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J. H. Siewerdsen and D. A. Jaffray, Proc. SPIE Physics of Medical Imaging (2003).
Three-Dimensional NEQ Transfer Characteristics
of Volume CT using Direct and Indirect-Detection Flat-Panel Imagers
J. H. Siewerdsen1 and D. A. Jaffray2
1,2
Ontario Cancer Institute, Princess Margaret Hospital, University Health Network
1
Department of Medical Biophysics and 2Department of Radiation Oncology, University of Toronto
Toronto, ON, Canada M5G 2M9
ABSTRACT
Foremost among the promising imaging performance characteristics of cone-beam CT using flat-panel imagers is the
ability to form volumetric images with soft-tissue contrast visibility in combination with sub-millimeter 3-D spatial
resolution. Each of these two essential characteristics is intimately related to the spatial-frequency-dependent signal and
noise transfer characteristics of the imaging system. Therefore a thorough, quantitative analysis of the 3-D noiseequivalent quanta (NEQ) and detective quantum efficiency (DQE) is essential to understanding the volumetric imaging
performance of such systems, identifying the factors that limit performance, and revealing their full potential.
This paper presents investigation of the 3-D NEQ and DQE for volume CT systems based on direct and indirectdetection flat-panel imagers (FPIs). Classical descriptions of image noise in transaxial CT are extended to the case of
non-ideal 2-D detectors and 3-D image reconstruction. Definitions of NEQ and DQE are extended to provide figures of
merit for 3-D imaging performance. A complex interplay between the system transfer functions, 3-D noise aliasing, and
the 3-D DQE is uncovered, revealing several important phenomena: 1.) 3-D NPS aliasing is a significant factor in the
reconstruction process affecting DQE; 2.) the degree of 3-D NPS aliasing is different for direct and indirect-detection
FPIs and is related in non-trivially to the detector MTF and reconstruction filter; 3.) the 3-D NEQ depends significantly
on the choice of reconstruction filter – in contrast to the classical notion that NEQ is independent of such – and the effect
is wholly attributable to 3-D NPS aliasing; and 4.) the 3-D DQE for volume reconstructions is asymmetric between
transverse and sagittal/coronal planes. Results for 3-D NEQ and DQE are integrated with 3-D spatial-frequencydependent descriptions of imaging task (e.g., ideal observer detection and/or discrimination tasks) to yield the 3-D
detectability index, helping to bridge the gap between NEQ and the performance of model observers.
Keywords: noise-power spectrum, noise-equivalent quanta, detective quantum efficiency, imaging task, computed
tomography, 3-D imaging, cone-beam CT, flat-panel imagers, imaging performance
1. INTRODUCTION
Flat-panel imagers have evolved to a level supporting investigation of several advanced applications, including real-time
fluoroscopy, dual-energy imaging, tomosynthesis, and cone-beam CT, with potential application in a variety of
diagnostic and image-guided procedures. An important part of this evolution was the development of a quantitative
understanding of the factors that govern the frequency-dependent signal and noise characteristics– viz., the modulation
transfer function (MTF), noise-power spectrum (NPS), detective quantum efficiency (DQE), and noise-equivalent
quanta (NEQ). For both direct-detection FPIs (e.g., employing a-Se, PbI2, HgI2, etc. as the x-ray converter) and indirectdetection FPIs (e.g., employing Gd2O2S:Tb or CsI:Tl converters), these properties have been well characterized through
measurements and theoretical models for the case of static 2-D projection imaging (e.g,. breast mammography or chest
radiography). As FPIs evolve toward advanced applications, so must the experimental and theoretical techniques that are
vital to quantifying imaging performance, identifying limitations, and revealing the full potential of the technology.
Cone-beam CT is a promising advanced application of FPIs, providing fully 3-D image reconstructions with soft-tissue
visibility and sub-millimeter, isotropic spatial resolution. Each of these characteristics is intimately related to the spatialfrequency-dependent signal and noise properties of the imaging system. This paper extends theoretical descriptions of
NPS and NEQ developed in the context of projection radiography and transaxial CT to the case of fully 3-D cone-beam
CT. Both direct and indirect-detection FPIs are considered, and the tradeoffs associated with different detector signal
transfer characteristics are investigated. Finally, the results are integrated with quantitative descriptions of imaging task
and detectability for a simple, ideal observer model under a variety of idealized detection and discrimination tasks.
Medical Imaging 2003: Physics of Medical Imaging. M. J. Yaffe and L. E. Antonuk Eds. Proc. SPIE Vol. 5030
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J. H. Siewerdsen and D. A. Jaffray, Proc. SPIE Physics of Medical Imaging (2003).
2. EXPERIMENTAL PROVING GROUND
FOR ADVANCED APPLICATIONS OF FLAT-PANEL IMAGERS
An enhanced experimental platform has been developed for investigation of flat-panel cone-beam CT and other
advanced applications of FPIs [Fig. 1(a)], offering greater performance and flexibility compared to the benchtop system
reported previously.1 The enhanced platform incorporates 7 servo-driven, precision linear tables (Parker-Daedal) with
range of motion up to 75 cm and a high-resolution, direct-drive rotation stage (Dynaserv) with rotation rate up to 1 Hz.
The eight axes are operated under computer control and are synchronized with firing of the x-ray tube (pulsed fluoro or
digital radiographic modes) and readout of the FPI (up to 30 fps). The position of both the x-ray tube and FPI can be
precisely controlled in 3-D during image acquisition, allowing investigation of advanced modes of image acquisition.
The system supports cone-beam CT imaging with complex trajectories (e.g., circle-and-line orbits or simulation of
geometric non-idealities). The system provides a robust test-bed for investigation of FPIs in fluoroscopy, tomosynthesis,
and dual-energy imaging, and the large range of motion is ideal for studies of x-ray scatter and optimal imaging
geometry. The detector mount allows the FPI to be interchanged (e.g., direct and indirect-detection FPIs), and the x-ray
tube mount can support multiple tubes. Finally, the flexible geometry allows the system to emulate the geometry of a
wide range of potential clinical embodiments (e.g., conventional CT gantries, medical linear accelerators, and C-arms).
Figure 1(a) shows the system configured with an indirect-detection FPI (Varian 4030A) at a source-to-detector distance
of ~160 cm (source-to-axis distance of ~100 cm). The flat-panel cone-beam CT system exhibits sub-mm spatial
resolution (e.g., ~0.4 mm full-width at half-maximum in images of a thin steel wire) and soft-tissue contrast visibility
[e.g., structures with ~5% contrast visible in (0.24x0.24x0.24) mm3 voxel reconstructions obtained at ~1-2 mGy dose to
the center of the phantom]. The quality of image reconstructions is illustrated in images of a New Zealand white rabbit
(euthanized for other purposes) shown in Fig. 1(b-d). These images demonstrate the two key imaging performance
characteristics of flat-panel cone-beam CT: sub-mm spatial resolution in 3D and visibility of soft-tissue structures. Since
these two essential characteristics are intimately related to the spatial-frequency-dependent signal and noise transfer
characteristics of the imaging system, a quantitative analysis of the 3-D noise-equivalent quanta (NEQ) and detective
quantum efficiency (DQE) is essential to revealing the full potential of such systems.
Figure 1.
(a) Experimental platform for
investigation of imaging performance in advanced
applications of flat-panel imagers, such as conebeam CT. The system incorporates an FPI, an xray tube, and the object to be imaged, all
synchronized within an eight-axis geometry. (b-d)
Example images of a rabbit acquired using the
system in (a). Images were acquired at 110 kVp at
low dose (63 mAs total; ~1-2 mGy dose to water
at center of reconstruction), and 300 projections
acquired through 360o. Volume images were
reconstructed at (0.24 x 0.24 x 0.24) mm3 voxel
size using a modified Feldkamp algorithm.
Medical Imaging 2003: Physics of Medical Imaging. M. J. Yaffe and L. E. Antonuk Eds. Proc. SPIE Vol. 5030
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J. H. Siewerdsen and D. A. Jaffray, Proc. SPIE Physics of Medical Imaging (2003).
3. NOISE TRANSFER CHARACTERISTICS FOR FLAT-PANEL CONE-BEAM CT
This section provides an overview and derivation of the noise transfer characteristics for flat-panel cone-beam CT. In
Sec. 3.1, image noise for conventional transaxial CT is briefly reviewed. In Sec. 3.2, the analysis is extended to the case
of non-ideal 2-D detectors (viz., direct and indirect-detection FPIs), and the propagation of the NPS through the process
of 3-D filtered backprojection is derived in Sec. 3.3 to yield a theoretical description of the 3-D volumetric NPS.
3.1 Background: The NPS for Conventional Transaxial CT
In the late 1970s, quantitative descriptions of image noise and NPS for 2-D transaxial CT were derived by Barrett et al.,2
Hanson,3 and Wagner et al.,4 showing the relationship between the (1-D) projection NPS, the filters applied during
reconstruction, and the 2-D reconstruction NPS. Following the notation of Hanson,3 we have for the reconstruction NPS:
2
§ m ·§ S · 2
2
2
¸¸ ¨ ¸ Tramp f Twin
f Tinterp
f S proj f S f ¨¨
(1a)
f
m
S
©
¹© ¹
where f is radial spatial frequency, m the number of views, Tramp the ramp filter, Twin the apodization window (e.g.,
Hanning), Tinterp the interpolation filter, and Sproj the normalized projection NPS. Such analysis was furthered by
Kijewski and Judy5 to include the effects of sampling and 2D aliasing on the reconstruction NPS. Early investigations
considered a detector with constant projection NPS (a good assumption for conventional CT detectors), with quantum
detection efficiency, K, and voxels of dimension axy and slice thickness az. The variance in voxel values, given by the
integral over the NPS, is found to be inversely proportional to dose, slice thickness, and the cube of the voxel size:
2
fN
2S
K xy2
Pe E
1 K xy
(1b)
V 2 ³ dT ³ df S f 3
3
Pd
NEQ a xy
0
0
Do e 2 Ua zK a xy
where NEQ is the (low-frequency) noise-equivalent quanta, and Do is the dose at the center of a cylindrical object
having attenuation P and diameter d. The factor Kxy is the bandwidth integral, as described by Wagner:4
K xy2
fN
3
2
2
2S a xy
³ df f Twin f Tinterp f (1c)
0
Such descriptions of NPS and NEQ provide essential relations of dose, noise, and spatial resolution in transaxial CT.
3.2 Extension to Non-Ideal 2-D Detectors
Classical treatment of NEQ in transaxial CT typically considered an “ideal” 1-D detector – i.e., no conversion noise, blur
(beyond that of the detector apertures), or additive noise – yielding a constant NPS determined simply by the aperture
and incident fluence.3 In considering flat-panel cone-beam CT, two extensions to the classical approach are required: 1.)
the detector is non-ideal (with significant sources of blur or noise possible at each stage of image formation); and 2.) the
projections are 2-D (yielding fully 3-D reconstructions). Theoretical models for the NPS of flat-panel imagers were
developed in the 1990s and found to agree well with measurements of 2-D NPS and DQE for both direct and indirectdetection FPIs.6,7. For the sake of brevity, we consider a simple model that unifies the most significant aspects of the
signal and noise transfer characteristics of direct and indirect-detection FPIs. Using notation common to linear cascaded
systems analysis (with numerical subscripts denoting stages in image formation), we have for the projection NPS:
S 7 u, v a x2 ap a z2 ap q g1 g 2 g 4 1 g 4 g 2 H g 2 T32 u , v T52 u , v III u, v; a x pix , a z pix S add u , v (2a)
>
>
@
@
where (u,v) are spatial-frequency coordinates in the detector plane, ax-ap and az-ap are the aperture dimensions, q is the
incident fluence, g 1 the quantum detection efficiency, g 2 the gain [secondary quanta (electrons or optical photons) per
interacting x-ray], Hg2 the Poisson excess in secondary quanta, T3 the MTF associated with spread of secondary quanta,
g 4 the coupling efficiency of secondary quanta, and T5 the MTF of the pixel apertures. The sampling matrix is III(u,v),
determined by the pixel pitch, ax-pix and az-pix, with ** denoting a 2-D convolution. Sadd comprises sources of additive
electronic and digitization noise. This form provides a reasonable approximation to the NPS for both direct and indirectdetection FPIs (e.g., with T3 approaching unity in the direct case and given by the scintillator MTF for the indirect case).
As in Eq. (1a), the normalized projection NPS is given by the detector NPS divided by the mean signal squared:
S 7 u, v (2b)
S proj u , v 2
a x a z q g1 g 2 g 4
This yields a useful starting point for consideration of the 3-D noise transfer characteristics, where the 2-D detector NPS
is generalized to provide a reasonable description of both direct and indirect-detection FPIs.
Medical Imaging 2003: Physics of Medical Imaging. M. J. Yaffe and L. E. Antonuk Eds. Proc. SPIE Vol. 5030
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3.3 Transfer Characteristics for 3D Filtered Backprojection
The signal and noise characteristics of projection images propagate through the process of image reconstruction
according to a series of deterministic mathematical steps. The process was described by Kijewski and Judy5 for the case
of 2-D transaxial CT and is extended here to the 3-D case. Figure 2 illustrates the propagation of the projection NPS for
indirect and direct-detection FPIs through a cascade of processes in the frequency domain. For brevity, we lump factors
of image scaling (e.g., gain and offset corrections, normalization, and log scaling) in the projection NPS at Stage 7 and
proceed to the step that imparts a change in frequency content – the ramp filter, |u|, applied in the lateral frequency
domain and written Tramp(u). For this deterministic process, the NPS is modulated by the square of the transfer function:
2
S 8 u , v S 7 u, v Tramp
u (3a)
Similarly, at Stage 9 high-frequency noise is attenuated through application of an apodization window (e.g., Hanning):
2
u S 9 u , v S 8 u, v Twin
(3b)
Next, the projection data are interpolated at Stage 10 (e.g., n-n or bilinear interpolation) prior to backprojection:
2
S10 u, v S 9 u, v Tinterp
u, v (3c)
Note the significant difference between the NPS for indirect and direct-detection FPIs illustrated in Fig. 2, particularly in
the longitudinal (v) domain. Only at Stage 10 is the longitudinal component of the NPS attenuated. In this sense, the
direct-detection case is analogous to the case of stacked-slice transaxial CT, spiral CT, or multi-detector CT (with or
without z-interpolation from row to row, where information from slice to slice may or may not be correlated).
Figure 3 illustrates the propagation of NPS for direct and indirect-detection FPIs through processes associated with 3-D
backprojection and sampling. At Stage 11, 2-D image data from the m projection views are back-projected along the
angle corresponding to each view. In the 3-D spatial-frequency domain, this corresponds to superimposing each 2-D
NPS along a vane at each angle through the origin. Superposition of the m NPS produces a radially discrete 3-D NPS
(analogous to the radial spokes in 2-D transaxial CT5). As in Eq. (1a), therefore, we have for the 3-D presampling NPS:
2
§ m ·§ S ·
S 2
2
2
¸¸ ¨ ¸ S10 f , f z f Tinterp
f , f z S proj f , f z S11 f , f z ¨¨
Tramp f Twin
(3d)
S
f
m
f
©
¹
¹
©
where f and fz are radial and longitudinal frequency coordinates. Finally, the image reconstruction is sampled according
to a 3-D sampling matrix at pitch axy (in the transverse domain) and az (in the longitudinal domain). This corresponds to
a 3-D convolution of the presampling NPS, S11, with the Fourier transform of the sampling grid, III(fx,fy,fz):
S12 f x , f y , f z
S11 f x , f y , f z III f x , f y , f z ; a xy , a z
(3e)
where (fx,fy,fz) are 3-D spatial-frequency coordinates. Associated with 3-D sampling is the potential for significant 3-D
NPS aliasing, wherein noise-power from frequencies greater than the cutoff frequency is deposited at lower frequencies.
In fact, since the ramp filter effectively zeroes the NPS at zero-frequency (aside from a small residual to maintain DC
values), noise at zero-frequency in the image reconstruction is entirely attributable to 3-D NPS aliasing.
For illustration below, we assume that the efficiency, gain, conversion noise, etc. are the same for the two detector types,
and that the “direct” and “indirect” detectors differ only in the frequency-dependent transfer characteristics [i.e., the
MTF, T3(u,v) above]. This allows examination of the fundamental differences associated with reconstruction of images
varying intrinsically in frequency content, without concern for scalar parameters of the detection material (e.g., CsI:Tl or
Gd2O2S:Tb versus a-Se or PbI2). Of course, such parameters are central to detector performance, but the point here is to
examine the effect of distinctly different NPS (band-limited for indirect, white for direct) on the reconstruction NEQ.
3.4 Relation to Voxel Noise and Scalar SNR
As examined previously,8 integration of the 3-D NPS to yield the voxel noise reveals a number of phenomena that are
distinct from the classical 2-D case. For example, incorporating “out-of-plane” correlations in the 3-D bandwidth
integral analogous to Eq. (1c) shows that the classical dependence of noise on voxel size is lost, and the concept of “slice
thickness” is largely obsolete. Furthermore, the parameter K is interpreted as the zero-frequency DQE. A unified
approach that integrates x-ray scatter in descriptions of 3-D SNR reveals strategies that allow knowledgeable “tuning” of
the SNR through adjustment of dose and spatial resolution. Furthermore, the effect of additive noise on the 3D NPS is
found to scale inversely with the number of views. An important fact to realize in light of these considerations is that
aliasing occurs twice – once at the sampling of the projection data, and once at the sampling of the reconstruction matrix
– with an increase in noise for each that depends on the transfer characteristics of the imaging system.
Medical Imaging 2003: Physics of Medical Imaging. M. J. Yaffe and L. E. Antonuk Eds. Proc. SPIE Vol. 5030
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J. H. Siewerdsen and D. A. Jaffray, Proc. SPIE Physics of Medical Imaging (2003).
Figure 2. Cascade diagram showing the spatial-frequency-dependent noise transfer characteristics for indirect-detection (left column)
and direct-detection (right column) FPIs from presampling (Stage 6) through interpolation of projection data (Stage 10) prior to
backprojection. The central column illustrates the transfer functions associated with each stage, and the symbols “x” and “*” denote
multiplication and convolution, respectively. The lack of presampling blur in the direct-detection case results in a projection NPS that
is nearly white (Stage 7) and a correspondingly increased high-frequency noise component in subsequent steps. Note that
interpolation (Stage 10) is normally the only mechanism for reducing very high frequency noise components.
Medical Imaging 2003: Physics of Medical Imaging. M. J. Yaffe and L. E. Antonuk Eds. Proc. SPIE Vol. 5030
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J. H. Siewerdsen and D. A. Jaffray, Proc. SPIE Physics of Medical Imaging (2003).
Figure 3. Cascade diagram (continued from Fig. 2) illustrating the noise transfer characteristics in 3-D backprojection and sampling.
As in Fig. 2, the left column shows the indirect case (band-limited projection NPS) and the right the direct case (white projection
NPS). The center column illustrates the transfer mechanism at each stage – viz., superposition of the 2-D NPS along vanes (Stage 11)
and convolution with the 3-D sampling matrix (Stage 12). The symbols 6 and * denote superposition and convolution, respectively.
4. THE THREE-DIMENSIONAL DQE AND NEQ
The generalized description of the 3-D NPS provides a starting point for analysis of the 3-D spatial-frequency-dependent
signal-to-noise characteristics of the imaging system, characterized by the 3-D DQE and NEQ. As noted by
Cunningham,9 the DQE can be defined in a number of equivalent forms, including the ratio of the “deterministic NPS”
(i.e., the “blur-only” NPS) to the actual NPS. For the cascade described above, the (normalized) deterministic NPS is:
1
S 2
2
S proj u, v T32 u, v T52 u , v and S12 det f , f z f Tint2 erp f , f z S proj det f , f z (4)
Tramp f Twin
det
f
q c g1
The “stochastic” definition of DQE is adapted to the 3-D case as:
S det f x , f y , f z
DQE12 f x , f y , f z
S12 f x , f y , f z
1
NEQ12 f x , f y , f z
(5a)
qc
Given this definition, the DQE can be interpreted as (the inverse of) the fraction by which the imaging system increases
the noise from that of the incident quanta. From the right-hand side of Eq. (5a), the DQE may be equivalently interpreted
as the effective fraction of quanta used in forming the reconstruction image, where the NEQ:
S det f x , f y , f z
NEQ12 f x , f y , f z
qc
(5b)
S12 f x , f y , f z
describes the effective number of quanta used in forming the reconstruction image. This description extends the
transaxial NEQ to the 3-D case and incorporates detector non-idealities (e.g., conversion noise, electronics noise, etc.)
and blur. It is important to note that although Stages 8-11 in the cascade represent deterministic filters, sampling the 3D
matrix at Stage 12 “freezes-in” the frequency-dependent noise characteristics. Therefore, the filters applied during
reconstruction (viz., ramp, apodization, and interpolation) do not cancel out in the NEQ, and contrary to classical
notions, the NEQ does depend on the choice of reconstruction filter. This effect is wholly attributable to 3-D aliasing.
The phenomenon is analogous to the case in projection imaging, where sampling the 2-D detector matrix “freezes-in”
the presampling NPS characteristics and irrecoverably increases the NPS; this is precisely the reason that a detector with
MTF determined solely by pixel apertures (e.g., direct-detection FPIs) exhibits a “white” projection NPS. The 3-D case
Medical Imaging 2003: Physics of Medical Imaging. M. J. Yaffe and L. E. Antonuk Eds. Proc. SPIE Vol. 5030
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J. H. Siewerdsen and D. A. Jaffray, Proc. SPIE Physics of Medical Imaging (2003).
Figure 4. Theoretical and measured 3-D DQE for flatpanel cone-beam CT images using an indirect-detection
(CsI:Tl-based) FPI. Transverse [(a) and (b)] and sagittal
[(c) and (d)] cuts of predicted (left) and measured (right)
DQE demonstrate reasonable agreement. The magnitude
of the DQE is given by the grayscale colorbar. (e)
Surface rendering of the predicted 3-D DQE at the 20%
isosurface, illustrating its asymmetric nature.
is a straightforward extension, where sampling the 3-D matrix “freezes-in” the transfer characteristics of the
reconstruction process and increases the NPS by aliasing; therefore, high-resolution (“sharp”) filters result in increased
3-D NPS aliasing.
Results are shown in Fig. 4, comparing the predicted [(a) and (c)] and measured [(b) and (d)] 3-D DQE. The geometry
was the same as in Fig. 1(a), with an indirect-detection FPI (PerkinElmer RID 1640A) used to form 3-D reconstructions
from 300 uniform projections (~400 PR per projection at the detector) at 120 kVp (2mm Cu added filtration). Detailed
description of the experimental methods is given in Ref. 10. In the transverse (fx,fy) domain, the DQE is radially
symmetric, reduced near zero-frequency (due to NPS aliasing) and tailing off at high frequency. In the sagittal (fy,fz)
domain, the DQE is asymmetric, exhibiting a monotonically decreasing characteristic in the fz direction. Discrepancy
near zero-frequency, particularly along fz, remains to be investigated and could be due to limitations in the experimental
and/or theoretical analysis techniques. From the grayscale it is seen that the 3-D DQE at low-mid frequency has
magnitude comparable to the 2-D projection DQE for this FPI (~45%), with reduction caused by 3-D NPS aliasing.
Figure 4(e) shows the DQE rendered at the 20% isosurface and illustrates the asymmetry between transverse and
sagittal/coronal domains.
Figure 5 shows the 3-D DQE evaluated for different choices of reconstruction filter (a cosine filter varied continuously
by the coefficient, hwin, equal to 0.5 for the “low-res” Hanning filter and 1.0 for the “hi-res” Ram-Lak filter). Figure 5(a)
compares measured and predicted DQE(fx) for the indirect-detection FPI as in Fig. 4, showing fair agreement at midhigh frequencies. For the low-res filter, the same discrepancy at low frequency is seen. In (b) and (c), DQE(fx) and
DQE(fz) are compared for direct and indirect-detection FPIs at two settings of reconstruction filter. As seen in Fig. 5(b)
for both direct and indirect-detection systems, the hi-res filter improves the DQE in the fx domain at high frequency, but
degrades the low-frequency DQE significantly due to NPS aliasing. In the fz domain [Fig. 5(b)] on the other hand, the hires filter degrades the DQE at all frequencies, since the filter does not change the signal transfer characteristics in this
domain, yet adds noise due to 3-D aliasing. For the low-res filter, the DQE for direct and indirect-detection cases are
comparable, with the DQE for the direct case slightly reduced due to increased 3-D NPS aliasing (bearing in mind that
these calculations consider only the differences due to detector signal transfer characteristics (i.e., MTF) and assume
equivalent efficiency, gain, etc. – see Secs. 3.3 and 6.). For the hi-res filter, the direct-detection system suffers to a
greater extent due to a high degree of NPS aliasing.
The 3-D DQE and NEQ are powerful figures of merit for evaluating volumetric imaging performance and are useful in
examining tradeoffs in system design (e.g., choice of detector and geometry), image acquisition (e.g., dose and kVp),
and image reconstruction (e.g., choice of reconstruction filter). Often, however, the resulting DQE indicate a cross-over
Medical Imaging 2003: Physics of Medical Imaging. M. J. Yaffe and L. E. Antonuk Eds. Proc. SPIE Vol. 5030
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Figure 5. 3-D DQE evaluated for various choices of reconstruction filter. (a) Comparison of predicted and measured values for the
system based on the same indirect-detection FPI as in Fig. 4. (b) Transverse [DQE(fx)] and (c) longitudinal [DQE(fz)] calculated for
direct and indirect-detection FPIs in cone-beam CT at two choices of reconstruction filter (Hanning and Ram-Lak).
[as in Figs. 5(a) and (b)], and the DQE and NEQ alone are insufficient to identify the superior configuration. Such
determination necessitates a quantitative description of the spatial frequencies of interest in the image – i.e., the imaging
task, as described in the next section for the simple case of an ideal observer under various detection and discrimination
tasks.
5. TAKING THE NEQ TO TASK
5.1 Three-Dimensional Imaging Task
As described in ICRU Report 54, imaging task can be quantified as a spatial-frequency-dependent task function11 that is
independent of imaging hardware and determined solely by the Fourier characteristics of the object and observer
response. This quantitative description of the frequencies of interest can be integrated with NEQ to evaluate system
performance in terms of detectability index. In projection imaging, complex backgrounds and anatomical noise make
formulation of realistic task functions challenging; however, in 3-D imaging, these limitations are alleviated as
background anatomical noise is largely removed. Hence, simple 3-D analytical descriptions of imaging task may provide
fairly realistic representations of tomographic imaging task.
We consider here the simple case of a model ideal observer performing various detection / discrimination tasks under
signal-known-exactly, background-known-exactly conditions.11 The task functions in Eq. (6) describe 3-D spatialfrequency representations of: i.) detection of a point-like structure (delta-function); ii.) detection of a 3-D gaussian
structure with characteristic width atask; and iii.) discrimination between two gaussian structures separated by a
characteristic width, atask:
f task f x , f y , f z
­
(Delta - Function)
°k1
2
2
2
2
° Satask f x f y f z (Detection)
®k 2 e
°
2
2
2
2
Sa f f f °k3 §¨1 e task x y z ·¸ (Discrimination)
¹
©
¯
(6)
as illustrated in Fig. 6(a). For simplicity all tasks are assumed to carry equal signal power, with the factors, ki,
normalizing the functions by the 3-D integral of the transform squared. Unless otherwise stated, the tasks below assume
a characteristic width of atask = 1 mm for the “Detection” task, and a characteristic separation of atask = 0.2 mm for
“Discrimination.” Higher-order imaging tasks and incorporation of the effect of non-uniform background structure are
the subjects of ongoing investigation.
Medical Imaging 2003: Physics of Medical Imaging. M. J. Yaffe and L. E. Antonuk Eds. Proc. SPIE Vol. 5030
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5.2 The Detectability Index in Fully 3-D Imaging
The ideal observer signal-to-noise ratio in performance of a given imaging task is given by the detectability index,11
which integrates the task-weighted NEQ (or DQE) over the frequency domain:
d
f xN
f yN
f xN
f yN
³ df x
³ df y
f zN
³ df z f task f x , f y , f z NEQ12 f x , f y , f z 2
(7)
f zN
The detectability index provides a useful metric for system optimization, particularly when the DQE alone is insufficient
to identify a superior configuration – e.g., regarding scintillator thickness,12 and imaging geometry.13 Figure 6 shows d
computed for cone-beam CT using direct and indirect-detection FPIs for various tasks and reconstruction filters. In Figs.
6(b) and (c), d is computed as a function of atask for “Detection” and “Discrimination” tasks. In each case, detectability
plummets when atask is less than the voxel size. The indirect case exhibits improved detectability by virtue of reduced
NPS aliasing and higher 3-D DQE. For the “Detection” task, the reduction in d at larger values of atask is a result of
reduced DQE at low frequencies (caused by NPS aliasing). Figure 6(d) shows d as a function of filter coefficient,
showing that higher-resolution filters improve performance of the “G-Function” and “Discrimination” tasks, but degrade
performance of the “Detection” task. A weak optimum is suggested (hwin~0.6) for the direct-detection FPI system and
the “G-Function” task.
Figure 6. Task functions and detectability index for direct and indirectdetection FPI-based cone-beam CT. (a) Example task functions for idealized
“G-Function,” “Detection,” and “Discrimination” tasks. (b) Detectability index
computed as a function of atask for the “Detection” task for cone-beam CT
systems based on direct and indirect-detection FPIs. (c) The same as (b), except
computed for the “Discrimination” task. The indirect case exhibits improved
detectability by virtue of reduced 3-D NPS aliasing and higher 3-D DQE. (d)
Detectability index computed as a function of reconstruction filter coefficient
(hwin=0.5 for Hanning; hwin=1.0 for Ram-Lak) for the three example imaging
tasks. For the “Detection” task, application of higher-resolution filters degrades
detectability, more so for the direct-detection case due to higher NPS aliasing.
For the “G-Function” and “Discrimination” tasks, d increases with apodization
window, with a slight optimum suggested (hwin~0.6) for the direct case.
6. DISCUSSION AND CONCLUSIONS
Extension of CT image noise theory to the case of non-ideal 2-D detectors and 3-D image reconstruction reveals a
complex interplay between detector spatial resolution, reconstruction method, and 3-D sampling. Generalization of
detector noise-power characteristics and propagation of such through the process of 3-D reconstruction yields analysis of
the 3-D NPS, which in turn provides a starting point for analysis of the 3-D NEQ and DQE. These figures of merit can
be defined for 3-D imaging in a manner directly analogous to accepted 2-D methods, providing the straightforward
interpretation that the NEQ (DQE) is the effective number (fraction) of quanta incident on the detector that are used in
forming the 3-D image. For both direct and indirect-detection FPIs, the 3-D NEQ is asymmetric between axial and
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sagittal/coronal planes, as shown in Fig. 4. This may have interesting, significant effects regarding the detectability of
fine and low-contrast structures visualized in axial versus sagittal/coronal planes, since the noise characteristics are
intrinsically different in these domains.
In applying a linear systems approach to 3-D image reconstructions, caveats abound regarding linearity, shift-invariance,
and stationarity. While it is difficult or impossible to verify such assumptions completely, and the analysis presented
may apply strictly near the center of reconstruction (where the beam is nearly parallel, and the density of back-projected
views is high), some basic measurements support the assumptions to a large extent and give confidence that the
techniques are applicable over a reasonable range of conditions. For example, the MTF in cone-beam CT reconstructions
was measured to be invariant to a high degree over ~8 cm fields of view.1 Similarly, image noise was found to be
stationary within 5% throughout a 15 cm field of view, provided that projection data were uniform - e.g., reconstructions
of “air” (with 2 mm Cu filter added to harden the beam) or reconstructions of a water cylinder (with a well-matched
bow-tie filter).
The investigation of NEQ and detectability reported above for direct and indirect-detection FPIs was designed to
examine specifically the differences associated with different detector transfer characteristics (i.e., MTF). The purpose
here was not to compare specific detector designs (e.g., a CsI:Tl-based FPI versus an a-Se-based FPI – though the
methods do support such a comparison if desired); rather, the purpose here is to consider quantitatively how the NPS
typical of each detector type (e.g., band-limited in the “indirect” case and constant for the “direct” case) imparts different
effects on the 3-D DQE. For the indirect-detection case, the x-ray converter and detector MTF were taken from
measurements using a well-characterized FPI (PerkinElmer RID1640). For the direct-detection case, the x-ray converter
MTF was assumed equal to unity (i.e., the detector MTF was described by a sinc function). In order to compare the
effect of just the detector band-pass characteristics on the 3-D DQE and allow straightforward comparison of the direct
and indirect cases, the quantum detection efficiency, gain, conversion noise, pixel pitch, etc. were taken to be equal in
each case, with values for each parameter taken from calculations and/or measurements for the CsI:Tl-based FPI.
Therefore, the magnitude of the NPS, DQE, NEQ, and detectability index reported for the “indirect” case are exactly
those of the FPI used in the measurements, whereas the “direct” case assumes a hypothetical direct-detection FPI with
equivalent efficiency, gain, etc., differing only in the converter MTF. Of course, the model presented is completely
general in terms of these parameters, and allows analysis of the 3-D DQE for other specific FPI designs if desired (e.g., a
direct-detection FPI with a given thickness of a-Se or PbI2, or a different design of indirect-detection FPI). The results
indicate that (all other parameters being equal – e.g., efficiency, gain, etc.), a cone-beam CT system based on a detector
with constant NPS will suffer increased 3-D NPS aliasing and corresponding reduction in 3-D DQE and detectability
index, unless care is taken to knowledgeably filter the projection data (particularly in the z-domain) in order to reduce
aliased noise.
Integration of the NEQ results with quantitative descriptions of imaging task provides a valuable figure of merit –
detectability index – that can identify superior system configurations when the DQE alone is insufficient [e.g., as in Fig.
5(a)] and provides a figure of merit for optimization.12,13 These techniques propel the analysis of frequency-dependent
NEQ and DQE forward by incorporating quantitative descriptions of imaging task, thereby helping to fill the gap
between detector performance and a more complete description that considers the imaging task and observer. While
these techniques point to a powerful methodology for analysis and optimization of performance in advanced
applications, they also raise a host of challenging issues, including realistic quantitation of complex imaging tasks,
incorporation of observer models for human and machine observers, and studies of imaging performance under
conditions of asymmetric signal and noise characteristics.
ACKNOWLEDGEMENTS
The authors extend their gratitude to Dr. D. J. Moseley for assistance with the experimental setup and data analysis and
to S. Ansell and G. Wilson for their expertise and assistance with software components of the imaging bench. Many
thanks to the scientists and clinicians at the Ontario Cancer Institute and Radiation Medicine Program for their
enthusiastic support of this project. This work was supported by National Institutes of Health Grants R01 CA89081-02
and R33 AG19381-02 and by an award from the University of Toronto Faculty of Medicine Dean’s Fund New Staff
Competition, Grant No. 72022001.
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REFERENCES
1.
D. A. Jaffray and J. H. Siewerdsen, “Cone-beam computed tomography with a flat-panel imager: Initial
performance characterization,” Med. Phys. 27:1311-1323 (2000).
2.
H. H. Barrett, S. K. Gordon, and R. S. Hershel, “Statistical limitations in transaxial tomography,” Comput. Biol.
Med. 6: 307-323 (1976).
3.
K. M. Hanson, “Detectability in computed tomographic images,” Med. Phys. 6:441-451 (1979).
4.
R. F. Wagner, D. G. Brown, and M. S. Pastel, “Application of information theory to the assessment of computed
tomography,” Med. Phys. 6: 83-94 (1979).
5.
M. F. Kijewski and P. F. Judy, “The noise power spectrum of CT images,” Phys. Med. Biol. 32: 565-575 (1987).
6.
W. Zhao and J. A. Rowlands, “Digital radiology using active matrix readout of amorphous selenium: theoretical
analysis of detective quantum efficiency,” Med. Phys. 24: 1819-1833 (1997).
7.
J. H. Siewerdsen, L. E. Antonuk, et al., “Signal, noise-power spectrum, and detective quantum efficiency of
indirect-detection flat-panel imagers for diagnostic radiology,” Med. Phys. 25: 614-628 (1998).
8.
J. H. Siewerdsen and Jaffray DA, “A unified iso-SNR approach to task-directed imaging in flat-panel cone-beam
CT,” SPIE Physics of Medical Imaging Vol. 4682: 245-254 (2002).
9.
I. A. Cunningham, “Analyzing system performance,” in The Expanding Role of Medical Physics in Diagnostic
Imaging, G. D. Frey and P. Sprawls, Eds. pp. 231-263 (Madison WI, 1997).
10. J. H. Siewerdsen, I. A. Cunningham, and D. A. Jaffray, “A generalized approach to n-dimensional image noisepower spectrum analysis,” Med. Phys. 29: 2655-2671 (2002).
11. ICRU Report No. 54, Medical Imaging – the Assessment of Image Quality (ICRU, Bethesda MD, 1996).
12. J. H. Siewerdsen and L. E. Antonuk, “DQE and system optimization for indirect-detection flat-panel imagers in
diagnostic radiology,” SPIE Vol. 3336: 546-554 (1998).
13. J. H. Siewerdsen and D. A. Jaffray, “Optimization of x-ray imaging geometry (with specific application to flat-panel
cone-beam computed tomography),” Med. Phys.27(8): 1903-1914 (2000).
*Corresponding author: Jeffrey H. Siewerdsen. Electronic mail: [email protected]; Phone: 416-946-4501
(ext. 5516); Fax: 416-946-6529; Ontario Cancer Institute, Princess Margaret Hospital, University Health Network, 610
University Ave., Toronto, ON, Canada M5G 2M9.
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