Download An SVD Study of Truncated Transmission Data in SPECT

IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 44, NO. 1, FEBRUARY 1997
107
An SVD Study of Truncated Transmission
Data in SPECT
Gengsheng L. Zeng, Member, IEEE, and Grant T. Gullberg, Senior Member, IEEE
Abstract— Even though a noiseless, band-limited function is
uniquely determined by its values in a local region, single photon
emission computed tomography (SPECT) projections are not
band-limited, and unmeasured projections may not be possible to
be exactly estimated from the measured data. Projections from
all views should be considered simultaneously and are modeled as
a set of linear equations. The singular value decomposition (SVD)
method is used to analyze and solve the equations. It is shown
that truncation does not always result in an underdetermined
problem, yet the problem may be ill-conditioned. An inaccurate
pixel model may cause reconstruction artifacts via mismatch
between the measured data and the modeled projections.
Index Terms— Incomplete data, SVD reconstruction, truncation.
I. INTRODUCTION
I
N SINGLE photon emission computed tomography
(SPECT), projections may be truncated because of the
relatively small detector size. The truncation problem is
especially severe during transmission imaging of the chest
by a three-detector SPECT system with fan-beam collimators
[1]. If the reconstruction of truncated projections is improperly
formulated, the problem results in solving an underdetermined
system of linear equations that leads to artifacts in the
reconstruction [2]–[7]. However, if band-limited truncated
projections are sampled sufficiently fine and reconstructed
into a sufficiently coarse grid using projection models with
minimum model mismatch between image and projection,
reconstruction artifacts from truncated projections can be
reduced.
Reconstruction from limited data sampling, such as the
reconstruction of truncated projections, has been studied in
many fields of engineering [13]–[16], including MRI [17].
Truncated projections were reconstructed initially by using
data extension or data extrapolation methods to “complete” the
projections [5]–[9]. In the case of the three-detector SPECT
system, some have modified the camera system by using fan
beam collimation on the simultaneous transmission/emission
detector and parallel collimation on the other two detectors
to obtain scatter information that can be used to determine
the truncation edge [11], [12]. Using information about the
Manuscript received December 11, 1995; revised April 1, 1996 and August
19, 1996. This work was supported in part by the NIH under Grant RO1
HL39792 and by Picker International.
The authors are with the Department of Radiology, University of Utah, Salt
Lake City, UT 84132 USA (e-mail: [email protected]).
Publisher Item Identifier S 0018-9499(97)01524-4.
truncation edge to estimate the patient contour, a filtered
backprojection [11] or an iterative ML-EM emission algorithm
[12] was employed to reconstruct the pixels within the contour.
The problem with this approach is the reduction in sensitivity
when parallel collimators are used. Another approach is to
keep the fan beam collimation but acquire two transmission
scans wherein, after the first scan, the patient is translated
horizontally in the bore of the imager. Another transmission
scan is acquired to fill in the missing projection rays [9]. Other
approaches involve using asymmetric fan-beam collimators
[10] or camera systems, such as the L-shaped camera [8]
design, that can obtain transmission data without truncation.
Our approach utilized the three-detector SPECT for cardiac imaging to determine the solution with the information
available by solving only the system of linear equations
given by the available truncated projections [23]. As the
accuracy of the support improved, so did the accuracy of
the reconstructed transmission image [1]. Here, a support is
a preset region where the pixel value is assumed to be zero
if the pixel is outside the region. However, in most cases
the accuracy of the support was not critical to calculate the
attenuation factors within the cardiac region. The attenuation
factors were exponential of negative partial line integrals of
the reconstructed transmission data. Thus these factors were
calculated with fair accuracy because the total line integrals of
the reconstructed transmission data closely approximated the
measured projection data.
This paper investigates the extent to which truncated projections cause underdetermined image reconstruction. The
research will show that when projections are truncated, a
band-limited image can be reconstructed under certain circumstances. The singular value decomposition (SVD) is used
to obtain the reconstructions from truncated projections; this
demonstrates that the reconstruction is not an underdetermined
problem if the data and reconstruction are sampled properly.
However, a fully determined problem can be ill-conditioned, as
determined by its condition number. It is demonstrated how the
condition number can be reduced to improve the reconstructed
images.
In SPECT the projections are not band-limited, thus the
truncated projections at one view cannot be exactly estimated
from the nontruncated part at the same view. Therefore, using
the projections from other views can help, by considering
a set of linear equations simultaneously (i.e., tomography).
An image can be reconstructed by solving the set of linear
equations. The SVD method is used to analyze and solve the
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IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 44, NO. 1, FEBRUARY 1997
equations with computer simulated projections and physical
phantom measurements.
II. METHODS
A. Linear Equation Method
(a)
For a band-limited signal the function values at any two
different points are correlated. For example, in MR imaging
the k-space signal is band-limited. However, in SPECT or
X-ray CT it cannot be assumed that the projections are the
samples of a band-limited signal because the imaging point
response function has a small finite spread. Therefore, the
truncated projections may not be recovered at one view from
the measured part in the same view, but the projections from
other views can be used to help by doing tomography. On the
other hand, the final goal in SPECT imaging is to reconstruct
the image from projections. The unmeasured projections do not
have to be estimated if an image can be reconstructed with the
locally sampled data (i.e., truncated projections). We believe
that reconstructing the image with truncated projections is
equivalent to reconstructing the image with extrapolated data,
because extrapolating does not increase information. This
paper uses square pixels with line-length weighting to set up
imaging equations, with the pixel values as unknowns. Image
reconstruction is achieved by solving the linear equations. The
singular value decomposition (SVD) method is used to analyze
and solve the equations in these studies.
Let
be a vector of projections and
be a vector
representing the image. The imaging equation is given as
(1)
where matrix is called projection matrix, modeling imaging
geometry and physics. The SVD of
is given by
(2)
where is an orthonormal matrix and is a diagonal matrix
with nonnegative diagonal elements, known as singular values
of
which are the square of the singular values of
The SVD provides a condition number to diagnose the
linear equation system. The condition number of the system
is defined as the ratio of the largest singular
value and the smallest singular value of
The condition
number indicates how sensitive the solution
is to the noise
in projections and the model mismatch in
A very large
condition number also indicates that the measurement may not
be enough, while a small condition number indicates that the
image may be properly reconstructed. If the condition number
is finite, the image can be reconstructed by
(3)
can also be reconstructed by the SVD of
Equivalently,
or of
Nevertheless, the formulation of (1) has some drawbacks:
1) A linear system model strongly depends on the pixel
characteristics (e.g., a uniform square or a Gaussian blob, etc.);
2) the nonnegativity constraint on the pixel values and some
other constraints are very difficult; and 3) the Poisson noise
is not modeled.
(b)
Fig. 1. (a) Ideal image. (b) Filtered backprojection reconstruction with
truncated data.
B. Computer Simulations for the Linear Equation Method
Computer simulations are used to demonstrate how the sampling interval size affects the reconstruction if the projections
are severely truncated with a fan-beam imaging geometry. In
all the simulations the fan-beam focal length was 65 cm. The
fan-beam detector size was 28.5 cm. The detector rotated in
a noncircular orbit with a minimal radius of 22.7 cm and
maximal radius of 28.8 cm. The reconstruction array was 60
pixels 60 pixels, with the pixel size of 0.712 cm. The ideal
image and the filtered backprojection reconstruction are shown
in Fig. 1. The pseudo-projection data were generated with a
line-length weighted projector.
The first set of reconstructions used a circular image support
(2642 pixels in a 60 60 image array), and the pixel values
were assumed to be zero if the pixels were outside the support.
The second set of reconstructions used a tight (but not exactly
the same as the object outline) image support (1908 pixels in
a 60 60 image array). In these studies the number of views
were set to 60 and 360, respectively, and the detector bin size
was set to the pixel size and half the pixel size, respectively.
Finally, reconstructions were performed from data corrupted
with random noise and deterministic noise, respectively. In all
but the very last study, a line-length weighted projector was
used to form the imaging equations which were solved by the
SVD method. The last study used an area-weighted projector
to form the linear equations. Because the projections were
generated with a line-length weighted projector, the mismatch
of these different projectors caused deterministic errors.
C. Physical Jaszczak Phantom Study
The Picker PRISM 3000 system was used to acquire transmission data of a Jaszczak phantom with a fan-beam detector
and a
line source (see Fig. 2). The system arrangement
was the same as in prior computer simulations. Projections
were acquired at 360 views over 360 Total scanning time
was 210 min. The projections were stored in 56 56 arrays
and were converted into line-integrals prior to reconstruction. Both the whole 360 views and a subset of 60 views
were used to reconstruct the images. A line-length weighted
projector/backprojector pair were used in the reconstructions.
III. RESULTS
A. Computer Simulations
The SVD reconstructions of the computer-simulated data are
shown in Figs. 3–5. These simulations showed the following:
1) reducing the number of unknowns by using a tighter support
ZENG AND GULLBERG: SVD STUDY OF TRUNCATED TRANSMISSION DATA
(a)
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(b)
Fig. 2. (a) The experimental arrangement with the Picker Prism 3000 system. (b) A scheme of a slice of the Jaszczak phantom.
Fig. 3. Reconstructions on a circular region (2642 pixels).
Fig. 4. Reconstructions on a tight support (1908 pixels).
improves the system condition significantly; 2) increasing the
sampling density in the linear and/or angular direction also
improves the system condition; 3) when the system is fairly
well conditioned, the random noise usually does not deteriorate
the object boundary, while the systematic errors in modeling
degrade the image.
B. Physical Jaszczak Phantom Study
Fig. 5 shows the reconstructions of one slice of the Jaszczak
phantom. Using more views did reduce the condition number,
but this approach was not as effective as using larger pixel
size and a tighter support. Reducing the number of pixels is
a very effective way to reconstruct the image with truncated
projections. The number of unknowns (pixels) is suggested to
be reduced so that the system of equations has a unique leastsquares solution, and if an iterative algorithm were used to
reconstruct the image, the algorithm would always converge.
Fig. 5. Reconstructions on a tight support (1908 pixels) with noise. An
area-weighted model was used in B.
Reducing the number of pixels by increasing the pixel
size reduces the image resolution but may cause model mismatch artifacts, which are better known as aliasing artifacts.
The model mismatch artifacts are severe in Fig. 5(B) and
Fig. 6. These artifacts do not necessarily indicate that the
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IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 44, NO. 1, FEBRUARY 1997
Fig. 6. Reconstructions of a slice of the Jaszczak phantom. Model mismatch artifacts are severe. A line-length weighted model was used in the
reconstructions.
projection data are insufficient, but that the nonoverlapped
square pixel model is not accurate enough to form imaging
equations.
IV. DISCUSSION
Truncation does not always result in an underdetermined
image reconstruction problem. It is known that an analytical
function can be uniquely determined by its values within
a small interval, and the function values elsewhere can be
evaluated by the Taylor series expansion. An example of
an analytical function is a band-limited function such as
the k-space signal in MRI measurements [17]. However, in
SPECT, the projection data cannot be assumed to be bandlimited or analytical because the object (i.e., the patient) to
be imaged has a finite support, and the band-width of SPECT
projections is infinite. Discrete samples of SPECT signal in
one region are insufficient to exactly estimate the signal in
a different region. In other words, exact data extension is
not possible. On the other hand, using projections at all
the views simultaneously (i.e., doing tomography) seems a
better approach than estimating unmeasured data at each
view individually. Tomographic image reconstruction can be
achieved by solving a set of linear equations. This paper has
shown that an accurate image can be obtained with truncated
projections, provided the projection model is accurate, that
is, the coefficients in the linear equations are accurate. The
modeling accuracy can be improved by using smaller sampling
intervals and smaller image pixel sizes.
Due to practical sampling limitations, when one is unable to
increase sampling density, one can use larger pixels, resulting
in fewer unknowns in the system of equations, to increase the
solvability of the system. However, if square pixels are used,
bigger pixels introduce larger model-mismatch artifacts. The
condition number indicates how sensitive the model-mismatch
errors are.
In reconstruction using truncated data, will more samples
help? The answer is “yes” if the imaging geometry (pixel
projection model, scatter, system geometric point response,
etc.) is modeled accurately (see results in Figs. 3 and 4). The
answer would be “some, not very effective” if the imaging
geometry is not modeled accurately, as in a realistic SPECT
study (see Fig. 6).
As demonstrated in computer simulations, if the imaging
geometry is exactly known, the linear equation method works.
However, for the real data, the imaging geometry is not
exactly known, thus the linear equation method does not work
very well for truncated data because the linear system is illconditioned and sensitive to modeling errors [also illustrated
in Fig. 5(B)].
Using a tighter support is equivalent to obtaining more
measurements (constraining some pixels to have a value
of zero). Therefore, using a tighter support helps in image
reconstruction. To ensure the convergence, one should always work with an overdetermined system. The number of
independent unknowns can be reduced by using better image
basis functions (i.e., alternative forms of pixels) or a priori
constraints (e.g., nonnegativity, smoothness etc.).
We believe that an image can be accurately modeled by
fewer pixels as long as the pixels (i.e., basis functions)
are optimally determined. Finding the optimal image basis
functions is an active research area [20]–[22]. It may be
promising to use overlapped, smooth-edged pixel models to
form the projector to eliminate or significantly reduce the
model mismatch artifacts.
In practice, if truncation is unavoidable, we recommend the
following: 1) increase the number of views while acquiring
data; 2) decrease the sampling interval on the detector; 3)
increase the pixel size of the image, sacrificing resolution for
smaller condition number; 4) use constraints such as supports,
nonnegativity, smoothness, and so on; and 5) use a good
projection model with accurate image basis functions to reduce
aliasing artifacts.
ZENG AND GULLBERG: SVD STUDY OF TRUNCATED TRANSMISSION DATA
ACKNOWLEDGMENT
The authors would like to thank the Biodynamics Research
Unit, Mayo Foundation for use of the Analyze software package. A grant of computer time from the Utah Supercomputing
Institute is gratefully acknowledged.
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