
WI
M- I Vb ¢I {3, Pl
for i = 1, 2, ... do
{3 Ui+l = P Vi -, ,Ui
=
=
=
f]
)1
L.. )
,=
=
=
=
=
=,
P= /Pt + (32
c
8
= pdp
= f31 P
step length & = c ¢d P
if(a> 0) set Cl'
min(&,
=
if(&
< O)setCl' =
=
max(&,
In [6][7] it was mentioned that by using a polar grid instead
of a cartesian grid for the image space, the memory and
computational requirements during the reconstruction can be
reduced. This concept was extended to the 3D case; Because
the TierPET scanner uses rotating pairs of planar detectors, the
whole system exhibits a circular symmetry. To adapt to this
geometry, a cylindrical reconstruction volume was chosen, as
shown in fig. 3.
tube of response (TOR)
min [-xi(n)lwi(n)])
----1-----------
w.(n)<O
----
min [xi(n)lwi(n)])
wi(n»O
+
Xi
Xi -1
Cl' Wi
if (a '# &) go to (*)
Vi+1 = pT Ui+1 -
f3 V i
, = JvTtl M-l Vi+l
Vi+l = (II,) Vi+l
Wi+l
M-l Vi+1 Pi+l = -c ,
¢i+l
8 ¢i
[I
n
[]
=
=
(8 ,Ip) Wi
.1 R
~~
Fig. 3 Schematic reconstruction volume
Figure 2: The PLSQR algorithm
method
storage
r
l.
o
[]
(1
U
work
(®,0)
(EB,e)
n
m
m
n
U
-
r,q
g,p
I
2
0
1
u
v,W
3
1
U
V,W
3
I
Table I: Storage and work per iteration
0
2
m
o
I
I
I
enddo
because every listed algorithms perfonns one forward- and
backward-projection per iteration.
[]
I
I
I
I
ML-EM
CO
LSQR
PLSQR
n
I
3
5
6
3
3
Compared to ML-EM, the PLSQR algorithm needs two
more n-vectors of storage. The number of measurements
m ~ 2 . 10 6 is much larger than the size of the image vector
n ~ 2· 10 5 for the 3D TierPET system. Therefore this
additional storage is usually not a problem. The PLSQR
storage requirements are still less than the ones needed by the
original CO method, which needs 2 m-vectors. Concerning the
work per iteration, the CO method is somewhat more efficient
than PLSQR. However, PLSQR is likely to obtain a more
accurate solution in fewer iterations if P is ill-conditioned,
which is typically the case for the reconstruction problem.
The ML-EM method is very economical concerning work and
storage. Aside from the operations listed in table 1, it appeared
that most of the work was involved in the matrix-vector
multiplications. Therefore, special storage techniques were
developed to decrease the memory consumption and speed up
the projection operations. These topics are discussed in the
next section.
=
Typical parameters for the TierPET are: ~R
1 mm,
2.5 0 , ~Z
1 mm. One advantage of this setup is
the higher density of volume elements (voxels) in the center of
the FOV, because the statistics and therefore also the resolution
is improving in this area. Secondly, there are two symmetries
which can be exploited while storing the elements of a tube
of response (TOR), which is defined as the connecting volume
between two special detector elements:
fiy;
=
=
• The rotational symmetry: If the detectors are rotated by
a multiple of ~<p, then only the indices of the voxels are
shifted; the weights connecting a special TOR with the
set of voxels is constant.
• The left-right symmetry within one TOR: Only one half
of the weights need to be stored.
A special storage technique (PRIS, polar relative indexed
storage) has been developed to take advantage of these
symmetries. The weights for one TOR are stored together with
their relative position to the entrance voxel, which is first "hit"
by a forward- or backward projection operation. This set of
parameters can then be used for axially shifted and/or rotated
TORs with the same axial tilt and the same distance to the
z-axis. These two values are determining a special parameter
set.
Because the weights for one TOR contain the nonzero
elements of one row of the system matrix P, this technique can
also be considered as a special sparse matrix storage method.
Although the number of rows m is typically ~ 1· 10 6 , the
resulting number of different TOR parameter sets is reducing
to several hundred. It is now possible to store all needed
TOR parameters in memory without the need to recalculate
\
.1
11997 International Meeting on Fully 3D Image Reconstruction
1711
the weights during every projection operation. Since this
calculation of all needed parameter sets is only performed once
(at the beginning of each reconstruction), one can implement
more precise and expensive models to determine the weight
values [8]. The projection operations (Px, pTb) for one TOR
now reduce to the following steps:
• Determine the start/end voxel.
• Calculate radial distance and axial tilt.
• Determine the correct set of TOR-parameters.
• Beginning at the startlend-voxel, apply the weights stored
in the PRIS format.
After the reconstruction process, the polar coordinates in
every axial slice are sampled on a cartesian grid to ease the
visualization of the result. This is done by a interpolation
method that is minimizing the usual sampling artifacts.
V. CONCLUSION
The overall quality of a high resolution PET system is
determined by the choice of the reconstl'lIction algorithm.
Iterative algebraic methods, which have some advantages
over analytical approaches, are usually demanding in terms
of memory and computer performance. By using acceleration
schemes and optimized data storage techniques, these methods
can be made efficient. Considering these aspects, a CO-based
reconstruction algorithm was implemented for the TierPET
scanner.
First reconstruction results, based on phantom
measurements, demonstrate the quality and performance of this
algorithm.
VI. REFERENCES
[1] S. Weber, A. Terstegge, R. Engels, H. Herzog, R. Reinartz,
P. Reinhart, F. Rongen, H. W. MOiler-GUrtner, and H. Halling,
liThe KPA TierPET: Performance Characteristics and
Measurements." IEEE Nuclear Science Symposium & Medical
Imaging Conference 1996.
[2] S. Weber, En/wickillng eines hochaufloselldcn Positl'OlIell-
IV. RESULTS
The results of two different real phantom measurements is
presented, which were acquired with the TierPET system and
reconstructed using the PLSQR method.
The first phantom is a plastic negative of a half walnut. A
picture of this phantom is shown in fig. 4(a). To fill the volume
with a liquid tracer (FDG), two drilling have been made which
can be closed with two small bolts. The 3.5 ml volume was
filled with an activity of 140 /-lei, and measured for 45 minutes.
The two pairs of detectors (distance 150 mm from crystal to
rotation axis) were rotated in 100 -steps, measuring data in
9 positions. A total of 4.5 . 10 6 events were measured, and
stored in a special Iistmode format. 20 iterations of the PLSQR
method were used to reconstruct the activity distribution. Every
iteration step took 3 minutes, resulting in a total reconstruction
time of 1 hour on a DEC AlphaStation 200 41233. Four
restarts of the PLSQR algorithm were encountered during the
reconstruction process. Only 152 TOR-parameter-sets with
a total of 27639 weight entries (16 bytes each) were used
during the reconstruction. The results are shown in fig. 4(b)
and fig. 4(c). They show iso-surface rendered views of the
reconstructed volume after the polarsrectangular interpolation
process. The two activity filled drillings (1.8 mm diameter) can
be recognized at the beginning of the nut volume.
The second phantom is a line phantom. A plastic cube (4·4·4
mm3 ) was used, and 4 drillings (1 mm diameter) were made. A
picture of the phantom is given in fig. 5(a). The four drillings
were filled with activity (FDG), and closed with adhesive tape.
The distance of the detectors to the z-axis was 120 mm, and
data was acquired at 6 positions, with the detectors rotating in
15 0 steps. A total of 4 . 105 events were stored in 4 minutes.
Again, 20 iterations of PLSQR were used for the reconstruction.
Each iteration took ~ 30 seconds. Only 125 TOR-parametersets with a total of 10223 weight entries were used. The 3D
reconstruction results are shown in fig. 5(b) and fig. 5(c).
11997 International Meeting on Fully 3D Image Reconstruction
[3]
[4]
[5]
[6]
[7]
[8]
Emissions-Tomographen mit kleillem Messvolumen - Das
DeteklOrsystem. PhD thesis, Forschungszentmm Jillich GmbH,
1996.
Y. Vardi, L. Shepp, and L. Kaufmann, "A Statistical Model
for Positron Emission Tomography," Journal of the American
Statistical Association, vol. 80, pp. 8-20, Mar. 1985.
L. Kaufmann, "Maximum Likelihood, Least Squares and
Penalized Least Squares for PET," IEEE Transactions Oil Medical
Imaging, vol. 12, no. 2, pp. 200-214,1993.
C. Paige and M. Saunders, "LSQR: An Algorithm for Sparse
Linear Equations and Sparse Least Squares," ACM Trans.
Mathematical Software, vol. 8, pp. 43-71, Mar. 1982.
K. Kearfott, "Comment: Practical Considerations," Journal of the
American Statistical Association, vol. 80, pp. 26-28, Mar. ]985.
L. Kaufmann, "Implementing and accelerating the EM algorithm
for positron emission tomography:' IEEE Transactions on
Medical Imaging, vol. 6, no. 1, pp. 37-51, 1987.
A. Terstegge, S. Weber, H. Herzog, H. W. MUller-Gartner, and
H. Halling, "High resolution and better quantification by tube of
response modelling in 3D PET reconstruction." IEEE Nuclear
Science Symposium & Medical Imaging Conference, 1996.
1721
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(a) The walnut-phantom
(a) The line-phantom
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(b) 3D Reconstruction
(b) 3D Reconstruction
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(c) 3D Reconstruction
Fig. 4 Reconstruction of the walnut-phantom
11997 International Meeting on Fully 3D Image Reconstruction
(c) 3D Reconstruction
Fig. 5 Reconstruction of the line-phantom
173\
3D-Reconstruction during Interventional Neurological Procedures
K. Wiesent (#), R. Graumann (#), R. Fahrig (*),
A.J. Fox (&), N. Navab (@), A. Bani-Hashemi (@)
(q~)
r.w.
Holdsworth (*),
Siemens AG, Medical Engineering Group, Erlangen, Germany
(*) Robarts Research Institute, London Ontario, Canada
(&) University Hospital, London Ontario, Canada
(@) Siemens Corporate Research, Princeton, N.J., USA
1. Introduction
Intravascular and minimal invasive techniques are of increasing
importance. They may benefit from a combination with 3D-imaging
technology during intervention. Technological an~ mathematical
problems are discussed. These are the distortion and intensity
corrections of the X-Ray Image Intensifier (XRII) , the mechanical
instability of the C-arm, and the additional reconstruction problems
related to partial rotation and truncated projections. Finally we
show results from simulations and from a prototype designed for
rotational angiography~
2. Clinical problems
A typical clinical problem is the endovascular therapy of
subarachnoid aneurysms' using detachable Guglielmi coils.
The success of this procedure critically depends on the packing
of the aneurysm. coils 'projecting into parent vessels may
cause thrombosis, while incomplete filling leads to regrowth
of the aneurysm. Clear visualization of the orifice will be very
helpful, but the current technologies, i.e. fluoroscopic
imaging and roadmapping must remain available. The only solution
of this problem is the usage of open C-arm mounted systems with
additional 3D-imaging capability.
3. Technological problems of C-arm systems
Because of the magnet~c field of the earth and the curvature of
the XRII entrance are~, position dependent correction is necessary. _
For this purpose we developed speci-al-calil:5ra.tion-procedures. -Results are shown for both distortion correction and
intensity correction. A future alternative is the usage of
flat panel detectors [1].
Furthermore, measurements of the mechanical instabilities of the
c-arm are presented. Reconstruction ignoring these instabilities
does not provide useful results.
4. Mathematical procedures
-----------------------~~-
There are two main methods available. Fahrig et al. [2J have
shown that XRI! distortions and gantry instabilities are reproducible
and can be corrected within subpixel accuracy. USing the information
of the calibration procedure measured data are corrected. The
result can be interpreted as 2D-images measured with a stable system
and well known reconstruction procedures like Feldkamp's algorithm
can be applied. N. Navab et ale [3] use a coded fiducial marker
system for dynamical determination of the geometry. The information
11997 International Meeting on Fully 3D Image Reconstruction
1741
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[]
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from this pose determination procedure is used by the reconstruction
algorithm. In the context of this work, we present some generalizations
of Feldkamp's algorithm. A sinogram weighting procedure corrects
data for incomplete rotation (ca. 200 degees) and inequidistant angular
increments. Truncation of projections is not critical. Two methods
are discussed: extrapolation of data and modification of the
convolution kernel. For use during intervention, reconstruction
time is a critical point. We therefore investigated several methods
to speed up the whole procedure.
5. Results
Very promising reconstruction results are shown from simulated data
as well as from measured data. Both the high contrast details of a
skull and the intracranial vessels of an anaesthetic living pig can
be seen nearly without artifacts.
6. Conclusions
3D-Reconstruction from a C-arm mounted system is possible with
diagnostic quality and can provide useful additional information
during therapeutic procedures.
Literature:
[la] Hoheisel et al., Amorphous Silicon.X-Ray Detectors,
9th International School on Condensed Matter Physics,
Varna/Bulgaria 9.-13.9.96, World Scientific
[lb] J. Chabbal et al., Amorphous Silicon X-Ray Image Sensor, 1996,
SPIE Vol. 2708, pp. 499[2] R. Fahrig et al., Characterization 'of aC-arm mounted
XRII for 3D image reconstruction during interventional
neuroradiology, 1996, SPIE Vol. 2708, pp. 351[3] N. Navab et al., Dynamic Geometrical Calibration for
3-D Cerebral Angiography, 1996, SPIE Vol. 2708, pp. 361-
[]
[]
[)
[)
11997 International Meeting on Fully 3D Image Reconstruction
1751
Developnlent of an Object.. Oriented Monte Carlo Shllulator for
3D Positron Tomography
H. Zaidi, A. Herrmann Scheurer* and C. Morel
Division of Nuclear Medicine, Geneva University Hospital, CH-1211 Geneva 4
1. Introduction
Monte Carlo simulation of 3D PET data is a very
powerful tool to check the performance of image
reconstruction algorithms and their implementations.
Since it allows to obtain separate images of prompt
and scattered events, it may help developing and
evaluating 3D attenuation and scatter correction
techniques. Furthermore, providing its design is
easily extendible, it represents an efficient tool to
study different 3D PET scanner configurations. We
present an object-oriented, extendible design for a
Monte Carlo simulator for 3D positron tomography.
Preliminary results from phantom simulation studies
including attenuation and scattering of the gamma
rays in the field-of~view are presented and future
prospects discussed.
2. tvIcthods
objects, or by escaping the PET scanner geometry
and field-of-view. Photoelectric absorption, as wcll
as incoherent and coherent scattering are taken into
account to simulate photon interaction within scatter
and detector objects. Interaction crossasections and
scattering distributions are computed from
parametrizations that were implemented in the
GEANT simulation package of CERN. Interaction
within scatter or detector objects can be switched on
and off interactively. In case interaction within
detector objects is switched off, any photon
impinging on a detector is assumed to deposit all its
energy in the detector crystal. Energy resolution of
the detector is simulated by convolving the deposited
energy with a Gaussian function. Photon pairs are
recorded in the sinogram object once two photons
resulting from one annihilation event have passed the
energy window set for discrimination.
2.1 Software description
The Monte Carlo simulator, Eidolon, was written
in Objective-C and runs under NextStep 3.3 on an
HP 9000 workstation. A graphical user interface
allows one to select scanner parameters such as the
number of detector rings, detector material and sizes,
discrimination thresholds and energy resolution. It
also allows to choose a set of simple 3D shapes, such
as parallelepiped, ellipsoid or cylinder, for both the
annihilation sources and the scattering media, as well
as their respective activity concentrations and
chemical compositions. One may view the reference
image and the sinograms as they are generated.
2.2 Design
In order to ease the job of incrementally adding
capabilities to the Monte Carlo simulator, a modular
design featuring dynamically loadable program
elements or bundles was adopted. The basic building
block is a model element class which allows elements
to be browsed, inspected, adjusted, created and
destroyed through a graphical inspector. This was
then used to implement simple parametric source,
detector and scatter classes and sinogram and image
classes to view and save the generated data in CTI
Matrix 6 format. A controller object oversees the
simulation process. The reference image and
sinogram displays are periodically updated.
The model assumes a cylindrical array of detector
crystals and known spatial distributions of
annihilation sources and scatter phantoms. Pairs of
annihilation photons are generated uniformly within
the source objects and are tracked until they expire,
either by interacting within scatter or detector
*Present address: Institute of Physiology, University of
Lausanne, CH -10 15 Lausanne.
11997 International Meeting on Fully 3D Image Reconstruction
3. Results
The time needed to perform a simulation study
depends on the complexity of the chosen sets of
source, scatter and detector objects, and on selected
interactions. The average time to track one
coincident detection for the ECAT -953B PET
scanner (16 detector rings, 256 sinograms, 96 views
of 128 elements each) is 1.15 ms without scattering
nor attenuation. It increases to 11 ms if photon
interaction is simulated within a single uniform
scatter object corresponding to a 20 cm diameter
cylinder filled with water, and to 15.2 ms if it is
simulated within both the scatter and the detector
objects.
Eidolon was used to obtain unscattered and
scattered energy distributions of coincident
detections (Fig. 1), as well as to study line-spread
functions (Fig. 2) and scatter fractions for the ECAT953B PET scanner. The scatter fraction is defined as
the ratio between the number of coincident
detections with at least one photon scattered in the
field-of-view and the total number of coincident
detections. Table 2 shows scatter fractions obtained
with Eidolon for three different radial positions of a
line source placed in a 20 em diameter cylinder filled
with water. Ten million annihilation events were
generated for each radial position of the line source.
The scatter fraction determined with the line source
in the centre of the phantom is 0.37. In the same
detection conditions, a real measurement of this
scatter fraction gave 0.42 [1] and it was estimated to
be 0.46 using a different Monte Carlo simulator [2].
The discrepancy between our value and the one
given in Ref. 2 results from the fact that photons
which are scattered only within the detector crystals
are not considered to be scattered regarding the
scatter fraction determination, as they are in Ref. 2.
1761
[]
7000
6000
5000
lj
4000
[]
3000
r~l
2000
1000
o
-_:
[. I
[]
[]
[J
f1
LJ
300
350
400
450
500
550
600
650
700
Energy (keV)
Figure 1: Energy distributions of coincident detections resulting from the simulation of a line source placed in the centre of a
20 cm diameter cylinder filled with water. Photons impinging on a detector are assumed to deposit all their energy in the
detector crystal. Energy resolution is proportional to the inverse square root of the deposited energy and is simulated by
convolving the deposited energy with a Gaussian function whose FWHM is 23% for 511 keY photons. Both photons of a
coincident detection have to pass an energy window set between 250 and 850 keY. Distributions of photons resulting from
exactly one or two successive Compton scatterings in the field-of-view are shown.
As for real measurements, the scatter fraction
decreases when the source moves off-axis.
Radial position
Eidolon Spinks
Michel
[mm]
[1]
[2]
o
0.37
0.42
0.46
40
0.36
0.40
80
0.29
0.30
Table 1: Comparison between Monte Carlo estimations
and real measurements of the scatter fraction for different
radial positions of a line source placed in a 20 cm diameter
cylinder filled with water. Same detection conditions as in
Figure 1 apply, except that interaction within detector
objects was switched on and energy window was set
between 380 and 850 keY for comparison with Ref. 1.
Eidolon was also used to simulate the Utah
phantom which is designed with a high degree of
inhomogeneity both transaxially and axially in order
to compare and test scatter correction techniques in
3D PET. Utah phantom data sets for the ECAT-953B
PET scanner were generated both with and without
scatter simulation (Fig. 3). The outer compartment of
the phantom which is generally used to provide
activity from outside the field-of-view is kept empty.
11997 International Meeting on Fully 3D Image Reconstruction
Generated data sets were reconstructed using four
different exact and approximate 3D reconstruction
algorithms implemented of a high performance
parallel platform [3]. Attenuation corrections were
applied before reconstruction to the data sets
generated with scatter simulation. The attenuation
correction files were created by forward projecting
the 3D density map estimated with a constant linear
attenuation coefficient of 0.096 cm- I .
4. Discussion and conclusion
Validation
of image
reconstruction
implementations and scatter correction techniques, as
well as design of new 3D PET systems using the
Monte Carlo method have received considerable
attention during the last decade and a large number
of applications have been developed. The objectoriented paradigm makes it possible to envision
incremental refinements to any of the elements
described in this extended abstract with maximum
code reuse by providing a framework for effectively
defining standards using the inheritance mechanism.
This approach streamlines development and
1771
6
10 ,.-----------------------------------------------.
Projection bin
Figure 2: Sum of one"dimensional transaxial projections resulting from the simulation of a line source placed in a 20 cm
diameter cylinder filled with writer. Same detection conditions as in Figure 1 apply.
improves reliability. It makes Eidolon a very
powerful tool that can be further modified to
investigate new possible designs of high performance
positron tomographs. Eventually, Eidolon will be
exploited to explore different sampling schemes of
the 3D X-Ray transform.
Although variance reduction techniques have
been developed to reduce the computation time, the
main drawback of the Monte Carlo method is that it
is extremely timeMconsuming. With the development
of parallel-processing computers, researchers have
turned their efforts towards the parallelisation of
Monte Carlo codes. An implementation of Eidolon
on a parallel system with 8 PowerPC-604 nodes that
was recently installed in our laboratory is being
presently undertaken.
Acknowledgements
This work was supported part by the
Schmidheiny Foundation, the Swiss Federal Office
for Education and Science under grant E3260 within
the European Esprit project HARMONY (CE 7253)
and the Swiss National Science Foundation under
project 2100-043627.95.
11997 International Meeting on Fully 3D Image Reconstruction
References
[1] T.J. Spinks, T. Jones, D.L. Bailey, et al., "Physical
performance of a positron tomograph for brain
imaging with retractable septa", Phys. Med. Bioi. 37
(1992) 1637"1655.
[2] C. Michel, A. Bol, T. Spinks, D. Townsend, D.
Bailey, S. Grootoonk and T. Jones, IIAssessment of
response function in two PET scanners with and
without interplane septa", IEEE Trans. Med. Imag. 10
(1991) 240-248.
[3] M.L. Egger, A.K. Herrmann Scheurer, C. Joseph and
C. Morel, IIFast volume reconstruction in positron
emission tomography: Implementation of four
algorithms on a high-performance scalable parallel
platform", to appear in Conj. Rec. 1996 IEEE Med.
Imag. Conj., Anaheim, CA, 1996.
[4] P.E. Kinahan and J.G. Rogers, "Analytic 3D image
reconstruction using all detected events", IEEE Trans.
Nucl. Sci. 36 (1989) 964-968.
[5] M. Defrise, D.W. Townsend and R. Clack, "Favor: a
fast reconstruction algorithm for volume imaging in
PET", in Conf. Rec. 1991 Med. Imag. Conj., Santa Fe,
NM, 1991, pp. 1919-1923.
[6] M. Defrise, "A factorization method for the 3D xRay transform", Inverse Problems 11 (1995) 883-994.
[7] M.E. Daube-Witherspoon and G. Muehllehner,
"Treatment of axial data in three-dimensional PET",
J. Nucl. Med. 82 (1987) 1717-1724.
1781
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112
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Piulbin
Figure 3: Eleventh slice reconstructions of Monte Carlo data sets of the Utah phantom generated without (left) and with
(right) scatter simulation. The reconstruction algorithms used are (from top to bottom): the Kinahan and Rogers reprojection
algorithm [4], the Fast Volume Reconstruction algorithm (FAVOR) [5], the Fourier Rebinning algorithm (FORE) [6] and the
Single-Slice Rebinning algorithm (SSRB) [7]. Same detection conditions as in Figure 1 apply. Seven million coincident
detections were recorded for both types of simulations. The maximum obliquity used for reconstruction corresponds to a ring
index difference of 11. No additional polar or azimuthal mashing was used. Horizontal profiles though the centre of the slices
are shown.
i
11997 International Meeting on Fully 3D Image Reconstruction
1791
BlockHIterative Techniques for Fast 4D Reconsttuction
Using A Priori Motion Models in Gated Cardiac SPECT
David S. Lalush and Benjanlin M. W. Tsui
Departlllent of Biolnedical Engineering and Department of Radiology
The University of North Carolina at Chapel Hill
Introduction
We investigate teclmiques for accelerating fhlly
four-ditnensional reconstruction algoritluns used
for gated cardiac SPEeT stl1dies. Gated SPECT
synchronizes the acquisition of tOlllography data
with the cardiac cycle, thus pennitting the
reconstnlCtion of a tune sequence of images
instead of a single 1110tiol1 blurred inlage. In effect,
the acquired data becomes four-dimensional, with
the fourth dinlension being the individual "time
fratnes" into which the data is binned. Each time
frame represents one segment of the cardiac cycle.
The time-sequenced images not only reduce the
effect of Inotion blurring, but also provide
impoliant cluneal information about wall motion,
wall thickening, cardiac volumes, and ejection
fraction. F or these reasons, gated SPECT studies
are finding ever widening use in nuclear medicine
clinics.
The primary disadvantage associated with gated
SPECT is a considerable increase in noise. Since
neither the patient dose nor the imaging time can be
significantly increased, each time frame of data has only
a fi'action of the counts which would be obtained in a
conventional ungated study. If linear filters are used to
smooth this noise, they may degrade the spatial and
temporal resolution we hope to gain by doing the gated
study.
We have investigated the use of "space-time" Gibbs
priors in a fully 4D MAP-EM algorithm [1, 2]. These
priors permit smoothing in the three spatial dimensions
as well as the time dimension, and they have been
shown to smooth noise with less degradation to both
spatial and temporal resolution as compared to 3D or
4D linear filtering. They also permit us to include prior
information about the actual motion of the heart, if such
information can be reasonably estimated. The MAP
algorithm does require that all of the time frames be
reconstructed simultaneously, resulting in significant
program memory' requirements.
Also, the EM
algorithm used is inherently slow, requiring at least 50
iterations and 12-24 hours of processing time to
e
complete the 4D reconstruction with compensation for
nonuniform attenuation and detector response.
In this paper; we consider two techniques for
decreasing the reconstruction time for these large 4D
datasets. The techniques are based on the Rescaled
Block-Iterative (RBI) EM procedure [3], which is
related to the popular Ordered"Subsets (OS) EM
algoritlun [4]. Both OS~EM and RBI-EM have been
shown to provide reconstructions with similar properties
to ML-EM, but requiring considerably fewer iterations
and significantly less processing time [5]. Neither OSEM nor RBI-EM has been translated into a satisfactory
MAP procedure, however, one where smoothing
constraints can be incorporated into the reconstruction
algorithm. The reason is that neither is a true
optimization algorithm, so there is no objective function
to maximize.
The two RBI-based algorithms include smoothing
constraints in the form of spaceMtime Gibbs priors. The
algorithms result from exploiting similarities between
the RBI-EM and ML-EM algorithms, to derive the
"correct" way to add smoothing constraints to the RBIEM algorithm. The first technique results from a direct
analogy to MAP-EM, and reduces the number of
iterations required from fifty for MAP-EM to five for
MAP-RBI-EM. The second method results from a
further modification to reduce processing time per
iteration by a factor of about seven.
The MAP-RBI-EM Algorithm
The MAP-RBI-EM algorithm results fi'om
interpreting the RBI-EM algorithm as an additive
update procedure with the same form as the additive
version of the ML-EM algorithm. In the full paper, we
present the details of the derivation. The resulting form
turns out to be a general reconstruction algorithm of
which RBI-EM, OS-EM, MAP-EM, and ML-EM are
special cases when there is no prior (RBI-EM, 08EM), a single subset (MAP-EM), or both (ML-EM).
The 4D form of this algorithm accommodates the fourdimensional motion model we apply to gated SPECT
reconstruction.
The MAP-RBI-EM algorithm reduces to five the
number of iterations required from fifty for the 4 D
11997 International Meeting on Fully 3D Image Reconstruction
1801
MAP-EM procedure.
The properties of the
reconstructed images are essentially the same in both
cases, for the same set of prior parameters. Because the
MAP-RBI -EM algoritlun requires evaluating the prior
term once per subset instead of once per iteration as in
MAP-EM, the time per iteration is much longer for
MAP-RBI-EM. Generally, there may be as many as
thirty-two subsets per iteration, so the time to compute
the prior term becomes significant. While the new
algorithm requires one-tenth the number of iterations of
MAP-EM, it requires about 75% of the processing time
when our most complex prior model is used. Thus, the
acceleration is not as significant as what is found in
non-MAP versions of ML-EM and RBI-EM.
[]
[1
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The Modified Algorithm
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To further decrease reconstruction time, we
determined that it would be necessary to reduce the
number of times the prior term is evaluated. Under
normal circumstances of MAP reconstruction, even
thirty-two evaluations of the prior term per iteration
might not add significantly to the processing time. In
the 4D context, however, the prior structures are
extremely complex, especially when motion is modeled
in the prior. Each time-space voxel in the 4D space is
linked with all of its nearest neighbors in four
dimensions, thus requiring a large number of nonlinear
calculations.
To address these problems, we have developed. a
modified algorithm which requires computing the prior
term only once per iteration, as in MAP-EM, but uses
the RBI approach to accelerate convergence. For the
modified algorithm, the prior term is computed once, at
the beginning of each full iteration, using the image
estimate at the end of the previous iteration. This
results in an inaccurate estimate of the prior tenn at
each sub iteration; however, as the iterated image
estimates come close to convergence, the image
estimates change slowly. Thus, the single computation
of the prior is a reasonable estimate for the whole
iteration as we approach convergence. At iterations one
or two, the results may be significantly different from
those of MAP-RBI-EM, but the two cases appear to
converge around iteration five to images with similar
properties of smoothing, noise, and spatial and temporal
resolution.
The modified algorithm requires the same number of
iterations as MAP-RBI-EM, but the processing time per
iteration is reduced to that of MAP-EM. Thus, it truly
requires about one-tenth the processing time of MAP-
EM, or· about 90 to 180 minutes for a complete 4D
reconstruction on a DEC AlphaStation 200 workstation.
Evaluation
The three reconstruction methods were evaluated on
gated SPECT data simulated from a version of the
Mathematical Cardiac Torso (MCAT) phantom [6]
which models the beating and rotating motion of the
heart. The phantom used included a cold lesion in the
inferiolateral wall of the left ventricle which moves with
the heart. Two datasets were simulated, one to emulate
a Tc-99m gated study using a LEHR collimator, the
other to emulate a TI-201 gated study using a LEGP
collimator. For each of the two agents, sixteen datasets
were simulated using Monte Carlo methods accounting
for effects of nonunifonn attenuation, detector response,
and scatter. Each of the sixteen represented one time
frame from the complete cardiac cycle. Noise was
simulated to approximate the count level from actual
patient Tc-99m and TI-201 studies.
Reconstructions of these data were performed using
each of the three 4D reconstruction procedures:';MAPEM, MAP-RBI-EM, and the hybrid algorithm. The
Gibbs priors used were those which apply prior
assumptions about the motion of the heart. Priors with
both correct motion and erroneous motion were applied.
The prior parameters were set to be the same except
where alteration of the global smoothing power was
necessary to achieve the same level of smoothing. Also,
non-MAP versions of each were applied, and followed
by 3D and 4D linear filters for comparison.
All of the algoritluns were evaluated on the basis of
their ability to recover activity levels in the region of the
cold lesion by region-of-interest (ROI) analysis, and on
the basis of spatial resolution recovery for the same
level of noise smoothing. Also, reconstruction times
were measured for each algorithm.
Results
Results were similar for both the Tc-99m study and
the noisier TI-201 study. We found that all three MAP
algorithms
produced
comparable
results.
Reconstructions using the 4D prior model in all cases
preserved spatial and temporal resolution better than
linear filtering, for the same level of noise smoothing.
Even with modest errors in the motion model, the MAP
methods outperformed their counterparts with linear
filtering.
The reconstructed images from five iterations of
each of the two new techniques were found to be
comparable to those from fifty iterations of the 4D
11997 International Meeting on Fully 3D Image Reconstruction
1811
MAPMEM procedure. However, the processing times
for the MAPwRBI-EM and modified algoritluns were
quite different, with MAP-RBI-EM requiring 75% of
the time of the MAP-EM algoritlull and the hybrid
requiring only 10%.
Conclusion
We conclude that the two 4D RBI~based
reconstruction algorithms introduced here produce
reconstructions that are cOlnparable to those from the
4D MAP"EM algoritlull, but require less processing
tiIne. The hybrid algol'itlun, however, would be the
favored of the two, since it is considerably faster than
either MApwEM or MAPMRBI ..EM.
The hybrid
approach can produce complete 4D reconstructions 'of
gated cardiac SPECT data, with compensation for
attenuation and 3D detector response, in 1w2 hours on
currentlYRavailable workstations, and Inay thus make
such reconstructions possible in clinical settings. We
conclude also that the 'priors incorporating assmnptions
about motion into the model are helpful for both Tc991n and TI-201 gated studies, that such priors offer
noise smoothing with better preservation of spatial and
temporal resolution than linear filters, and that motionbased priors are not sensitive to errors in the motion
assunlptions used.
presented at Conference Record of the 1996 IEEE
Nuclear Science Symposium and Medical Imaging
Conference, Anaheim, California, 1996.
[6]
B. M. W. Tsui, J. A. Terry, and O. T.
Gullberg, "Evaluation of cardiac cone-beam Single
Photon Etnission Computed Tomography using
observer perfonnance experiments and receiver
operating characteristic analysis," Invest Radiol, vol.
28, pp. 1101-1112, 1993.
Table 1: Processing times for the three algorithms.
Reconstructions were from simulated TI·20 1 gated SPECT
data for 64 views over 180 degrees from 45° RAO to 45°
LPO.
Sixteen time frames were reconstructed
simultaneously. There were 64 bins per slice per view, and
24 slices were reconstmcted on 64x64 grids for each time
frame. NonunifOim attenuation and 3D detector response
were modeled in the reconstructions. The RBI techniques
used 32 subsets with two views per subset. Processing times
. 200 workstation.
.
were measunLcf on a DEC' AllpjhaStatlOn
Algorithm
MAP-EM RBI-MAP- Modified
EM
RBI
50
Iterations
5
5
Processing
25.1
time
18.1
2.5
(hours)
'-"
700
References
I
600 <~lSJ 0
[1]
D. S. Lalush and B. M. W. Tsui, "Space-time
~500 ''v~~
Gibbs priors applied to gated SPECT myocardial
'S;
perfusion studies," in Three-dimensional Image
.- 400
'0
Reconstruction in Radiology and Nuclear Medicine.
c( 300
Dordrecht, Netherlands: Kluwer Academic Publishers,
(5
-+-MI-EM
1996, pp. 209-224.
a: 200
o RBlsEM
[2]
D. S. Lalush and B. M. W. Tsui, "A priori
100
motion models for fourwdimensional reconstruction in
gated cardiac SPECT," presented at Conference Record
o o~--------~------------------~
5
10
15
of the 1996 IEEE Nuclear Science Symposium and
Medical Imaging Conference, Allaheitn, California,
Frame Number
1996.
Figure 1: Plot of activity in a region of interest (ROJ) for the
[3]
C. L. Byrne, "Block-iterative methods for sixteen time frames in images reconstructed by each of the
itnage reconstruction from projections," IEEE three algorithms: MAP-EM, RBI-MAP-EM (represented by
Transactions on Image Processing, vol. 5, pp. 792- RBI-EM in the legend), and Modified RBI-MAP-EM (Mod
794, 1996.
RBI in the legend), as well as ML-EM followed by a 4D
[4]
H. M. Hudson and R. S. Larkin, "Accelerated filter. The ROI is placed so that the defect rotates into it
image reconstruction using ordered subsets of' (note the drop in activity at frame 5) and then back out. The
projection data," IEEE Trans Med 1m, vol. 13, pp. 601- noise-free ML-EM reconstruction is shown as a basis for
609, 1994.
comparison, since it is the best we could hope to do with the
[5]
D. S. Lalush and B. M. W. Tsui, "Convergence given projection model. All three MAP algorithms are
and resolution recovery of block iterative EM able to recover the motion similarly, especially the
algorithms modeling 3D detector response in SPECT," transition from frames 5 to 10. Temporal resolution is
maintained better than the filtered ML-EM result.
11997 International Meeting on Fully 3D Image Reconstruction
1821
\--1
I
1._,
[]
11
[-1
11,j
[]
[-1
J
1-!J1
0
Figure 2: One slice taken from sixteen frames of MCAT
phantom reconstructions from noise-free data using 50
iterations of ML-EM, considered the best result possible
with the given model. The inferiolateral defect rotates into
this slice and back out again.
Figure 4: One slice taken from sixteen frames of MCAT
phantom reconstructions from noisy TI-20 1 data using 5
iterations of RBI-MAP-EM with the true motion modeled in
the prior.
Figure 3: One slice taken from sixteen frames of MCAT
phantom reconstructions from noisy TI-201 data using 50
iterations of MAP-EM with the true motion modeled in the
prior.
Figure 5: One slice taken from sixteen frames of MCAT
phantom reconstructions from noisy TI-201 data using 5
iterations of modified RBI-MAP-EM with the true motion
modeled in the prior.
[-l
I
I
\
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I
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0
[]
[]
I
[:1
[
f
I
L
0
[
'I
Figures 2 through 5 show the results of reconstructing the simulated TI-201 gated SPECT data with the three
reconstruction algorithms.
Both nonunifonn attenuation and 3D detector response are modeled in the
reconstructions. The three algorithms provide similar, though not identical, smoothing properties when the
same prior is used. The RBI algorithms require greater relative weighting on the prior than MAP-EM
because the noise tends to increase more quickly. All are effective at smoothing noise while preserving spatial
and temporal resolution, even with very noisy TI-201 gated data, if heart motion can be estimated a priori.
r -1
t
I
i
11997 International Meeting on Fully 3D Image Reconstruction
1831
Ora ft \. ~r:\ i l.lI1
15
SPECT Reconstruction .. The
Jal~lIar~
19~)-
~l to' ~Iodel
Donald L. Gunter
Rush Presbyterian St. Luk~':; ), [~d ica l C~nter
Chicago. IIlinoi!i
.A.bsrract
l\ realistic model of the imaging process in SPECT acquisitions is proposed that
incorporates constant attenuation (~l) within the patient body, a depth .. dependent
point.. source response function (t), and the intrinsic resolution (0') of the gamma ..
ray cmnera. The resulting integral equation represents the fully three .. dimensional
SPECT imaging system. An analytic inversion algoritlun is derived for this model
that does not have a low spatial.. frequency cutoff and, therefore, is not equivalent to
the Tretiak..Metz inversion even in the limit of the exponential Radon transform
(t=cr=O).
l. Introduction
In single ..photon emission computed tomography (SPECT) imaging, trace amounts of
radiopharmaceuticals are injected into a patient. These radiopharmaceuticals are absorbed in tissues
of physiologic interest and subsequently emit gamma rays that are imaged with a gamma camera.
The gamma camera rotates around the patient and two~dimensional (2D) proj ected images are
acquired from many directions. The mathematical problem posed by SPECT is the determination
of the three .. dimensional (3D) distribution of activity within the body from these projected 2D
images.
The standard inversion technique used in SPECT is filtered backprojection, which is based on
the ~nalytic inversion of the two "dimensional (2D) Radon transform. Unfortunately, the 2D Radon
transform, which assumes that image intensity is proportional to a line integral of the activity
distribution through the body of the patient, is a rather crude approximation of the SPECT imaging
process. Consequently~ the reconstruction often introduces artifacts that can signific:antly_affectthe
diagnostic results. The five most important physical factors affecting the SPECT imaging process
(in approximate order of importance) are (1) the position dependent pointwsource response function
(PSRF) produced by the camera collimator, (2) the attenuation of the radiation \vithin the tissues of
the patient'S body, (3) the noise caused by quantum counting statistics, (4) the scattering of
radiation within the patient and collimator, and (5) the intrinsic spatial resolution of the gamma
camera.
In this paper, a realistic mathematical model is proposed for data acquisition in SPECr and a
reconstruction algorithm is derived. The proposed model, called the ~'t'cr model, contains three
parameters which characterize the attenuation (M), the PSRF (t), and the intrinsic resolution (cr).
Ho\vever, this model treats noise naively and ignores scattering completely. The proposed
11997 International Meeting on Fully 3D Image Reconstruction
1841
reconstruction algorithm provides an exact analytic inversion for the exponential Radon
transrom1ation (the special case -r=O: ~l,G¢O) and produces an approximate inversion for non-
II
vanishing values of 1'.
[]
for SPEeT reconstruction and have substituted various iterative methods. Two significant
In recent
years~
researchers in nuclear medicine have generally abandoned
anal~ lie
techniques
developments motivated this shift in emphasis. First, Tretiak and iVletz (1980) derived a closedform analytic inversion for the exponential Radon problem (in the· notation of this paper t=cr=O,
~~O).
Unfortunately, the analytic inversion proposed by Tretiak and NIetz amplifies noise and is,
therefore, unsuccessful in clinical applications. Concurrent with the failure of this exact analytic
[]
technique, other researchers began using maximum likelihood and other iterative techniques that
were more easily implemented on computers. The results of these iterative techniques are generally
D
superior to both filtered backprojection (that does not compensate for any of the physical processes
[1
circumstances, most researchers concluded that iterative algorithms were the appropriate tool for
involved in SPECT imaging) and the Tretiak-Metz algorithm (that enhances noise). Under these
SPECT reconstruction.
Recently, Metz and Pan (1995) reexamined the Tretiak-Metz algorithm and found a';new
D
method that minimizes the noise amplification. Metz and Pan utilize various symmetries imposed
by reality conditions (Le., both the source distribution and the projected images must he real
D
functions in the spatial domain) on the projected images and source distribution in the frequency
D
domain) in a way that minimizes the effects of noise. This method successfully suppresses much
[~
reconstruction. Unfortunately, the Metz-Pan method ignores the fundamental cause of noise
[J
[J
domain. U sing these symmetries, they combined signals that should be equal (in the frequency
of the noise that is amplified by the Tretiak-Metz inversion. and has renewed interest in analytic
amplification in the Tretiak-Metz algorithm.
One of the striking features of the Tretiak-Met.z inversion is the introduction ofa low-frequency
cutoff in the reconstruction. Data from low spatial-frequencies «J..l) are discarded and ·only data
from frequencies larger than the attenuation coefficient are included in the reconstruction. Because
of this low-frequency cutoff, the least noisy data are discarded and only the comparatively noisy
high-frequency data are used in the reconstruction. Intuitively, one expects that the SPEeT
[I
projection data at low spatial-frequencies contain significant information about the source
[J
high spatial-frequencies and, in the absence of noise, the data from the high spatial-frequencies is
distribution. The Tretiak-Metz algorithm demonstrates that this information is duplicated in the
sufficient for reconstruction. However, in noisy imaging systems like SPECT, the high-frequency
data are not equivalent to the data at the low spatial frequencies. Consequently, an algorithm that
D
[J
uses data from low spatial-frequencies should be more immune to noise than an algorithm that uses
exclusively high-frequency data.
A number of researchers working on exponential Radon
(
I
L
11997 International Meeting on Fully 3D Image Reconstruction
1851
transfotm attempted to remove the low-frequency cutoff introduced by Tretiak and Nletz. but none
succeeded.
The ~t 'CO' luodel was originally formulated as a realistic and mathen1atically simple
representation of the SPEeT imaging process in three dimensions. Surprisingly, hovvever, the
analytical singularities that plague the inversion of the exponential attenuation (t=O'=O) model and
produced the 10wHfrequency cutoff are ameliorated by the introduction of a depth· dependent PSRF
(t¢O). As a direct consequence, the low-frequency cutoff can be eliminated from the
reconstruction algorithm. The removal of this cutoff is the major result of this paper. Whether this
algorithm is practical for clinical applications is cUlTently under investigation.
11997 International Meeting on Fully 3D Image Reconstruction
1861
Strategies for Fast Implementation of Model-Based Scatter Compensation in
Fully 3D SPECT Image Reconstruction
Dan 1. Kadnnas l, Eric C. Freyl,2, and Benjamin M.W. Tsui l,2
IDepartment of Biomedical Engineering and 2Department of Radiology
University of North Carolina-Chapel Hill, Chapel Hill, N.C.
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Abstract
Iterative reconstruction-based scatter compensation (RBSC) is a technique in which the scatter response
function is modeled during the reconstruction process. It can be very accurate and has good noise properties;
however, substantial computational effort is required to model the complex process of photon scatter. When
using RBSC for fully 3D SPECT image reconstruction, most of the time required for reconstruction is spent in
the scatter model calculation. In this work we investigate strategies of implementing model-based scatter
compensation that limit the number of iterations that use a. projector incorporating the scatter model. The
scatter component of SPECT projection data tends to contain mostly low frequencies. When using iterative
reconstruction algorithms, the scatter estimate converges in relatively few iterations. We propose to use scatter
models only during a limited number of iterations, holding the scatter estimate constant for the other iterations.
The approach was evaluated by reconstructing Monte Carlo simulated projection data of the MCAT torso
phantom. When using accelerated iterative algorithms, it was found that accurate scatter compensation can be
achieved with as few as two or three scatter model calculations. This approach may ultimately bring modelbased scatter compensation reconstruction times down to the realm of being clinically realistic.
1. INTRODUCTION
Accurate reconstruction in SPECT requires compensation for the effects of attenuation, detector response,
and scatter. Accurate and efficient methods of compensating for attenuation and detector response have
already been developed. However, compensating for the effects of scatter is more difficult. A promising
approach to scatter compensation, iterative reconstruction-based scatter compensation (RBSC), involves
modeling the scatter response function (SRF) in the projector (and backprojector) of an iterative
reconstruction algorithm [1]. In effect, scatter compensation is perfonned by mapping scattered photons back
to their point of origin. Iterative RBSC has been found to result in images with less bias and reduced variance
as compared to subtraction-based scatter compensation methods.
The major shortcomings of RBSC are that the scatter models are very computationally intensive, and
iterative recovery of image features is slowed when scatter is modeled (hence, more iterations are required).
These two effects result in greatly increased reconstruction times, even in the 2D case. When inter-slice scatter
is included, reconstruction times become prohibitive. To overcome this problem, several researchers have
attempted to develop faster scatter models with some success. However, due to the complexity of the scattering
process, it is unlikely that very fast models will be developed without sacrificing accuracy. In past experience
with these methods, we have observed that scatter effects in the image estimate are reduced in a small number
of iterations, but resolution loss due to detector response blurring takes many iterations to recover.
Based on these observations, we have developed strategies of implementing model-based scatter
compensation that limit the number of times scatter estimates must be calculated. We call this approach
Intennittent RBSC, since scatter-modeling is performed intermittently during the iterative reconstruction
process. The rate of iterative convergence of modeled scatter estimates was first examined, and the results were
used to determine the minimum number of iterations at which the scatter estimate must be updated. Several
implementations of the approach were then tested and evaluated by comparison to standard RBSC methods.
II. METHODS
A. Simulated Phantom Experiment
The Intennittent RBSC methods are evaluated by reconstructing Monte Carlo simulated projection data
of the MCAT torso phantom shown in Figure 1. In order to limit the computational requirements of the
experiment, the phantom was chosen to be uniform in the axial direction. This allowed us to reconstruct a
single slice of the phantom while including both inter-slice and intra-slice scatter. The SIMIND [2] Monte
Carlo program was used, and the parameters of the simulated SPECT acquisition are given in Table 1. A large
number of photon histories were simulated to obtain essentially noise-free data. The data were later scaled and
Poisson noise added to simulate a scan which acquired 1.9xl05 counts total.
11997 International Meeting on Fully 3D Image Reconstruction
1871
Figure 1. MCAT phantom activity distribution (left) and attenuation map (right)
B. Reconstruction Methods
The data were reconstructed using the fast rescaled block iterativeaexpectation nlaxinlization (RBIMEM)
algorithm [3,4]. This algorithm is an accelerated relative of MLEM similar to orderedHsubsets (OS-EM).
Models for non .. uniform attenuation and detector response were included in both the projector and
backprojector at each iteration, and the effective source scatter estimation model [5] was used as indicated.
Other researchers have shown that scatter need not be modeled in the backprojector, and faster convergence
results if it is only modeled in the projector. For our standard methods, we model scatter in (1) both the
projector and backprojector, and (2) in the projector only. We refer to these as Full RBSC and Forward RBSC,
respectively. For the Intennittent RBSC methods, the scatter model, when used, appears in the projector only.
For comparison, we have also reconstructed images without modeling scatter (No Scatter Compensation).
Descriptions of the methods used are given in Table II.
For the Intermittent RBSC methods, scatter compensation is achieved by incorporating an es#mate of the
scatter component of the projection data into the iterative algorithm. This is similar to subtracting the scatter
estimate prior to reconstruction, but has the advantage that is preserves the Poisson statistics of the projection
data. Intermittent RBSC differs from Forward RBSC in that the scatter estimate is not updated at each iteration.
The scatter estitllate is only updated intermittently, being held constant for the other iterations.
Table I. Parameters of the simulated
SPECT acguisition.
Tracer: Tc-99m Sestamibi
" 643 Image matrix, 0.625 cm pixels
" LEGP parallel hole collimator
- Energy resolution 11% FWHM at 140 keY
- 20% wide photopeak energy window
- 64 x 64 Projection matrix, 0.625 cm bins
- 64 Projection views evenly spaced over 3600
- Simulated effects of nonunifonn attenuation
and detector response
RIO Orders of scatter simulated,
both coherent and incoherent
8
Table H. Descri tion of the Intermittent RBSC methods.
Name
Scatter Mqdeled in Scatter modeled at
projector &
All iterations
Full RBSC
backprojector
projector
All iterations
Forward RBSC
Intermittent RBSC Methods:
Every Other
projector
1,3,5,7, ...
Doubling
projector
1,2,4,8, ...
projector
First 3
1,2,3 only
1;3-only
1&3
projector
No Scatter Comensation
N/A
N/A
--"
III.
RESULTS
A. Iterative Convergence of Scatter Estimates
Figure 2 shows the projected scatter estimate sinograms for several iterations of Full RBSC. The dark
bands on the sinogram for the fIrst iteration arise because the inaccurate quantitation of the initial image
estimate affects the early subsets the most. This gross quantitative difference is resolved by the first few subsets
of the first iteration, hence the scatter estimates" projected after the first" few subsets are much closer to the true
values. In fact, the scatter estimates are nearly fully converged after only two iterations of RBI-EM
(Figure 2b). The Intennittent RBSC methods developed in this work are designed to exploit this fast
convergence of the scatter estimate.
11997 International Meeting on Fully 3D Image Reconstruction
1881
[J
10
( a)
(b)
20
30
40
50
60
Projection Bin number
Figure 2. Scatter estimate sinograms (a) as projected at RBI-EM iterations I (top left); 2 (top right), 3 (bottom left),
and 50 (bottom right). The dark bands on the sinogram for the first iteration are explained in the text. Horizontal
profiles at the angle indicated are shown at the far right (b). The projected scatter estimates are very nearly converged
within 2-3 iterations.
B. Application of Intermittent RBSC
[]
Figure 3 shows noisy reconstructed images for each of the methods described in Table II. Image
quality is very similar for all of the RBSC images, and the No Scatter Compensation image has typical artifacts
associated with scatter.
No Scatter Compensation
EC>
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J:
Q)
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(I
U
,--'
I I
>
~
Q)
a:
(a)
(b)
10
20
30
40
50
60
Pixel Number
Figure 3. (a) RBI-EM reconstructed images (noisy data, Butterworth post-reconstruction filter) at 8 iterations:
Forward RBSC (top left); Intennittent RBSC methods: Every Other (top center), Doubling (top right), First 3
(bottom left), 1&3 (bottom center); and No Scatter Compensation (bottom right). Horizontal profiles across each
of the images (at position indicated) are also shown (b).
L)
LI
[]
L
Each of the methods was evaluated using single parameter measures of image quality. The bias of
each reconstructed image was measured by calculating the mean squared error (MSE) at each iteration. To
separate bias due to model inaccuracy, the "true" image used for this calculation was taken as the noise-free
Full RBSC reconstructed image at 50 iterations. In addition to MSE, which measured overall image bias, the
left ventricle (L V)-to-blood pool (BP) contrast ratio was calculated as follows: the mean pixel value was
determined for ROIs drawn over the LV and BP, and the contrast was calculated as (LV-BP)/(LV+BP).
The MSE and Contrast are plotted as functions of iteration in Figure 4. All of the· RBSC methods
provide similar improvements in MSE and Contrast as compared to No Scatter Compensation. In terms of
reconstruction time required, the various RBSC methods behaved quite differently. Typical per-iteration.
reconstruction times are in the ratio of 9: 1 with scatter modeling in the projector, as compared to
reconstructing with only attenuation and 3D detector response compensation (17: 1 with scatter modeling in the
projector and backprojector).
To emphasize the differences in reconstruction times, we have plotted the
Contrast as a function of Relative Reconstruction Time in Figure 5. The time required to recover good image
contrast is reduced when using the Intennittent RBSC methods, resulting in speed-up factors of 2x and 5x as
compared to Forward RBSC and Full RBSC, respectively.
11997 International Meeting on Fully 3D Image Reconstruction
1891
1&3
o· ~
~
0.9
.E
8
0.8
~
0.7
r
~~~..-.HINIPtft~r.iI~'~i~l;;;
,
(Pull RBSO)
.
~··-··.--F-u-IIR-B-S-C--~
I"'- - - - - - - - _ . Forward
:> 0.6
-Doubling
~
--=1&3
-
-
20
30
40
Evory Olhor
___ First 3
0.5
10
RBse
~
...,.
- - • No Scnller
Compensation
so
I
I
10
20
30
40
so
Iteration
Iteration
(b)
(a)
Figure 4. MSE (a) and LVstoaBP Contrast (b) plotted as a function of iteration for each of the methods studied.
1.0,....-~----~~--~---...,
1;S 0.9
~
<:>
Co)
~
CQ
0.8
~
....:l 0.7
25
50
75
100
125
150
Relative Reconstruction Time
Figure 5. LV-to-BP Contrast plotted as a function of Relative Reconstruction Time. Each time unit is the time
required for one iteration without scatter modeling, but with attenuation and 3D detector response compensation.
IV. Summary and Conclusions
When using iterative RBSC, the scatter estimate converges in relatively few iterations. We have
exploited this using a new strategy for implementing model..based scatter compensation, which we call
Intermittent RBSC. The scatter update is updated intennittently and, once converged, is held constant for the
remaining iterations. The result is a similar rate of iterative recovery of image features as compared to
standard implementations of RBSC, but in substantially reduced reconstruction times. The Intennittent RBSC
methods recover good image contrast in reconstruction times that are 2x and 5x faster than Forward RBSC and
Full RBSC, respectively. When using Intermittent RBSCmethods with fast scatter models, fully· 3D SPECT
reconstructions can be performed within the realm of clinically realistic times.
vn. Acknowledgment
This work was supported by a grant # R29NCA63465 from the National Cancer Institute and by an. academic research grant
from the North Carolina Supercomputer Center. Its contents are solely the responsibility of the authors and do not necessarily
represent the official views of the National Cancer Institute or the NCSC.
VllI. REFERENCES
[1]
[2]
[3]
[4]
[5]
B.C. Frey and B.M.W. Tsui, "A practical method for incorporating scatter in a projectotNbackprojector for accurate scatter
compensation in SPECT," IEEE. Trans. Nucl. Sci., vol. NS-40, nO. 4, pp. 1107-1116, 1993.
M. Ljungberg and S.-E. Strand, "A Monte Carlo program for the simulation of scintillation camera characteristics,"
Compo Meth. Pr'!g. Biomed., vol. 29, pp. 257 N272, 1989.
C.L. Byrne, "Block-iterative methods for image reconstruction from projections," IEEE Trans. 1m. Proc., vol. 5,
pp. 792-794, 1996.
.
D.S. Lalush and B.M.W. Tsui, "Convergence and resolution recovery of block iterative algorithms modeling 3D detector
response in SPECT~" Conference Record of the 1996 IEEE Medical Imaging Conference, Anaheim, CA, 1996, in press.
E.C. Frey and B.M.W. Tsui, "A new method for modeling the spatially-variant, object-dependent scatter response
function in SPECT," Conference Record of the 1996 IEEE Medical Imaging Conference, Anaheim, CA, 1996, in press.
11997 International Meeting on Fully 3D Image Reconstruction
1901
['
i
I
I
f
l.1
3D Tomographic Reconstruction Using Geometrical Models
X. L. Battle, G. S. Cunningham and K.M. Hanson
Los Alamos National Laboratory, MS D454
Los Alamos, New Mexico 87545 USA
[1,
Keywords: Bayesian analysis, tomographic reconstruction, geometrical models, adjoint differentiation
r-)
LJ
1. INTRODUCTION
We address .the issue of reconstructing an object of constant interior density in the context of 3D tomography, where
there is prior knowledge about the unknown shape. We explore the direct estimation of the parameters of a chosen
geometrical model (e.g. a triangulated surface defining a closed volume), rather than performing operations (e.g.
segmentation) on a reconstructed volume. This model-based approach fits well in the framework of Bayesian analysis,
where the likelihood of a set of measurements is integrated with prior information about the models used.
2. THE BAYES INFERENCE ENGINE
The Bayes Inference Engine (BIE) provides a general framework to conduct Bayesian analysis. Given some measurements of an object, one wants to estimate the parameters of a chosen model that describes the unknown object.
The measurement process is simulated on the BIE by a set of transformations that produce a set of predicted measurements. This succession of transformations will be referred to as the forward transformation. The comparison of
the real measurements with the predicted ones leads to a minus log likelihood function (cp). A maximum likelihood
(ML) or maximum a posteriori (MAP) criterion determines the estimate of the chosen model that best matches the
measured data.
[]
The first implementation of 3D functionalities within the BIE follows the scheme described above. A tessellated
surface represents the external boundary of an object of constant interior density. This model is the input of
the forward transformation that models the measurement process. Its output is a set of images representing the'
projections of the object in given directions. The estimation of the parameters of the surface (the positions of the
vertices) gives an estimate of the shape of the object.
3. A COMMON REPRESENTATION OF 3D OBJECTS
Modularity and reusability are certainly the most important characteristics of the BIE. Based on an object-oriented
approach, the BIE represents the successive transformations by a data-flow diagram. Each block in the diagram is
independent from the others and any combination can be represented. Ensuring that the description of the measurement process is independent of the choice of the geometrical model requires the use of a common representation
for 3D objects. As a pixelated image was chosen in the 2D case, an array of voxels has been chosen for the 3D case.
The tessellated surface describing an object of constant interior density is projected onto an array of voxels. The
simulation of the measurement process is performed on the voxel array.
[j
f
1
"oj
The memory requirement of such a data structure is an issue. For the objects considered in this study, a run-length
encoded description of the volumetric dataset helps to decrease the amount of memory used.
This approach ensures that both the object model and measurement model can be coded independently. When a
new object model is added, one is only required to write its conversion to an array of voxels, rather than a complete
neW measurement process. In this paper, we will focus on the conversion of a triangulated surface to a voxel density
grid.
u
u
11997 International Meeting on Fully 3D Image Reconstruction
1911
4. TI-IE CONVERSION ALGORIT1IM
The conversion of a volume described by its tessellated external surface into an array of voxel computes the volume
overlap of each voxel. The algorithm is based on the superposition principle. Each facet of the external surface is
processed independently, and contributes to the voxel array. The overlap in the voxels are the sum of the contributions
of all the surface elements. A single facet contributes to a fairly small portion of the volume. Therefore, only a portion
of the voxel array is updated when a single facet is converted.
Seen in a chosen direction, each facet determines two sets of voxels : the voxels that are in front of the facet,
referred to as the front voxels, and the voxels that are behind the facet, referred to as the back voxels. The normal
of the facet gives its orientation, and determines if the front voxels (resp. back voxels) are inside (resp. outside)
the volume. A positive contribution is added to the inside voxels, whereas a negative contribution is added to the
outside voxels. This method is general and can project any closed surface, including cavities in a time proportional
to the number of facets. Figure 1 shows the result of the conversion of the volume described by the surface on the
left into a voxel array.
In this study, we consider triangulated surfaces, as they are the most common representations. Using other
tessellation is still possible since any tessellated surface can be converted to a triangulated surface. The general scheme
of the algorithm is to subdivide a single triangle into a set of smaller facets (a set of triangles and trapezoids), referred
to as sulrfacets. Each sub-facet is totally contained in a single voxel, so that it is' easy to compute the contribution
to the voxel described volume.
A recursive algorithm successively "slices" each triangle along each of the three axes. Each slice has a dimension
less than the voxel dimension along the current axis. At the end of the recursion, which is attained with a depth of
three, every sub~facet is contained in a single voxel. Figure 2 shows the successive subdivisions. First (Figure 2a),
the facet is sliced along Z. Then the subdivisions along axes Y and X follow, leading to the final subdivision (Figure
2c). The contributions of the different sub-facets are computed in a closed form (integral of a bilinear function on a
trapezoidal region), and added to the overlap volume.
5. ADJOINT DIFFERENTIATION TECHNIQUE
An optimization of <p is performed in order to determine the MAP estimate. The derivatives of <p with respect to
the parameters of the model (Le. the coordinates of the vertices) must be determined. Considering the number of
parameters (from thousands to millions) the estimation of the gradient of <p can become costly with the conventional
perturbation methods.
The adjoint differentiation technique computes the derivatives needed to optimize <p with a computational cost
comparable to the one of the forward calculation. The application of this chain rule like method to the succession
of transformations computes the derivatives of <p with respect to the input parameters. This computation is done in
the reverse direction compared to the forward transformation, and will be referred to as the adjoint transformation.
The forward and adjoint transformations are very similar. For example, in the case of a linear forward transformation representable by a matrix, the adjoint is just the transpose of the matrix. The adjoint counterpart cfeach
transformation can be written and will have a comparable complexity.
It is very useful to design the forward and adjoint transformations at the same time. The two algorithms can
share the same branching and the same data structures. Recursive algorithms, very useful to handle multidimensional
datasets, can also be implemented in this fashion. We will focus on the precautions that must be taken when writing
forward and adjoint transformations, such as the importance of the parameterization and the means to avoid useless
computations.
Great care should be taken when choosing the parameterization that will be used in the forwa.rd and adjoint
transformations. Over-determined and non-independent parameterizations tend to be easier to handle in the forward
transformation, but lead to incoherent formulations in the adjoint transformation.
In the case of a non-sequential algorithm (Le. there is at least one "if'), the branching has a tree structure. While
the forward transformation computes its results going down the tree, the adjoint transformation should compute its
values going up the tree. This ensures that only the useful adjoint quantities are computed.
11997 International Meeting on Fully 3D Image Reconstruction
1921
r-;
, J
(b)
(a)
Figure 1. Conversion of a triangulated surface (a) into a voxel array (b). The surface contains 320 triangles. The
resolution of the voxel grid is 128x128x128.
Constant Y Planes
/
[J
I:
"
.
x
I
\
l)
(a)
~
(b)
(c)
Figure 2. The triangular facet is sliced with planes normal to the Z axis (a). Then each "sub-facet" is sliced with
planes normal to the Y axis (b), and recursively with the planes normal to the X axis (c). In this figure, the different
colors correspond to different types of sub-facets (first sliced in light grey, intermediate in grey and last sliced in
black).
[J
II
L __
(1
I
J
L-.1
1
1
j
(a)
(b)
Figure 3. Original object (a) obtained by warping a sphere. Reconstruction (b) obtained after thirty three iterations,
starting with a sphere.
11997 International Meeting on Fully 3D Image Reconstruction
1931
6. RESULTS AND CONCLUSION
The complete algorithm of conversion and its adjoint counterpart have been implemented. The accuracy of the
derivative calculation has been tested by comparison with a perturbation method. This revealed the power of
the adjoint differentiation technique: the derivatives of cp computed by the adjoint code were obtained in a time
comparable to the forward calculation, whereas the derivatives computed by the perturbation method were obtained
in a time thirty to sixty times longer,
A first reconstruction has been performed on simulated data. A beanMlike object (figure 3a) obtained by warping
a sphere (320 triangles, 162 vertices) is the object to be reconstructed. After conversion to a voxei array (64x64x64),
ten noiseless parallel projections (64x64), equally spaced around 180 degrees, became the measured data.
Starting with a sphere (320 triangles, 162 vertices), we optimize the positions of the vertices describing the
surface to minimize cp (Le. to best match the measured data). Figure 3b shows the MAP estimate obtained after
thirty three iterations. The surface of the reconstruction appears to agree with the original to better than one voxel
width everywhere, since the difference volume (original object projected onto volume minus the reconstructed object
projected onto volume) has no values greater in absolute strength than 0.2.
The system response of the University of Arizona FAST SPECT machine is being integrated into the BIE. This
setting will allow us to show that geometrical models can be efficiently used to determine the unknown shape of
objects of nearly constant interior such as the heart ventricle. The application of this method to dynamic heart data
is very promising.
11997 International Meeting on Fully 3D Image Reconstruction
fi
!l I,
Symmetry properties of an imaging system and consistency condif1
tions in image space
i J~
L
Eric Clarkson and Harrison Barrett
Department of Radiology, University of Arizona, Thcson, AZ 85724
Abstract
Consistency conditions on the data in tomographic imaging systems are important for noise removal and object reconstruction algorithms. These conditions
r'\
\
I
L_ )
are usually expressed by the vanishing of scalar products (in some Hilbert space)
between arbitrary image vectors or functions and vectors or functions in some
specified set. The purpose of this paper is to show that the symnietries of a tomographic system are intimately connected with this set and that this fact can
(i'
Ll
expedite the search for consistency conditions.
Consider an imaging system descriqed by a linear map H : U
-T
V, where U is
the object space and V is the image space, each of which may be finite or infinite
dimensional. Suppose that
r is a group, which may be finite or infinite.
representation of r as linear operators on U, then write
corresponding to I E
1r'Y
for this system if there exist representations
L
for the operator on U
r. Similarly, for p a representation of r
on V, write P'Y for the actual operators. We will say that
11997 International Meeting on Fully 3D Image Reconstruction
1r
r
If 1r is a
as linear operators
is a symmetry group
and p such that H 1r'Y f
= P'YHf for
1951
every lEU. In the special case where U and V are Hilbert spaces and
1r
and p
are unitary representations, then ('!r,/I' '!r,/2)U = (/l,/2)U and (p,gl, p,,(g2) V =
(gI,g2)V for all 11,12 E U and 91,g2 E V. If HTis the adjoint of II with respect
to these inner products, then these relations imply that '!r,HT
turn gives '!r,I{T II = fIT H
:=
HT p,. This in
1r, and p,H HT = H HT P, for all 'Y E r. This may be
a more familiar notion of symmetry for the system described by H. The above
definition for a symmetry group of a system is thus a generalization of this idea.
These last two equations will not be true in general when one or both of the
representations '!r and P are not unitary.
Suppose that there are ill.iler products on both spaces as above and that
(P,91,P,g2)V = t('Y) (gl,g2)V' where t('Y) is a scalar. If t('Y)
=1, then p is
a unitary representation, otherwise it is called a conformal representation. A
consistency condition in image space is a vector or function 'if; which satisfies
(HI,'if;)v =
a for all lEU. We will say that 'if; E H(U)l., the orthogonal comple-
ment to H(U), the image of U under H, also called consistency space. In other
words 'if; is an element of inconsistency space for this system. Now, since p is
conformal, 'if; E H(U).L implies that P,'if; E H(U).L for all 'Y E G. Therefore H(U)
and H(U).L are both invariant subspaces for the representation p . . (Similarly, if
'!r is unitary or conformal, then N(H) and N(H).L are
11997 International Meeting on Fully 3D Image Reconstruction
1r
invariant.) The aim of
1961
this work is to show that for such representations, there is a connection between
the symmetries of the system and structure of the space H(U)..L.
The simplest case is when U and V are finite dimensional,
()
r is finite, and 7r and
I
I
i
P are unitary representations. If N(H) is the null space of H, then 7rIN(H), the
I.. .. J
representation 7r restricted to N (H), must be a sum of irreducible representations
of
r.
This is also true for 7rIN(H)..L, pIH(U) and pIH(U)..L. Furthermore the
decompositions of 7rIN(H).l and pIH(U) into irreducible representations must
be exactly the same. These facts are all a consequence of Schur's lemma and
rI \
i
lJ
they can be used to determine which irreducible representations must appear in
pIH(U)..L. The search for consistency conditions then reduces to finding vectors
that transform according to each of these representations that are also elements of
H(U)..L. Some examples will illustrate this point. Similar reductions are possible
when U is allowed to be an infinite dimensional Hilbert space and" the other
conditions are kept the same.
'!
When U and V are both in:fi.n1te dimensional the situation can be more complicated. This will be illustrated first for the case where H is the Radon transform
r1
tl
in two dimensions and U is the space of square integrable functions with compact
support. It is well known that, if )...(p, ¢) is the Radon transform of an arbitrary
fEU and
a~
k < l, then J~oo Jo27r )...(p, ¢)pkei1cpd¢dp
=
O. (Note that)... also
f 1
L.J
r·
I
(
11997 International Meeting on Fully 3D Image Reconstruction .
1971
has compact support so that there is no difficulty in evaluating the integral.) In
other words, if 'l/Jkl(P, ¢) = pkeil 1>, then 'l/Jkl E H(U)J.. for 0 ~ k < 1. This result can be related to the representations in image space of the group containing
scale transformations, rotations and translations in object space. In particular,
the form of the function 'l/Jkl arises from scale and rotational symmetries, while
the condition on k is related to translational symmetry_ A similar analysis can
be done for the three-dimensional Radon transform and for the two-dimensional
Radon transform at discrete angles. There are also similar consistency conditions
for the attenuated Radon transform in two dimensions which may also be related
to group theoretical properties of this operator_
11997 International Meeting on Fully 3D Image Reconstruction
1981
Experience with Fully 3D PET and Implications for Future High Resolution 3D Tomographs .
Dale L.Bailey, Matthew Miller, Terry J.Spinks, Peter M. Bloomfield, Lefteris Livieratos,
Helen Youngt and Terry Jones.
MRC Cyclotron Unit and t Department of Imaging, Hammersmith Hospital, London. UK.
r~
(
;
We have two 3D only human tomographs in our institute: the EXACT3D, a full ring whole body 3D PET
tomograph with a ring diameter of 82 cm and axial field of view of 23.4 cm, and the ECAT ART, a partial
ring rotating tomograph of 82 cm ring diameter and 16.2 cm axial field of view. In addition, we have a
2D/3D neuro-tomograph, the ECAT 953B, which we operate exclusively in 3D mode apart from
transmission scanning (all from CTI, Knoxville, TN, USA). All of these systems use bismuth germanate
(BaO) block detectors. The aim of this paper is to report on unique aspects of 3D data acquisition in the
context of the implications these have for next-generation 3D tomograph designs.
I
The majority of studies that have investigated or utilised 3D PET have been for studies of the brain. Concerns
about scattered radiation arising from inside or outside the field of view have been addressed and various
strategies exist for producing reconstructed data which are at least as accurate, but with much better signal
to noise ratios, than that produced by 2D data acquisition(Cherry et al, 1993; Rakshi et al, 1996; Townsend et
al, 1996; Trebossen et al, 1996). The tomographs used in these studies had large ring diameters (86 -102 cm),
limited axial fields of view (~10.8 em), and thick end shields to eliminate radiation arising from outside the
field of view. However, the trend to decrease the ring diameter and extend the axial field of view of later
generation devices combines to greatly increase the solid angle for detection of photons from outside the
coincidence field of view.
r- -:-
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[)
KEY
[[[I] Detectors
I
Side shields
Coincidence FoV
_ . Boundary of SinglesFoV
ri
,I
1J~1
\
LJ
Cross-sectional diagram (to scale) of a 1st-generation whole body 2D /3D PET scanner (ECAT 951R) aeft)
with a 102 cm diameter detector ring, 10.8 cm axial field of view, and side shielding which left a subject port
of approx. 62 cm. The EXACT 3D (right) is the first full-time 3D BGO based full ring tomograph. It has an 82
cm diameter detector ring, 23.4 cm axial field of view, and side shielding that leaves a subject port also of
approx. 62 cm. This has a dramatic effect on the field of view for single events that can irradiate the
detectors.
r
c.r.
,,'r
r
The single greatest limiting factor in the operation of these tomographs in 3D is the moderately slow
phosphorescence· decay time for BaO (300 nsec) leading to coincidence timing windows of 12 nsec. The
increased sensitivity to events within the field of view of 3D
2D meant that lower doses needed to be
administered to realize the maximum sensitivity gain of 3D, due to dead time. Contrary to what is often
thought, random coincidence rates arising from within the coincidence field of view are decreased in 3D
2D at a given count rate. Scattered radiation was adequately dealt with on the ECAT 953B (and similar)
tomographs as it was predominated by events recorded within the field of view. However, the full-time 3D
systems accept many more scattered events from outside of the coincidence field of view.
1
I \
"- ,J
c.r.
The increased sensitivity of the EXACT 3D and dramatic impact that single photons from outside the
coincidence field of view have on the EXACT 3D performance are illustrated in the next graphs. Count rate
performance was compared between these different tomographs using a 20 cm diameter cylinder. To assess
the effect of out of field of view activity, a water cylinder containing 11C was placed just outside one end of
the field of view and count rates recorded over time. This was compared with the same experimental set up,
but after end shields (8mm lead) to restrict the patient aperture to 35 cm had been added. In clinical studies
the randoms rates immediately after injection of -200 MBq of [l1C}-Diprenorphine were approximately
equivalent to the trues rates without end shields in place, but reduced to 30% of the trues rates after the end
shields added.
11997 International Meeting on Fully 3D Image Reconstruction
1991
200000
- 1 - NEG 953B 2D
- 0- • NEG 953B 3D
• - ••. NEG HR++
-H-NEGART
,
150000
Of
,,
~
,
I
U100000
,,
I
~
50000 _ f
I
~
0- -
_0 __ -
. ,
P
..
..n
.-0-
...-.-H
.(t I)' .().
~ G.o -
-0- -
-
-
0=-0-
~ro-D H----=.~--:,.---·,·-
o ~._.tIDI~~.I-"""-t-t-I--I=-r
_".-.-'
10
o
20
I.,
I
•
~~_~-N-=-I----
,
,
•
I.
•
30
40
AClivily in roY (MDq)
I,
The noise equivalent count rate curves for a
20 cm diameter water-filled cylinder are
shown for the 3D tomographs that have
been discussed, as well as the ECA71 953B in
2D. It can be seen that the EXACT 3D
tFlR++') scanner has extremely high
sensitivity, but limited dynamic range. This
is due to the characteristics of light emission
of BGG. The discontinuity for the EXACT
3D at -25 MBq in the field of view is most
lillely due to dead tin1e in the electronics
,I
60
50
150000
~ 100000
-
~
.g
~
50000
..,/ - l - Rnndoms (cps) no shields
/'
Randoms rates from activity outside the
coincidence field of view for the EXACT 3D
tomograph with no end shields and with 8
mm of lead shielding, which reduced the
patient aperture from 62 cm to 35 cm.
- .... Randoms (cps) Smm
_0----0---
O~~~~~~~~~_~-~D-_-~~~-_-~~~~~~
o
2
4
6
8
10
12
Cline (kBq/ml)
'l'ransmission scanning without septa produces a large scattered component. This biases the attenuation
correction factors lower than expected. The 1.1. values measured with a single photon transmission source on
the EXACT 3D are -0.06 0.07 cm- 1 for soft tissue (0.095 cm- 1 expected). Single photon transmission for PET
produces very high count rates, but the bias from scatter needs to be removed. This can be effected by
collimating the source, scatter correcting the transmission data, or using a hybrid measured/segmented
attenuation approach to producing unbiased estimates of the attenuation correction factors. Future
tomograph designs will need to evaluate the best options from these for optimal attenuation correction.
8
Proposals for tomographs based on faster scintillators such as lutetium oxyorthosilicate (LSO) have been
suggested with greatly reduced ring diameters to improve sensitivity, improve resolution (lower nons
collinearity error), and decrease costs (less detectors required). The experience with the ART and EXACT 3D
suggest that the gains of a faster detector in terms of effective sensitivity (detector live time -and lower
randoms rates) may be compromised by the large photon flux impinging on the detectors from activity
outside the field of view. A wider ring diameter with side shielding might prove to give much higher quality
data as the m~ority of photons arriving at the detectors will have arisen from within the field of view.
References
Cherry SR, Woods RP, Hoffman EJ, and Mazziotta JC (1993): "Improved Detection of Focal Cerebral Blood
Flow Changes Using ThreeNDimensional Positron Emission Tomography" J Cereb Blood Flow Metab 13(4):
630-638
Rakshi J, Bailey DL, Morrish PK, and Brooks DJ (1996): "Implementation of 3D Acquisition, Reconstruction
and Analysis of Dynamic Fluorodopa Studies". In: Myers R, Cunningham VJ, Bailey DL and Jones T,
Quantification of Brain Function Using PET. San Diego: Academic Press, pp. 82-87
Townsend DW, Price JC, Mintun MA, et al (1996): "Scatter Correction for Brain Receptor Quantitation in 3D
PET". In: Myers R, Cunningham VJ, Bailey DL and Jones T, Quantification of Brain Function Using PET.
San Diego,: Academic Press, pp. 76-81
Trebossen R, Bendriem B, Fontaine A, Frouin V, and Remy P (1996): "Quantitation of the [18F]F1uorodopa
Uptake in the Human Striatum in 3D PET with the ETM Scatter Correction". In: Myers R, Cunningham VJ,
Bailey DL and Jones T, Quantification of Brain Function Using PET. San Diego: Academic Press, pp.88-92
11997 International Meeting on Fully 3D Image Reconstruction
2001
Inter-Comparison of Four Reconstruction Techniques for
Positron Volurne Imaging with ~otating Planar Detectors
r--'
\
I
,
j
t
J
A. J. Reader, D. Visvikis, R. J. Ott, M. A. Flower
Joint Department of Physics, Institute of Cancer Research, Royal Marsden NHS Trust, Downs Road, Sutton, Surrey SM25PT
r)
UK
Abstract
I
Measured data have been used to evaluate four reconstruction techniques for positron volume
imaging (PVI) scanners based on rotating planar detectors (RPDs). The four techniques compared are
the backproject then filter (BPF) method using constrained deconvolution, 3-D re-projection (3-D
RP), Fourier rebinning (FORE) and ordered-subsets expectation-maximisation (3-D OSEM). OSEM
gave the best spatial resolution, BPF the best contrast, 3-D RP the best signal to noise ratio and FORE
gave a good all round performance.
[I
\.
J
Introduction
PETRRA [1], a hybrid BaF2 - TMAE detector system soon to be installed at the Royal
Marsden Hospital, will offer uniform and accurate azimuthal, polar, axial and transaxial sampling,
and a very large axial field of view (FOV) with> 10 12 lines of response (LaRs). This paper presents a
comparison of four volumetric image reconstruction methods implemented for large RPD systems
such as PETRRA. As well as PETRRA, the comparison is relevant to double-headed gamma cameras
with coincidence electronics.
The four methods compared are BPF [2,3], 3-D RP [4], FORE [5] and 3-D OSEM [6]. This
selection includes at least one from each of the main categories of reconstruction techniques: Fourierbased methods, fast rebinning methods and iterative methods. The BPF method, which fully utilises;
the accurate angular and spatial sampling of the acquired data, has been used routinely for clinical
studies at this hospital and thus forms a base-line for comparison with the more recently implemented
methods. 3-D RP has been regarded as the gold-standard for other PET systems and so is included in
the comparison. FORE is a relatively accurate, noise-free, one-step rebinning method, requiring no
axial filtering. 3-D OSEM was selected because of the speed of the algorithm and its reputation for
being a good method for sparse data.
r~)
i
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r-,
i
"\
~
!
Methods
Data Acquisition
The current positron camera at this hospital (MUP-PET [7]), consisting of two large-area
planar multi-wire chamber detectors, was used for the experiments. Owing to the low sensitivity of the
available camera, scans were carried out for much longer acquisition times than normal in order to
simulate the statistics expected from PETRRA.
For 3-D RP, FORE and 3-D OSEM the list-mode data, offormXl,Yl,Zl,X2,Y2,Z2 (figure 1),
were rebinned into 2-D parallel projections p(y',z',<I>,6). Each projection had 128x128 bins (3mm
side); ·the azimuthal (<I» sampling interval was 1.41 0 (128 samples) and the co-polar (6) sampling
interval was 3.390 (9 samples). The BPF method used the list-mode data directly, thus avoiding any
[!
z
'f.l
Z1
rt
C1
L
D1
~
AXIS OF
ROTATION
Z
~x'
........................ ............. :z'
Z2
'.
1\
...........•
y'
,,, ..... j
.' I
.. '"
D2
.
!
.
y
~
".,.,.,.,. ., i /.//
Xl
S
..................................................................:..':::.....1........
Figure 1: Acquisition and reconstruction geometry. L=60cm, W=30cm .and S=87cm giving a maximum co-polar angle of 19u
(effective maximum 15.25U). The list-mode data co-ordinate frames are shown on the detectors, and the reconstruction frame is shown
in the centre and on the right hand side (note the orientation change for clarity).
I
"
l_ ;
11997 International Meeting on Fully 3D Image Reconstruction
loss in sampling accuracy due to the comparatively coarse rebinning. All four methods were used to
reconstruct image volumes of diameter 38.4cm within matrices of 128x128x128 voxels, 3mm side.
Recollstruction Algorithms
1) The BPF algorithm operates directly on the list mode LOR data giving a backprojected
8
image g(x). The backprojected image is filtered by constrained deconvolution in frequency space
Fl.
G(k)H* (k)
t( <) = H(k)H* (k)+y(2nlkl)4
(1)
where H(I<) is the Fourier transform of the point response function, defined as the backprojected
imuge of a point source. The Lagrange multiplier 'Y was chosen to be 8xlO-8 [2,3].
2) The 3-D RP method uses the completely measured projections (specified by 8=0) to
reconstruct an initial estimate of the image:
J
fo=o(x) == pF(fi,x-(x.fi)fi)dfi for fi. E {filS
=O}
(2)
where pF (u,s) are the Colsher filtered (Hamming windowed, Nyquist cut-off) 9=0 projection data, the
u
unit vector specifies the projection direction and s :::
y'n + z/b is a 28D vector on the projection plane
(a and 6 are orthogonal unit vectors (figure 1)). The oblique projection data were then completed by
forward-projection of the noisy image:
f
p(fi,s) "" fe=o(S + n1)dr
for Is.hl>
i
cose - ~ sine
(3)
The complete projections were then reconstructed using a Hamming windowed Colsher filter, with
varying cut-off frequencies.
3) FORE rebins the four-dimensional projection data set p(y',z',<j>,S) into a three-dimensional
data set in frequency space according to
P(ky"k~,Zdirect) ~ P(kyl,z',k~,e)
(4)
where ky' is the y' frequency, k$ is the Fourier series integer and Zdirect is the axial index of the directplane sinogram, given by
z'
ZeJirect = - -
cosS
k
2nky'
+ --~-tanS
(5)
The single-slice rebinning approximation (whereby the second term in (5) is omitted) was
used for the DC component, but only for the smallest non-zero co"polar angle. The rebinned
sinograms were reconstructed with a Hamming windowed ramp filter with varying cut-off
frequencies.
4) The 3-D OSEM method consists of sub-iterations l:::.1 to N where N is the number of
subsets. The sequence of image estimates is given by
frk.l+l
J
=
~k.l ~ [p;ay ]
~
£..J
£.a y ieSN
(6)
k.1
q;
ieSN
where al} is the probability of an emission in voxel j being binned into projection element i, p are the
measured projection data,
J
qik,1
- ~a fk.1
-£..iijj
j=l
(7)
is the projection of the current estimate and the initial image is assumed to be uniform. Eight
interleaving subsets (Sj. .. S8) were used [6], consisting of 2 co-polar angle subsets each containing 4
azimuthal angle subsets. The image volume and projection data were both modelled as continuous [8].
The system matrix al} was modelled by the forward-projection process, (bin-driven, calculating
contributions at 3mm steps along each LOR with tri-linear interpolation), and the backprojection
process (voxel-driven, using bi-linear interpolation of the projections to determine contributions).
11997 International Meeting on Fully 3D Image Reconstruction
2021
(-'
;l I
J
r ')
1 '
\
I
I
I
[,
\
\
1)
Experimental method
The figures of merit (FOMs) looked at for each method were: Point Spread Function (PSF), Contrast Recovery Coefficient (CRC) and Signal to Noise Ratio (SNR).
A 22Na point source was located at 6 different positions in the FOV: (0,0,0), (7,0,0),
(14,0,0), (0,0,7), (7,0,7),(14,0,7) «x,y,z) in cm). For each position 3.5x105 events were acquired. The
list-mode data were then added together to create one file of 2.1xl0 6 events for the 6 point sources.
The mean axial and tangential full widths at half maximum (FWHMs) and tenth maximum (FWTMs)
were calculated for each method (cubic spline interpolation of the profiles was used).
A cylindrical phantom of 680a (18.9cm diameter, 11.6cm long), with 3 cold cylindrical
inserts (4cm, 3cm and 1.5cm in diameter and length), was located in the centre of the FOV and 6x107
events were acquired. The CRC was found for each cold insert by (B - C) / B ; where C is the mean
count in the cold insert and B is the mean count in a region of interest (ROI) of the same size and
radial location as the insert. The SNR was found by B / 0' ; where B is the mean count in the largest
background ROI and 0' is the standard deviation.
Neither scatter nor attenuation correction were carried out for any of the methods, as only
relative FOMs were required to inter-compare the methods. Correction for these effects would require
differing implementations (BPF is an image space based reconstruction) which would possibly
complicate the comparison,
Results
Axial FWHM
u
rI
i
Axial FWTM
15----------------------~
14
E 13
eE
E 12
:r 11
3-D RP
:c 10
~9r_-----"--'~~~~-=-.:=-.:....::....1 FORE
8
7+-----------r---------~
0.5
-
28
26
24
22
== 20
~ 18
3-DR?
F:::::---------l
u. 16
14
12
BPF
0.75
BPF
0.5
Cut-off frequency (fraction of Nyquist)
Tangential FWTM
i
3-~
RP
FORE
12
11
10
BPF
0.5
0.75
0.75
Cut-off frequency (fraction of Nyquist)
Tangential FWHM
18
17
E 16
E 15
14
:c 13
~
FORE
35
33
_ 31
E 29
-5.27
== 25
~ 21
23
19
17
15
3-D RP
FORE
------BPF
0.5
1
0.75
Cut-off frequency (fraction of Nyquist)
Cut-off frequency (fraction of NyqUist)
OSEM profiles
19~------------------------.
I'--IJ'
1_
18
17
-16
E 15
§. 14
=13
:2 12
3: 11
=§ 10
u.
~
- - - - -
Tangential FWTM
--+---~~-:--:~-:-::-:r Axial FWTM
~-- - - - -
:L.:-=-...;;..-o_-Io---=--=--;-"'"--.....
- ...:.=-,....;- Tangential FWHM
7
- - - - -
Axial FWHM
6+--------r------~r_----~
1
3
2
4
Iteration
Figure 2: Comparative values ofPSF widths (each datum point represents mean of6 measurements for 6 different positions)
11997 International Meeting on Fully 3D Image Reconstruction
2031
15 mm diameter oyllnders
30 mm diameter cylInders
0.8 .,. .............. ,...................................,.. ,..... ,........".........,...........................
0.7..----0.6 ..
0.6 ..
0.7
0.6
~ 0.4"
0 0.5
00.3 ..
0.2 ..
0 0.3
0::
0.4
0.2
0.1
0.1 ..
O·
o
0.6
0.76
0.5
0.75
Cut·off frequency (fraction of NyquIst)
Cut.aft frequency (iraci/on of Nyquist)
40 mm diameter cylinders
OSEM Contrast Recovery Coofflclont
0.0 ,......,..........,...... ,..,........".........,...............................................................,
0.64 ..,. .............."......................................,............. ,...................... ,' ............. .
0.62
0.5
0.6
0 0.4
~ 0.66
{30.3
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0.2
0.64
0.52
0.5
0.1
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0.5
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Cut-off frequency (fraction of NyquIst)
4
3
Iteration
Signal to Noise RatIo
OSEM Signal to NoIse RatIo
12~----------------------~
10
,
.. , .............. , ....
12
10
8
~ 6
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6
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Cut-off frequency (fraction of Nyquist)
3
4
IteratIon
Figure 3: Comparative values of CRC and SNR
Method
BPF
OSEM 4
3-D RP
FORE
'Rank'
1
1
3
3
PSF
2
1
4
3
SNR
4
2
1
3
eRe
. . . . . . . . . . , ' , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U •• M . .U . . . . . . . . . . . . . .
1
4
3
2
Table 1: Perfommnce order for each of the FOMs
Discussion
Overall, BPF and OSEM are the better algorithms of the four inter-compared, yet BPF has
the worst SNR and OSEM the worst eRe of the four. OSEM is a dramatically more time consuming
method than BPF, but to ensure shift-in variance of the PSF the BPF method requires application of an
angular constraint which results in reducing the axial FOV and rejection of .... 25% of events. The eRe
for OSEM steadily increases with iterations implying for later iterations that OSEM would have the
best contrast of the four methods which would be consistent with its superior resolution.
References
1) Visvikis D, Wells K, Ott R J, Stephenson R, Bateman.1 E, Connolly J, Tappern G (1995) IEEE Trans Nucl Sci 42:1030-1037
2) Chu G, Tam K-C (1977) Phys Med Bioi 2:245-265
3) Webb S, Ott R J, Bateman J E, Flesher A C, Flower M A, Leach M 0, Marsden P, Khan 0, McCready V R (1984) Nucl Inst
Meth 221:233"241
4) Kinahan P E, Rogers J G (1989) IEEE Trans Nucl Sci 36:964"968
5) Defrise M, Kinahan p, Townsend D, 1995 Proceedings of the International Meeting on Fully Three-Dimensional Image
Reconstruction in Radiology and Nuclear Medicine, pp235-239
6) Hudson H M, Larkin R S (1994) IEEE Trans Med 1m 13:601-609
7) Cherry S R, Marsden P K, Ott R J, Flower M A, Webb S, Babich J W (1989) Eur J Nucl Med 15:694-700
8) Carson R E, Yuchen Y, Chodkowski BA, Yap T K, Daube-Witherspoon M E (1994) IEEE Trans Med 1m 13:526-537
11997 International Meeting on Fully 3D Image Reconstruction
2041
A FULLY-3D, LOW COST. PET CAMERA USING HIDAC DETECTORS
WITH SUB.-MILLIMETRE RESOLUTION FOR IMAGING SMALL ANIMALS
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A.P. Jeavons, R.A. Chandler, C.A. Dettmar
Oxford Positron Systems, Oxford, UK.
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INTRODUCTION
1!
The HIgh Density Avalanche Chamber (HIDAC) was invented [1] in 1973 at CERN,
Geneva, Switzerland. The detector consists of a MultiWire Proportional Chamber
(MWPC) with the addition, for gamma-ray conversion and localisation, of a
laminated plate containing lead sheets and perforated with a dense matrix of
small holes. Ionisation resulting from photon interactions with the lead is
trapped by, amplified in, and extracted from, the holes into the MWPC, and the
result is precise, two-dimensional localisation of the impinging gamma-rays.
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Two HIDACs operated in coincidence comprise a Positron Camera which was used for
Positron Emission Tomography (PET) at Geneva Hospital [2J. During an eight year
period, fully 3D, high-reso~ution (3mm FWHM) PET was demonstrated [3,4,5].
However, by 1986 state-of-the-art crystal cameras were in routine use and the
competitive position of HIDAC-PET was that, despite its excellent intrinsic.
spatial resolution of 1.5mm, its applicability was limited [6] by low
sensitivity, and high noise from accidental coincidences, to studies of small
objects permitting long acquisition times (e.g. human thyroid). Recent work, at
the Christie Hospital NHS Trust in Manchester, has aroused particular interest
in HIDAC-PET using 18-FDG for research on tumours grown in mice [7], where' the
high spatial resolution can also be realised: see figure 1.
In recent years, a HIDAC technological improvement programme has been carried
out at Oxford Positron Systems to achieve sub-millimetre spatial resolution for
beta-rays for digital autoradiography [8]. This new technology has been licensed
to the Packard Instrument Company of Downers Grove, Illinois, USA and
commercialised under the name "InstantImager" [TM] [9]. It has been well proved
as over 300 cameras are now installed in 25 countries world-wide.
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THE NEW HIDAC CAMERA
In view of the particular interest in HIDAC-PET for studies of mammary
carcinomas using mice, a dedicated camera is being constructed incorporating the
technical advances made for autoradiography. These are:
1. A finer hole matrix of 0.4 mm diameter holes on 0.5 mm pitch.
2. The use of thinner (0.05 mm) lead sheets in the converters.
3. The use of thinner (2.5 mm) converters.
4. The tripling of the number of converters by using a new stacking method.
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The full design specification for the camera is:
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Number of detectors:
Detector separation:
Detector active volume:
Number of converters/detector:
Converter thickness:
Converter hole pattern:
Number of holes/converter:
Number of holes/detector:
Electronic channels/detecto~:
2, with rotation
150 mms
250 mms x 210 mms X 80 mms deep
12
2.5 mm
hexagonal, 0.4 rom diameter on 0.5 mm pitch
210,000
2,520,000
282 {132 X + 144 Y + 6 Z}
Singles detection efficiency:
Resolving time (2t)
Intrinsic spatial resolution:
16%
30 nsec
<1 mm (3D isotropic and uniform)
,
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11997 International Meeting on Fully 3D Image Reconstruction
2051
Absolute sensitivity:
Maximum coincidence counting rate:
8.5 cps/Kbg (central point source in air)
100 Khz from -10 MBg (incl 50% randoms)
RESULTS AND DISCUSSION
Currently, (January 1997), detectors with two converters each, are in operation
and rotating. First measurem~nts confirm that the design data will be achieved.
Sub-millimetre spatial resolution: Due to the reduced hole size and detector
separation, and implicit depth of interaction measurement (Z) as each converter
has separate readout, sub~millimetre spatial resolution is achieved for 22-Na
line sources in air [10]. Figure 2 shows the variation in resolution (FWHM) for
various detector spacings and acceptance cone angles. 'Figure 3 shows images of
line sources spaced, centre to centre, at 2mm and 1mm.
Detection efficiency-/Sensitiyity~ Net detection efficiency for singles is nearly
tripled compared to the Geneva camera. This comes principally from the doubled
surface area of the converter holes: 1.6% is measured, for each converter, from
the counts in a 3 x 3 mm region of the image of a 1 mm point source located
centrally on the camera mid-plane. This allows for all counting errors due to
photon scattering, noise, electronic rejection and accidentals. Although this
efficiency is modest compared to BGO cameras, the large (33%) solid angle
achieved affords an absolute sensitivity nearly two-thirds that of, for example,
the ART camera. More modules could be employed to double either or both solid
angle and efficiency to give up to an eightfold sensitivity improvement.
The converters are manufactured by standard printed circuit techniques.
The modest timing requirements means that low-cost, low-power electronics is
appropriate and a coding scheme reduces the total number of electronic channels
substantially from 'amplifier-per-wire' techniques. The basic manufacturing cost
is well established by years of experience at around $100,000.
~
Relationship to crystal cameras: Obviously, the detection efficiency and time
resolution are both drastically down (x5) on the best BGO/LSO systems. However,
as each hole is resolved by the readout, the camera may be considered as
composed of over five million, individual, 0.5 mm diameter x 2.5 mm long
'crystals', of low-density lead, stacked 12 deep, at a cost of one cent per
'crystal'. This provides ultra-high spatial resolution at a very modest cost.
The count rates will be adequate for the particular mice studies and, it is
anticipated, for many other applications using small animals.
The camera will be completed by April 1997 and its full performance parameters
will be presented. A software reconstruction package is being developed in
collaboration with D. Townsend and P. Kinahan of the University of Pittsburgh
and 18-FDG mice images with a 3D resolution of 1 mm should be available.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
Jeavons A, Charpak G, Stubbs R. NIM ~ (1975) 491-503.
Jeavons A, Townsend D, et al. IEEE Trans. Nucl. Sci. ~ (1978) 1,164-173.
Jeavons A, Hood K, et al. IEEE Trans. Nucl. Sci. ~ (1983) 640-645.
Townsend D, Clack R, et al. IEEE Trans. Nucl. Sci. ~ (1983) 594-600.
Townsend D, Frey P, Jeavons A, et al. J. Nucl. Med. 2a (1987) 1554-1562.
Townsend D. NIM ~ (1988) 443-450.
Acton P, Jeavons A, Hastings D, et al. Eur. J. Nucl. Med. 21 (1994) 872.
Jeavons A. Patents (1988). UK: 2220548 (1993}i USA: 5,138,168 (1992}i
5,434,468 (1995) i European: 0428556 (1995) 1 Japanese: 1-507869 (pend).
9. Englert D, Roessler N, Jeavons A, Cell. Mol. BioI. ~, 1, (1995) 57-64.
10. Jeavons A. "Sub-millimetre PET with the HIDAC Camera". Workshop bn PET
Instrumentation for Animal Imaging, UCLA, USA. (Oct 1995).
11997 International Meeting on Fully 3D Image Reconstruction '
2061
Figure 1:
Imm ~agittal sections of a mouse following injection of 18-FDG.
Bin size ImlTI x Imm
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(FWHM) and the full width tenth max. (FWTM) of an image
of a O.4mm diameter 22 ...Na line sealed in 10mm of plastic.
The data was obtain~d from mid-plane images produced
from the average co-ordinates of coincident photon pairs.
11997 International Meeting on Fully 3D Image Reconstruction
2081
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AUTHOR INDEX
Y. AIno
R. Badawi
D. Bailey
A. Bani-I-Iashelni
20
56
97,199
M. Barlaud
1t:'n
H. Barrett
R. Basko
X. Battle
F. Beeknlan
T. Beyer
L. Blanc-Feraud
W. Blass
P. Bloomfield
R. Carson
R. Chandler
A. Chatziioannou
J. Cheng
S. Cherry
R. Clack
E. Clarkson
N. Clinthorne
C. Conltat
G. Cunningham
P. -E. Danielsson
1. Darcourt
M. Daube-Witherspoon
C. Dawson
M. Defrise
L. Desbat
C. Dettmar
P. Dupont
P. Edholm
F. Emert
J. Eriksson
K. Erlandsson
R. Fahrig
T. Farquhar
M. Flower
A.Fox
E. Frey
S. Glick
P. Grangeat
R. Graumann
S. Green
1. Gregor
G. T. Gullberg
D. Gunter
H. Halling
K. Hanson
174
LJO
195
16
191
134
154
158
93
97,199
63
205
28
77
28
162
138,195
12
154
191
141
158
63
117
32,85,154
145
205
85
141
93
141
166
174
28
166,201
174
67,105,134,187
130
71
174
63
89
16,40,67,113,
121,150
184
170
191
D. Harrington
R. Harrison
S. Haworth
D. Haynor
A. Hennann Scheurer
A. Hero
H. Herzog
D. I-Ioldsworth
Y.-L. I-Isieh
H.Hu
R. Huesman
F. Jansen
R.laszczak
A. Jeavons
R. Johnson
T. Jones
D. Kadrmas
C. Kamphuis
M. Kaplan
S. Karimi
1. Karp
P. Kinahan
M.King
V.Kohli
P.-M. Koulibaly
H. Kudo
V.La
D. Lalush
I. Laurette
R. Leahy
T. Lewellen
R. Lewitt
J. Li
Z. Liang
1. Linehan
L. Livieratos
M. Magnusson Seger
P. Marsden
S. Matej
C. Michel
M. Miller
J. Missimer
R. Miyaoka
C. Morel
L. Mortelmans
E. Mumcuoglu
H. MOller-Gartner
N. Navab
F. Noo
1. Nuyts
11997 International Meeting on Fully 3D Image Reconstruction
77
52,75,101
117
52
176
12
170
174
150
117
121
101
126
205
117
97,119
187
134
75
67
24
154
130
130
158
36
71
180
158
28
52,75,101
24
77
77
117
199
81,141
55
24
154
97,199
93
101
176
85
28
170
174
32,162
85
2101
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T. Oakes
R.Ott
T.-S. Pan
J. Qi
F. Rannou
A. Reader
B. Reutter
A. Rodriguez
W. Rogers
T. Roney
T. Ruth
T.Saito
S. Samarasekera
F. Sauer
A. Sauve
M. Silver
M. Smith
V. Sassi
T. Spinks
P. Suetens
W. Swan
K.Tam
E. Tanaka
A. Terstegge
D. Townsend
B. Tsui
S. Vannoy
D. Visvikis S. Weber
T. White
K. Wiesent
C.Wu
J. Ye
H. Young
H. Zaidi
L. G. Zeng
59
166,201
130
28
89
166,201
121
93
12
162
59
36
48
48
12
44
126
59
97,199
85
75,101
48
20
170
154
67,105,180,187
75,101
201
170
162
174
109
77
199
176
16,40,113,
121,150
[]
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11997 International Meeting on Fully 3D Image Reconstruction
2111
3D97
Jesper Andersson
PARTICIPANT LIST
Uppsala University
PET-Centre
Uppsala 751 85
Sweden
[email protected]
Ramsey D. Badawl
Guy's and St. Thomas' Clinical PET Centre
Division of Radiological Sciences
Lower Ground Floor, Lambeth Wing
Lambeth Palace Road
London SE1 7EH
U.K.
[email protected]
Dale Bailey
Medical Research Council
Cyclotron Unit
Hammersmith Hospital
Du Cane Road
London W12 ONN
U.K.
dale @wren.rpms.ac.uk
Harrison Barrett
University of Arizona
Department of Radiology
P. O. Box 245067
1502 North Campbell Avenue
Tucson AZ 85724-5067
U.S.A
[email protected]
Roman Basko
University of Utah
Department of Radiology
MIRL AC211 School of Medicine
50 N. Medical Drive
Salt Lake City UT 84132
U.S.A
[email protected]
11997 International Meeting on Fully 3D Image Reconstruction
2121
Xavier L. Battle
Los Alamos National Laboratory
Physics Division, Biophysics Group
MS D454
Los Alamos NM 87545
U.S.A.,
Freek Beekman
Utrecht University Hospital
Department of Nuclear Medicine
Room E02-222, P.O. Box 85500
Utrecht 3508 GA
The Netherlands
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[email protected]
Thomas
Beyer
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University of Pittsburgh Medical Center
Department of Radiology, PET-Facility
Room B-938, PUH
200 Lothrop Street
Pittsburgh PA 15213-2582
U.S.A
[email protected]
[}
Yen-Wei Chen
CJ
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chen @tec.u-ryukyu.ac.jp
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Rolf Clack
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University of Ryukyus
Faculty of Engineering
1 Senbaru
Nishihara
Okinawa 903-01
Japan
University of Utah
Department of Radiology
MIRL AC213 School of Medicine
50 N. Medical Drive
Salt Lake City UT 84132
U.S.A.
ro/[email protected]
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11997 International Meeting on Fully 3D Image Reconstruction
2131
J
Erlc Clarkson
University of Arizona
Department of Radiology
P. O. Box 245067
1502 North Campbell Avenue
Tucson AZ 85724-5067
U.S.A.
Claude D. Comtat
University of Pittsburgh Medical Center
Department of Radiology, PET-Facility
Room 8"938, PUH
200 Lothrop Street
Pittsburgh PA 15213-2582
U.S.A.
[email protected]
Per-Erik Danlelsson
Linkoping University
Department of Electrical Engineering
Arbetateg. 50
Linkoping 58183
Sweden
ped@;sy.liu.se
Margaret E. Daube-Witherspoon
National Institutes of Health
PET Department
Building 10, Room 1C-497
10 Center Drive MSC 1180
Bethesda MD 20893-1180
U.S.A.
daube-witherspoon @nlh.gov
Michel
Oefrlse
Free University of Brussels
Division of Nuclear Medicine, AZ-VUB
Laarbeeklaan 101
Brussels 1090
Belgium
michel@vub. vUb.ac.be
Laurent Desbat
TIMC-rMAG
UJF-UMR CNRS 5525-CHUG
La Tronche 38706
France
[email protected]
11997 International Meeting on Fully 3D Image Reconstruction
2141
Frank DiFilippo
L]
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Picker International
Nuclear Medicine Division
595 Miner Road
Highland Hts. OH 44143
U.S.A.
frankd@ nm.pieker.com
I
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Paul Edholm
Linkoping University
Department of Electrical Engineering
Arbetateg. 50
Linkoping 58183
Sweden
[email protected]
[1
Frank Emert
r\L.J
Paul Scherrer Institut
PET Program
Villigen 5232
Switzerland
frank. [email protected]
Jan Eriksson
[:J
Linkoping University
Department of Electrical Engineering
Arbetateg. 50
Linkoping 58183
Sweden
janer@ isy.liu.se
[]
Lars Eriksson
CTI, Inc.
[]
810 Innovation Drive
Knoxville TN 37932
U.S.A.
-.1
[. J
eriksson @eti-pet.eom
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Jeff Fessler
University of Michigan
4240 EECS Building
1301 Beal Avenue
Ann Arbor MI 48109-0552
U.S.A.
[email protected]
L.J
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11997 International Meeting on Fully 3D Image Reconstruction
2151
Erlc Frey
The University of North Carolina at Chapel Hill
Department of Biomedical Engineering
Campus Box 7575
152 MacNider Hall
Chapel Hill NC 27599-7575
U.S.A
[email protected]
Cllffreda Gilreath
CTI PET Systems, Inc.
810 Innovation Drive
Knoxville TN 37932
U.S.A.
[email protected]
Gene
Glndl
SUNY at Stony Brook
Department of Radiology
Stony Brook NY 11794
U.S.A.
[email protected]
Pierre
Grangeat
LETI (CEA - Technologies Avancees)
Department Systemes, SCSI
17 rue des Martyrs
Grenoble Cedex 9 38054
France
[email protected]
Michael Grass
Philips Research Hamburg
Division Technical Systems
Roentgenstrasse 24-26
Hamburg 22335
Germany
[email protected]
Jens Gregor
The University of Tennessee
Department of Computer Science
107 Ayres Hall
Knoxville TN 37996-1301
U.S.A.
11997 International Meeting on Fully 3D Image Reconstruction
2161
Eugene Gualtieri
UGM Laboratory, Inc.
3611 Market Street
Philadelphia PA 19104
U.S.A.
[email protected]
Grant T. Gullberg
University of Utah
Department of Radiology
MIRL AC215 School of Medicine
50 Medical Drive N.
Salt Lake City UT 841 32
U.S.A.
[email protected]
Donald L. Gunter
[l
Rush-Presbyterian St. Luke's Medical Center
Department of Medical Physics and Diagnostic Radiology
1653 W Congress Parkway
Chicago IL 60612
U.S.A.
[email protected]/mc.edu
-1
J
[
James Hamill
CTI,lnc.
810 Innovation Drive
Knoxville TN 37932-2571
U.S.A.
Kenneth M. Hanson
Los Alamos National Laboratory
MS P940, DX-3, Hydrodynamics
Los Alamos NM 87544
U.S.A.
[]
[]
kmh @/anJ.gov
[J
Robert L.
[J
,
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Harrison
University of Washington
Department of Radiology
Box 356004
1959 Pacific Avenue NE
Seattle WA 98195-6004
U.S.A.
[email protected]
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11997 International Meeting on Fully 3D Image Reconstruction
2171
Erlc G. Hawman
Siemens Medical Systems, Inc.
Nuclear Medicine Group
2501 N. Barrington Road
Hoffman Estates IL 60195-5203
U.S.A.
[email protected]
David R. Haynor
University of Washington
Imaging Research Laboratory
Box 356004
Seattle WA 98195
U.S.A.
haynor@u. washington. edu
Sophie Henry
Centre Hospitalier Universitaire de Liege
Service de Medecine Nucleaire
Sert Tilman B35
Liege 1 4000
Belgium
Dominic J. Heuscher
Picker International
595 Miner Road
Highland Hts OH 44026
U.S.A.
[email protected]
Yu .. Lung Hsieh
University of Utah
Department of Radiology
MIRL AC213 School of Medicine
50 N. Medical Drive
Salt Lake City UT 84132
U.S.A.
[email protected]
Hui Hu
GE Medical Systems
P.O. Box 414, NB922
Milwakee WI 53201
U.S.A.
[email protected]
11997 International Meeting on Fully 3D Image Reconstruction
2181
Ronald Huesman
Lawrence Berkeley National Laboratory
Center for Functional Imaging
One Cyclotron Road #55-121
Berkeley CA 94720
U.S.A.
[email protected]
Roberto A. Isoardi
[1
Fundaci6n Escuela de Medicina Nuclear de Mendoza
Garibaldi 405
Mendoza 5500
Argentina
risoardi@ raiz. uncu. edu.ar
Filip Jacobs
[1
University of Ghent
Department of Electronics and Information Systems, Medical
Image and signal Processing research group
Sint Pietersnieuwstraat 41
Gent 9000
Belgium
jacobs @petultra.rug.ac.be
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Alan P. Jeavons
Oxford Positron Systems Ltd
Weston Business Park, Weston-on-the-Green
5 Landscape Close
Oxfordshire OX6 8SY
U.K.
alan @oxpos.co.uk
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Roger H. Johnson
Marquette University
Biomedical Engineering Department
P.O. Box 1881
Milwaukee \NI 53201-1881
U.S.A.
[email protected]
Dan J. Kadrmas
The University of North Carolina at Chapel Hill
Department of Biomedical Engineering
Campus Box 7575
152 MacNider Hall
Chapel Hill NC 27599-7575
U.S.A
[email protected]
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11997 International Meeting on Fully 3D Image Reconstruction
2191
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Chrls Kamphuls
University Hospital Utrecht
Department of Nuclear Medicine
Room E02 222, P.O. Box 85500
Utrecht 3508 GA
The Netherlands
[email protected]/
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Joel S. Karp
University of Pennsylvania
Department of Radiology
110 Donner Bldg., HUP
3400 Spruce Street
Philadelphia PA 19104
U.S.A.
joe/@goodman.pet.upenn.edu
Paul E. Kinahan
University of Pittsburgh Medical Center
Department of Radiology, PET Facility
Room 8 938, PUH
200 Lothrop Street
Pittsburgh PA 15213·2582
U.S.A.
a
8
pau/@pet.upmc.edu
Michael King
University of Massachusetts Medical School
Department of Nuclear Medicine
55 Lake Avenue N
Worcester MA 01655
U.S.A.
king@ lightseed. ummed.edu
Vandana Kohli
University of Massachusetts Medical Center
Department of Nuclear Medicine
55 Lake Avenue N
Worcester MA 01655
U.S.A.
kohli@ lightseed. ummed. edu
Hlroyuki Kudo
University of Tsukuba, Japan
University of Brussels, Division of Nuclear Medicine
Laarbeeklaan 101
Brussels 1090, Belgium
hkudo@vub. VUb.Bc.be
11997 International Meeting on Fully 3D Image Reconstruction
2201
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Claire Labbe
[-)
Geneva University Hospital
Division of Nuclear Medicine
Geneve 4 1211
Switzerland
clabbe @ulipn.unil.ch
David S. Lalush
[1
The University of North Carolina at Chapel Hill
Department of Biomedical Engineering
Campus Box 7575
152 MacNider Hall
Chapel Hill NC 27599-7575
U.S.A
lalush @bme.unc.edu
Ivan Laurette
University of Nice-Sophia Antipolis
Faculte de Medecine, Laboratoire de Biophysique et de
Traitement de I'lmage
Avenue de Valombrose 28
Nice Cedex 2 06107
France
la urette @ unice. fr
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Richard Leahy
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University of Southern California
Department of Electrical Engineering-Systems, Signal and
Image Processing Institute
Electrical Engineering Building, Room 400
3740 McClintock Avenue
Los Angeles CA 90089
U.S.A.
gloria @ sipi. usc. edu
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Robert M. Lewitt
University of Pennsylvania
Department of Radiology / MIPG
Blockley Hall, 4th Floor
423 Guardian Drive
Philadelphia PA 19104-6021
U.S.A.
[email protected]
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11997 International Meeting on Fully 3D Image Reconstruction
2211
Jerome Z. Liang
State University of New York at Stony Brook
Department of Radiology
4th Floor, Room 0921HSC
Stony Brook NY 11794 8460
U.S.A
jz/@clio.rad.sunysb.edu
8
Jelh .. San
Llow
University of Minnesota
Department of Radiology
PEr Imaging (11 P), VA Medical Center
One Veterans Drive
Minneapolis MN 55417
U.S.A
je/@pet.med. va.gov
Marla Magnusson Seger
Link<>ping University
Department of Electrical Engineering,
Image Processing Group
Arbetateg. 50
Link5ping 58183
Sweden
maria @isy./iu.se
Paul Maguire
Paul Scherrer Institute
PET Program
Villingen 5232
Switzerland
nlagu/re @psl.ch
Ronald E. Malmln
Siemens Medical Systems, Inc.
Nuclear Medicine Group
2501 N. Barrington Road
Hoffman Estates IL 60195
U.S.A.
[email protected]
11997 International Meeting on Fully 3D Image Reconstruction
2221
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Samuel Matej
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University of Pennsylvania
Department of Radiology, Medical Image Processing Group
4th. Floor, Blackley Hall
423 Guardian Drive
Philadelphia PA 19104-6021
U.S.A
[email protected]
Christian Michel
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Universite Catholique de Louvain
PET Laboratory
Chemin du Cyclotron, 2
Louvain-Ia-Neuve 1348
. Belgium
[email protected]
Christian Morel
Geneva University Hospital
Division of Nuclear Medicine
Geneve 4 1211
Switzerland
Christian. Morel @ipn.unil.ch
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Gerd
Muehllehner
UGM Medical Systems
3611 Market Street
Philadelphia P A 1 91 04
U.S.A.
[email protected]
[J
Tom Nichols
[j
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Carnegie Mellon University
Department of Statistics
5000 Forbes Avenue
Pittsburgh PA 15213
U.S.A.
[email protected]
Douglas C. Noll
University of Pittsburgh Medical Center
Department of Radiology, M.R.I Research Center
Room 8-804, PUH
200 Lothrop Street
Pittsburgh PA 15213
U.S.A.
[email protected]
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11997 International Meeting on Fully 3D Image Reconstruction
2231
Frederic Noo
University of Liege
Institute of Electricity Montefiore
828
Rue R. Sualem 135
Jemeppe 4101
Belgium
[email protected]
Ronald Nutt
CTI,lnc.
810 Innovation Drive
Knoxville TN 37932
U.S.A.
[email protected]
Johan Nuyts
Katholieke Universiteit Leuven
Department of Nuciear Medicine
U.Z. Gasthuisberg
Herestraat, 49
Leuven 3000
Belgium
[email protected]
Terry R. Oakes
University of British Columbia / TRIUMF PET Centre
4004 Wesbrook Mall
Vancouver Be V6T 2A3
Canada
[email protected]
Anne
M. J. Paans
Gronlngen University Hospital
PET~Center-
P.O. Box 30.001
Gronlngen 9700 RB
The Netherlands
[email protected]
Tinsu Pan
GE Medical Systems
General Electric Company
P.O. Box 414, NB~922
Milwaukee WI 53201
U.S.A.
[email protected]
11997 International Meeting on Fully 3D Image Reconstruction
2241
Roland Proska
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Philips Research Hamburg
Division Technical Systems
Roentgenstrasse 24-26
Hamburg 22335
Germany
[email protected]
Ij
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Jinyi Qi
[]
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University of Southern California
Department of Electrical Engineering-Systems, Signal and
Image Processing Institute
Electrical Engineering Building, Room 400
3740 McClintock Avenue
Los Angeles CA 90089-2564
U.S.A.
gloria @sipi.usc.edu
Fernando Rannou
The University of Tennessee
Department of Computer Science
107 Ayres Hall
Knoxville TN 37996-1301
U.S.A.
[email protected]
[]
Andrew J. Reader
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U.K.
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Institute of Cancer Research
Joint Department of Physics
Royal Marsden NHS Trust
Downs Road
Sutton Surrey SM2 5PT
[email protected]
Trudy D. Rempel
Siemens Medical Systems, Inc.
Nuclear Medicine Group
2501 N. Barrington Road
Hoffman Estates IL 60195
U.S.A.
[email protected]
[]
11997 International Meeting on Fully 3D Image Reconstruction
2251
Alberto F. Rodriguez
The University of Tennessee
Center for International Networking Initiatives
2000 Lake Evenue
Knoxville TN 37996
U.S.A.
alberto @aurora.phys.utk.edu
Christopher Ruth
Analogle
8 Lentennial Drive
Penbody MA 01960
U.S.A.
oruth@analog/e.oom
Anne Claire Sauve
University of Michigan
EECS Department, Systems
407 N. Ingalls Street, #A8
Ann Arbor MI 481 04
U.S.A.
asauve@eng/n.umioh.edu
Matthias Schmand
CTI PET Systems, Inc.
810 Innovation Drive
Knoxville TN 37932
U.S.A.
[email protected]
Michael D. Silver
Bio-Imaging Research, Inc.
425 Barclay Boulevard
Lincolnshire IL 60069
U.S.A.
Mark F. Smith
Duke University Medical Center
Department of Radiology
Box 3949
162 Bryan Research Building
Durham NC 27710
U.S.A.
[email protected]
11997 International Meeting on Fully 3D Image Reconstruction
2261
[]
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Vesna Sossi
[]
University of British Columbia / TRIUMF PET Centre
4004 Wesbrook Mall
Vancouver BC V6T 2A3
Canada
[email protected]
Ii
Terry J. Spinks
Medical Research Council
Cyclotron Unit
Hammersmith Hospital
Du Cane Road
London W12 ONN
U.K.
[email protected]
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Wendy L. Swan
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University of Washington
Department of Radiology
Room NW040
Box 356004
Seattle WA 98195-6004
U.S.A
wendy_swan @oscar.rad. washington. edu
Kwok C. Tam
Siemens Corporate Research, Inc.
CT Research and Development
755 College Road E.
PrincetonNJ 08540
U.S.A.
[email protected]
Eiichi Tanaka
Hamamatsu Photonics K.K.
Mori-Bldg, n° 33, 5F
3-8-21, Toranomon, Minato-ku
Tokyo 105
Japan
tanaka @hq.hpk.co.jp
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Andreas Terstegge
Forschungszentrum JOlich GmbH
Zentrallabor fOr Elektronik'
Leo Brand StraBe
JOlich 52425
Germany
A. [email protected]
[]
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11997 International Meeting on Fully 3D Image Reconstruction
2271
Krls Thielemans
Medical Research Council
Cyclotron Unit
Hammersmith Hospital
Du Cane Road
London W12 ONN
U.K.
[email protected]
Christopher
Thompson
Montreal Neurological Institute
Research Computing Laboratory
3801 University Street, #798
Montreal QC H3A 284
Canada
ehris@ rclvax.medcor. mcgill. ca
David W. Townsend
UniV6isity of Pittsburgh Medicai Center
Department of Radiology, PET-Facility
Room 8-938, PUH
200 Lothrop Street
Pittsburgh PA 15213-2582
U.S.A.
[email protected]
Benjamin M. W. Tsui
The University of North Carolina at Chapel Hill
Department of Biomedical Engineering
Campus Box 7575
152 MacNider Hall
Chapel Hill NC 27599-7575
U.S.A.
[email protected]
Heang K. Tuy
Picker International
595 Miner Road
Highland Hts OH 44026
U.S.A.
[email protected]
Paul Vaska
UGM Laboratory, Inc.
3611 Market Street
Philadelphia PA 19104
U.S.A.
[email protected]
11997 International Meeting on Fully 3D Image Reconstruction
2281
[I
I
[J
Charles Watson
[ :I
CTI PET Systems, Inc.
810 Innovation "Drive
Knoxville TN 37933
U.S.A.
[email protected]
Daniel Wessell
The University of North Carolina at Chapel Hill
Department of Biomedical Engineering
Campus Box 7575
1'52 MacNider Hall
Chapel Hill NC 27599-7575
U.S.A
[email protected]
Klaus Wienhard
Max Planck Institute of Neurological Research
Gleueler StraBe 50
K61n 50931
Germany
Klaus. [email protected]
Karl Wiesent
SiemensAG
Medical Engineering Group
Med GT 1
P. O. Box 3260
Erlangen 91050
Germany
karl. [email protected]
Chunwu Wu
Positron Corporation
16350 Park Ten Place
Houston TX 77084
U.S.A.
[email protected]
Guofeng Yin
[]
Toshiba America MRI, Inc.
Nuclear Medicine Engineering
280 Utah Avenue
South San Francisco CA 94080
U.S.A.
[email protected]
11997 International Meeting on Fully 3D Image Reconstruction
Habib Zaidi
Geneva University Hospital
Division of Nuclear Medicine
Geneve 4 1211
Switzerland
[email protected]
Larry G. Zeng
University of Utah
Department of Radiology
MIRL AC211 School of Medicine
50 N. Medical Drive
Salt Lake City UT 84132
U.S.A.
[email protected]
George Zubal
Yale University
Department of Diagnostic Radioiogy
BML 332
333 Cedar Street
New Haven CT 0651 0
U.S.A
[email protected]
11997 International Meeting on Fu"y 3D Image Reconstruction
2301