# SVD-Based Evaluation of Multiplexing in Multipinhole SPECT Systems

Authors: Aaron K. Jorgensen; Gengsheng L. Zeng
Journal: International Journal of Biomedical Imaging
 (2008)

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## Abstract

Multipinhole SPECT system design is largely a trial-and-error process. General principles can give system designers a general idea of how a system with certain characteristics will perform. However, the specific performance of any particular system is unknown before the system is tested. The development of an objective evaluation method that is not based on experimentation would facilitate the optimization of multipinhole systems. We derive a figure of merit for prediction of SPECT system performance based on the entire singular value spectrum of the system. This figure of merit contains significantly more information than the condition number of the system, and is therefore more revealing of system performance. This figure is then compared with simulated results of several SPECT systems and is shown to correlate well to the results of the simulations. The proposed figure of merit is useful for predicting system performance, but additional steps could be taken to improve its accuracy and applicability. The limits of the proposed method are discussed, and possible improvements to it are proposed.


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## Body

## 1. Introduction

Small-animal SPECT imaging provides the opportunity for advanced monitoring and analysis of cancer drug tests in laboratory animals. In order to be effective, a small-animal SPECT system must have high spatial resolution and high sensitivity. The design of multipinhole systems involves many subtle factors which affect both resolution and sensitivity in ways that are difficult to model. Currently, systems are designed based only on general principles; optimization is not a part of the design procedure. Once an aperture is designed, it is tested and analyzed. A method whereby system performance could be predicted, and therefore optimized, in the design phase would allow system designers to experiment with a wider range of design possibilities and to achieve better design results overall.The main problem in deriving a system performance predictor is the definition of system
performance. An optimal system obtains a balance between high spatial resolution and low system noise. Therefore, an objective error predictor must favor both system characteristics equally; an optimal system, as defined by the error predictor, must give low noise and allow for detection of small lesions. We present an error predictor which is shown to account for both spatial resolution and noise, and therefore correlates to image quality in terms of usefulness to the clinician.In addition, the error predictor provides an objective measure of system performance. Current evaluations of SPECT systems include simulation and actual physical imaging. Of those performing physical experiments, some use laboratory animals [1, 2], some use phantoms [3–5], and some use both [6, 7]. Some
of those using simulation to evaluate their systems image a single point or a
homogeneous sphere, instead of the type of complex system that would be
encountered in clinical use. In
addition, current evaluation methods are not completely thorough or standardized,
either. Systems can be evaluated in
terms of signal-to-noise ratio (SNR) [1], contrast-to-noise ratio,
mean square error (MSE) [4, 5], or other performance indicators [3, 8]. Others give no quantitative results and rely on visual comparison of results [2, 7, 9].In order to
obtain this error prediction, the singular value spectrum must be
calculated. The matrix-based
representation of clinical systems is far too large to store the entire system
matrix in the computer memory. We show that the application of
the power method to the analysis of SPECT imaging systems is valuable because of
the ability to use simulation to find the singular value spectrum of a
system. This allows for a
frequency-based analysis of systems involving attenuation, photon scattering,
and other complex and random phenomena, for which the creation of a system
matrix would be complicated.The theoretical
background of the proposed error estimate is presented in Section2. The generalized SPECT system is presented as
a matrix algebra problem. The singular
value decomposition is used to analyze the system in terms of frequency
content. The relationship between
iteration of the minimal-residual (MR) algorithm and frequency content of the
reconstructed image is discussed briefly.
The singular-value-based analysis is used to create an estimate of the
error inherent in the imaging system.
The power method is explained and used to determine singular values of
the system.In Section3, experimentation is presented to verify the error estimate derived in Section 2. Methodology is presented for a
two-dimensional and a three-dimensional SPECT simulation. A two-dimensional and a three-dimensional
phantom are imaged to a noisy detector and the projections are
reconstructed. The results of these
simulations are analyzed in terms of error, and these error measurements are
compared to the error estimates. The
results are discussed, and the efficacy of the error estimate is examined.Section4 presents a summary of the work and the conclusions drawn from the results. Possibilities for future work are discussed,
including areas into which research must be extended to qualify the proposed
error estimate for use in design of clinical systems.


## 2. Theory

### 2.1. Matrix Representation of Projection-Backprojection

For any given
system, a matrixB can be defined,
which defines the translation of the object x (arranged as a column vector of length m) to a set of projections p:
(1)Bx=p.Several
algorithms exist for solving this type of problems [10–12]. However, for an ill-posed problem, Bx=p is usually inconsistent due to noise and a
solution does not exist. To solve this
problem, let us define the transposed matrix BT as the backprojection operation, that
is, a map from the projection set p to the backprojected image vector BTp.
A projection-backprojection matrix can then be defined as
(2)A=BTB,and image reconstruction can be performed by solving
(3)Ax=BTp.In this form, the
problem is always consistent, and a unique, least-squares solution can be
found, which is also the least-squares solution ofBx=p.
An evaluation of the imaging system can be performed by examining
pertinent characteristics of A. Such a figure of merit would provide an
objective means by which multipinhole apertures could be evaluated and optimized.


### 2.2. Singular Value Decomposition of System Matrix

The singular value decomposition (SVD) of the projection-backprojection matrixA can be represented as
(4)A=UΛUT.BecauseU is a unitary matrix, UTU=I, or UT=U−1. Thus, if all
singular values are nonzero, we have
(5)A−1=[UΛUT]−1=[UT]−1Λ−1U−1=UΛ−1UT.This can be used to solve the original problem:(6)Ax=BTp,x=A−1BTp=UΛ−1UTBTp.From the singular
value decomposition,Λ is a diagonal
matrix which contains, in descending order, the singular values of A, which are all nonnegative:
(7)Λ=[λ10⋯⋯00λ20⋯0⋮0⋱⋱⋮⋮⋮⋱⋱000⋯0λm],Λ−1=[1λ10⋯⋯001λ20⋯0⋮0⋱⋱⋮⋮⋮⋱⋱000⋯01λm].For any imaging
system, there is therefore a threshold for acceptable values ofλk. Values of λk below this threshold
will add more noise to the system than image data, and thus all values of λk below the threshold are truncated, as in (8).
The threshold is defined by the variable n, such that all inverted singular values corresponding to k>n are set to zero. We define
(8)Λ˜−1=[1λ10⋯00⋱⋮1λn⋱⋮⋮⋱0⋮⋱00⋯00].It follows that forU=[U1,U2,…,Um] and Uk=[u1,k,u2,k,…,um,k]Twe can
define a generalized inverse of A as
(9)A†=UΛ˜UT=[U1λ1U2λ2⋯Unλn0⋯0][U1TU2T⋮UmT]=1λ1U1U1T+1λ2U2U2T+⋯+1λnUnUnT.The regularized solutionx′ can thus be written as the following summation:
(10)x′=∑k=1n1λkUkUkTBTp.


### 2.3. Noise Analysis

The reconstructed imagex′ is expressed as a linear combination of image
components UkUkTBTp, with scaling factors 1/λk. As in a Fourier transform, each of these
image components contains frequency content of the total image
x′, with frequency increasing with k.The
projection-backprojection and reconstruction process can be visualized as in
Figure1. In this conceptual system, a
noise signal ξ is introduced in the
projection operation. The image
components are “filtered” in the A operation by the corresponding scaling factors
λk. Each component must therefore be amplified by a factor of
1/λk in the inverse operation, A†. The noise in
x′ is therefore a function of the noise ξ and the singular values
λk in A†.Figure 1
Block diagram of projection-backprojection operation. Noise is introduced at the projection operation,B. Image components which are “attenuated” in B and BT must be “amplified” in A†, increasing the noise in x′.In SPECT, the
projection data noise is Poisson distributed, that is, its variance equals its
mean. We can assume that in the
backprojected image, the noiseξ is
Poisson distributed. Let N be the
total photon count in a projection data set, then the signal uncertainty, or
the square root of the noise variance divided by the mean, can be approximated as
(11)NN=1N.
Considering that the noise power of each image component in (10) is
(12)1λk2,
if we assume that the noise power is uniformly distributed over the
entire singular-value-decomposition spectrum, then the susceptibility of the
system to noise is the Poisson noise uncertainty multiplied by the sum of the
noise powers
(13)1N∑k=1n1λk2,
which is hereafter referred to as the “noise amplification factor.” Because it involves all singular values, this error estimate describes the general ability of a system to reproduce image
data at all frequencies.A less accurate
but less computationally expensive estimate of the noise amplification involves
the condition numberK(A), the ratio
of the largest to the smallest singular value of A. In this way, the
condition number can be calculated for a real-world system and the uncertainty
can be estimated as
(14)K(A)N.Although not as
precise as the noise amplification factor, the condition number does relate to
how well-posed the problemAx=BTp is.
However, systems with different singular value spectra can have
identical condition numbers, even though their performance is not the
same. For this reason, the noise
amplification factor is a more revealing estimate of system performance.


### 2.4. Power Method

For real-world
systems, it is not feasible to obtain singular values fromA due to its large size. The
power method is an iterative algorithm which can estimate the dominant singular
value of a system indirectly; only access to the matrix operation is needed [13]. The method of deflation is used to find nondominant singular values, that is, the
second singular value of A is equal
to the dominant singular value of
(15)A2=A−λ1U1U1T,
and so on, so that the kth singular value of A is found by
estimating the dominant singular value of
(16)Ak=Ak−1−λk−1Uk−1Uk−1T,Of course, for
large systems, access toA is not
available, so an equivalent operation—simulation of the projection and
backprojection operations—is performed on the vectors employed
in the power method algorithm in order to compute singular values. For an image of size N×N×N, the computational complexity of the projection (or backprojection) is
O(N3) per projection, or O(N4) for the entire image,
assuming that the number of projections is O(N). Therefore, the computational complexity of
computing a condition number (two terms) with the power method is O(N4),
and the complexity of computing the entire noise amplification factor (N3 terms) with the power method is O(N7). It is not realistic to compute an entire SVD
spectrum for each system to be analyzed.
In practice, only a small number of singular values are computed, so
that the overall computational complexity remains O(N4). On our 2 GHz Windows-based computer, a
MATLAB-based projection-backprojection operation requires approximately two
minutes to run. In order to be useful
for calculations of many SVD spectra, the simulation would have to be
optimized, but this was not done for this paper.


## 2.1. Matrix Representation of Projection-Backprojection

For any given
system, a matrixB can be defined,
which defines the translation of the object x (arranged as a column vector of length m) to a set of projections p:
(1)Bx=p.Several
algorithms exist for solving this type of problems [10–12]. However, for an ill-posed problem, Bx=p is usually inconsistent due to noise and a
solution does not exist. To solve this
problem, let us define the transposed matrix BT as the backprojection operation, that
is, a map from the projection set p to the backprojected image vector BTp.
A projection-backprojection matrix can then be defined as
(2)A=BTB,and image reconstruction can be performed by solving
(3)Ax=BTp.In this form, the
problem is always consistent, and a unique, least-squares solution can be
found, which is also the least-squares solution ofBx=p.
An evaluation of the imaging system can be performed by examining
pertinent characteristics of A. Such a figure of merit would provide an
objective means by which multipinhole apertures could be evaluated and optimized.


## 2.2. Singular Value Decomposition of System Matrix

The singular value decomposition (SVD) of the projection-backprojection matrixA can be represented as
(4)A=UΛUT.BecauseU is a unitary matrix, UTU=I, or UT=U−1. Thus, if all
singular values are nonzero, we have
(5)A−1=[UΛUT]−1=[UT]−1Λ−1U−1=UΛ−1UT.This can be used to solve the original problem:(6)Ax=BTp,x=A−1BTp=UΛ−1UTBTp.From the singular
value decomposition,Λ is a diagonal
matrix which contains, in descending order, the singular values of A, which are all nonnegative:
(7)Λ=[λ10⋯⋯00λ20⋯0⋮0⋱⋱⋮⋮⋮⋱⋱000⋯0λm],Λ−1=[1λ10⋯⋯001λ20⋯0⋮0⋱⋱⋮⋮⋮⋱⋱000⋯01λm].For any imaging
system, there is therefore a threshold for acceptable values ofλk. Values of λk below this threshold
will add more noise to the system than image data, and thus all values of λk below the threshold are truncated, as in (8).
The threshold is defined by the variable n, such that all inverted singular values corresponding to k>n are set to zero. We define
(8)Λ˜−1=[1λ10⋯00⋱⋮1λn⋱⋮⋮⋱0⋮⋱00⋯00].It follows that forU=[U1,U2,…,Um] and Uk=[u1,k,u2,k,…,um,k]Twe can
define a generalized inverse of A as
(9)A†=UΛ˜UT=[U1λ1U2λ2⋯Unλn0⋯0][U1TU2T⋮UmT]=1λ1U1U1T+1λ2U2U2T+⋯+1λnUnUnT.The regularized solutionx′ can thus be written as the following summation:
(10)x′=∑k=1n1λkUkUkTBTp.


## 2.3. Noise Analysis

The reconstructed imagex′ is expressed as a linear combination of image
components UkUkTBTp, with scaling factors 1/λk. As in a Fourier transform, each of these
image components contains frequency content of the total image
x′, with frequency increasing with k.The
projection-backprojection and reconstruction process can be visualized as in
Figure1. In this conceptual system, a
noise signal ξ is introduced in the
projection operation. The image
components are “filtered” in the A operation by the corresponding scaling factors
λk. Each component must therefore be amplified by a factor of
1/λk in the inverse operation, A†. The noise in
x′ is therefore a function of the noise ξ and the singular values
λk in A†.Figure 1
Block diagram of projection-backprojection operation. Noise is introduced at the projection operation,B. Image components which are “attenuated” in B and BT must be “amplified” in A†, increasing the noise in x′.In SPECT, the
projection data noise is Poisson distributed, that is, its variance equals its
mean. We can assume that in the
backprojected image, the noiseξ is
Poisson distributed. Let N be the
total photon count in a projection data set, then the signal uncertainty, or
the square root of the noise variance divided by the mean, can be approximated as
(11)NN=1N.
Considering that the noise power of each image component in (10) is
(12)1λk2,
if we assume that the noise power is uniformly distributed over the
entire singular-value-decomposition spectrum, then the susceptibility of the
system to noise is the Poisson noise uncertainty multiplied by the sum of the
noise powers
(13)1N∑k=1n1λk2,
which is hereafter referred to as the “noise amplification factor.” Because it involves all singular values, this error estimate describes the general ability of a system to reproduce image
data at all frequencies.A less accurate
but less computationally expensive estimate of the noise amplification involves
the condition numberK(A), the ratio
of the largest to the smallest singular value of A. In this way, the
condition number can be calculated for a real-world system and the uncertainty
can be estimated as
(14)K(A)N.Although not as
precise as the noise amplification factor, the condition number does relate to
how well-posed the problemAx=BTp is.
However, systems with different singular value spectra can have
identical condition numbers, even though their performance is not the
same. For this reason, the noise
amplification factor is a more revealing estimate of system performance.


## 2.4. Power Method

For real-world
systems, it is not feasible to obtain singular values fromA due to its large size. The
power method is an iterative algorithm which can estimate the dominant singular
value of a system indirectly; only access to the matrix operation is needed [13]. The method of deflation is used to find nondominant singular values, that is, the
second singular value of A is equal
to the dominant singular value of
(15)A2=A−λ1U1U1T,
and so on, so that the kth singular value of A is found by
estimating the dominant singular value of
(16)Ak=Ak−1−λk−1Uk−1Uk−1T,Of course, for
large systems, access toA is not
available, so an equivalent operation—simulation of the projection and
backprojection operations—is performed on the vectors employed
in the power method algorithm in order to compute singular values. For an image of size N×N×N, the computational complexity of the projection (or backprojection) is
O(N3) per projection, or O(N4) for the entire image,
assuming that the number of projections is O(N). Therefore, the computational complexity of
computing a condition number (two terms) with the power method is O(N4),
and the complexity of computing the entire noise amplification factor (N3 terms) with the power method is O(N7). It is not realistic to compute an entire SVD
spectrum for each system to be analyzed.
In practice, only a small number of singular values are computed, so
that the overall computational complexity remains O(N4). On our 2 GHz Windows-based computer, a
MATLAB-based projection-backprojection operation requires approximately two
minutes to run. In order to be useful
for calculations of many SVD spectra, the simulation would have to be
optimized, but this was not done for this paper.


## 3. Experiments and Results

### 3.1. Setup

Preliminary simulations were run for a
two-dimensional phantom with a one-dimensional detector. The phantom was a31×31 pixel modified
Shepp-Logan phantom. The phantom was imaged
at 120 angles to a 60-pixel detector, using apertures of 1, 3, 5, 7, 9, 11, 13,
and 15 evenly spaced pinholes. Poisson
noise was added to the projections, and the images were reconstructed using the
MR algorithm.The MR algorithm
is used in place of the more popular ML-EM algorithm because of its natural
applicability to the singular value decomposition. Although the ML-EM algorithm models Poisson
noise properly [14, 15], it cannot be analyzed with a simple algebraic
method, and so is not suitable for this analysis. Under the assumption that photon count per
detector bin is sufficiently high (greater than 10), the Poisson noise can be
approximated as Gaussian, and so the MR algorithm can be used.Because of the
relatively small system size, the errors in these images were compared to two
error predictors: one based on the condition number of the system, and the
other based on thenoise amplification
factor, which is based on the entire singular-value spectrum. As discussed earlier, a function of the
condition number K(A) can be
substituted for the noise amplification factor.
The square of the condition number, (K(A))2, seems to be a good estimate of the noise
amplification factor, and so the condition-number-based error estimate used in
these experiments is (K(A))2/N. This is, of course,
an empirical fit and not based on any rigorous mathematical principle.Final simulations were run for a three-dimensional phantom with a
two-dimensional detector. The phantom
used (Figure2) is a custom 64×64×64 voxel phantom, which has been made to
resemble the Shepp-Logan phantom. The Shepp-Logan phantom is used frequently in
analysis of medical imaging systems, and is designed to resemble a head. The phantom was imaged at 60 angles to a 128-by-128
pixel detector through apertures with varying numbers of holes and varying
arrangements of holes. The apertures are
illustrated in Figures 3(a)–3(f), and will be referred to as apertures A through F, respectively. Poisson noise
was added to these projections.Figure 2
Custom phantom used in simulation.Apertures used in final simulation.
(a)(b)(c)(d)(e)(f)The noisy
projections were backprojected to create the image vectorBTp. These results were then reconstructed using
the MR algorithm. The reconstructed
images were compared to the original phantom and the error in each image was
calculated. Because this system is
large, calculation of the entire singular value spectrum is not feasible. The
error was therefore compared only to the condition-number-based noise
prediction.The acceptance
angle of all pinholes in both experiments is60°, meaning that photons may
enter a pinhole at an angle of up to ±30° from perpendicular (Figure 4). In both sets of simulations, the aperture, phantom,
and detector were placed as close together as possible while allowing emitted
photons from all points in the phantom to pass through the aperture and strike
the detector.Figure 4
Illustration of acceptance angle.


### 3.2. Results

The error plots of the preliminary simulations
are shown in Figure5. The plots in
Figure 5 show the normalized error predictions based on condition number,
(17)ξK=(K(A))2N,
error predictions based on the full singular-value spectrum,
(18)ξSVD=1N∑k=1n1λk2,
and true error, defined as the standard deviation of the error in the image pixels,
(19)ξtrue=∑i=1m3(x−A−1BTp)j2.Figure 5
Normalized error predictions and actual error of two-dimensional systems.The reconstructed
two-dimensional images are shown after (a) 1, (b) 9, (c) 17, and (d) 25
iterations in Figure6. The rows
represent results for the apertures with 1, 3, 5, 7, 9, 11, 13, and 15 holes,
from the top down. Figure 6(e) shows the
image reconstructed as A−1BTp. The original phantom is
shown in Figure 6(f) for comparison.
Note that each set of error predictions and of actual errors is
normalized; the error predictors, as currently defined, are useful only for
comparison between systems and do not represent any absolute real-world value.Comparison of reconstructed images in two-dimensional
simulation after (a) 1, (b) 9, (c) 17, and (d) 25 iterations. (e) The image reconstructed asA−1BTp. (f) The original phantom.
(a)(b)(c)(d)(e)(f)The reconstructed images from the final (three-dimensional) simulations are shown in Figure7. Figures 7(a)–7(f) show slices from the
reconstructed images from systems A through F, respectively, at the stage in the MR algorithm at which they are closest to the original phantom (Figure 7(g)). The number of iterations used for the systems is 9 (system
A); 11 (system B); 11 (system C); 12 (system D); 20 (system E); 15 (system F). Although most systems use a fixed number of
iterations in MR reconstruction, it is not unreasonable to use a different
number of iterations for each system.
The normalized error predictions for the three-dimensional simulations
and the normalized true error are shown in Figure 8. The reconstructed images
from the final (three-dimensional) simulations are evaluated both in terms of
noise and in terms of lesion detection (an indirect measure of spatial
resolution). In order to evaluate the
comparative performance of the systems, a composite error is used. This error composite is defined as the sum of the mean squared error of the entire image,
(20)ξ1=∑i=1m3(x−A−1BTp)j2
and the square root of the noise power along the profile of the three
small lesions in the bottom half of the phantom,
(21)ξ2=∑j(x−A−1BTp)j2
as shown in Figure 9. Sensitivity
is measured in the noise parameter, ξ1, and resolution is measured
in the small-lesion profile, ξ2.
Figure 10 compares profiles along the line shown in Figure 9. Using this error composite, the reconstructed images can be evaluated in terms of noise
and lesion detection, or sensitivity and resolution.Figure 7
Comparison of reconstructed images from three-dimensional simulations.Figure 8
Predicted and measured errors for three-dimensional simulations.Figure 9
Illustration of line used for profile comparison.Comparison
of system profiles for (a) original phantom, (b) system A, (c) system C, and
(d) system E.
(a)(b)(c)(d)


### 3.3. Analysis

The two-dimensional simulations show
the condition number and singular values to be useful in determining relative
uncertainty in reconstructed images.
Both error estimates (17) and (18)) perform well in estimating error,
but the error estimate which involves the entire singular value spectrum—the noise amplification factor—more accurately predicts system
performance. However, the
characteristics of any small two-dimensional system are much closer to ideal
than those of a real-world system, and are easier to model.The composite error for the
three-dimensional system, as previously defined, was created in order to
measure both system sensitivity and spatial resolution. For example, systems A and B are able to
resolve the three lesions in the bottom half of the phantom, but contain
substantial amounts of noise, as evidenced by the noisy reconstruction of the
large bright circle at the top half of the phantom (Figures7(a) and 7(b)). Systems D and E contain relatively low amounts of noise (Figures 7(d) and 7(e)), but the three small lesions at
the bottom are almost indistinguishable.
System F (Figure 7(f)) has the worst reconstructed image; although the
background is mostly homogeneous, the bright spot at the top is not well defined,
and there is a large artifact in the center of the image. This artifact is most likely due to poor
placement of pinholes in the aperture.
System C is a good compromise between the high-noise problems of the
one- and two-pinhole apertures (systems A and B) and the poor resolution of the
nine- and ten-pinhole apertures (systems E and F). It also has the lowest error prediction. Note that systems C and D contain the same
number of holes, yet have observably different performance, as reflected in the
error predictions, the actual composite error, and the reconstructed images.Note that
although the condition-number-based error predictor described in (14) and the
composite error measurement described in (20) and (21) show some correlation in
this simulation, the error predictor cannot predict exact performance for a
particular phantom. The error predictor
is derived from the projection and backprojection equations, but has no
relation to the phantom in question. It
can therefore be used to predict performance generally, but cannot predict
performance exactly for a specific phantom.
On the other hand, the composite error measurement used above is
significant only for this particular phantom, as it relies partially on a profile whose location was specifically
selected to match the location of the lesions to be detected. It should be taken as most significant, then,
that the systems which performed well in simulation were generally likely to
also have low error predictions.


## 3.1. Setup

Preliminary simulations were run for a
two-dimensional phantom with a one-dimensional detector. The phantom was a31×31 pixel modified
Shepp-Logan phantom. The phantom was imaged
at 120 angles to a 60-pixel detector, using apertures of 1, 3, 5, 7, 9, 11, 13,
and 15 evenly spaced pinholes. Poisson
noise was added to the projections, and the images were reconstructed using the
MR algorithm.The MR algorithm
is used in place of the more popular ML-EM algorithm because of its natural
applicability to the singular value decomposition. Although the ML-EM algorithm models Poisson
noise properly [14, 15], it cannot be analyzed with a simple algebraic
method, and so is not suitable for this analysis. Under the assumption that photon count per
detector bin is sufficiently high (greater than 10), the Poisson noise can be
approximated as Gaussian, and so the MR algorithm can be used.Because of the
relatively small system size, the errors in these images were compared to two
error predictors: one based on the condition number of the system, and the
other based on thenoise amplification
factor, which is based on the entire singular-value spectrum. As discussed earlier, a function of the
condition number K(A) can be
substituted for the noise amplification factor.
The square of the condition number, (K(A))2, seems to be a good estimate of the noise
amplification factor, and so the condition-number-based error estimate used in
these experiments is (K(A))2/N. This is, of course,
an empirical fit and not based on any rigorous mathematical principle.Final simulations were run for a three-dimensional phantom with a
two-dimensional detector. The phantom
used (Figure2) is a custom 64×64×64 voxel phantom, which has been made to
resemble the Shepp-Logan phantom. The Shepp-Logan phantom is used frequently in
analysis of medical imaging systems, and is designed to resemble a head. The phantom was imaged at 60 angles to a 128-by-128
pixel detector through apertures with varying numbers of holes and varying
arrangements of holes. The apertures are
illustrated in Figures 3(a)–3(f), and will be referred to as apertures A through F, respectively. Poisson noise
was added to these projections.Figure 2
Custom phantom used in simulation.Apertures used in final simulation.
(a)(b)(c)(d)(e)(f)The noisy
projections were backprojected to create the image vectorBTp. These results were then reconstructed using
the MR algorithm. The reconstructed
images were compared to the original phantom and the error in each image was
calculated. Because this system is
large, calculation of the entire singular value spectrum is not feasible. The
error was therefore compared only to the condition-number-based noise
prediction.The acceptance
angle of all pinholes in both experiments is60°, meaning that photons may
enter a pinhole at an angle of up to ±30° from perpendicular (Figure 4). In both sets of simulations, the aperture, phantom,
and detector were placed as close together as possible while allowing emitted
photons from all points in the phantom to pass through the aperture and strike
the detector.Figure 4
Illustration of acceptance angle.


## 3.2. Results

The error plots of the preliminary simulations
are shown in Figure5. The plots in
Figure 5 show the normalized error predictions based on condition number,
(17)ξK=(K(A))2N,
error predictions based on the full singular-value spectrum,
(18)ξSVD=1N∑k=1n1λk2,
and true error, defined as the standard deviation of the error in the image pixels,
(19)ξtrue=∑i=1m3(x−A−1BTp)j2.Figure 5
Normalized error predictions and actual error of two-dimensional systems.The reconstructed
two-dimensional images are shown after (a) 1, (b) 9, (c) 17, and (d) 25
iterations in Figure6. The rows
represent results for the apertures with 1, 3, 5, 7, 9, 11, 13, and 15 holes,
from the top down. Figure 6(e) shows the
image reconstructed as A−1BTp. The original phantom is
shown in Figure 6(f) for comparison.
Note that each set of error predictions and of actual errors is
normalized; the error predictors, as currently defined, are useful only for
comparison between systems and do not represent any absolute real-world value.Comparison of reconstructed images in two-dimensional
simulation after (a) 1, (b) 9, (c) 17, and (d) 25 iterations. (e) The image reconstructed asA−1BTp. (f) The original phantom.
(a)(b)(c)(d)(e)(f)The reconstructed images from the final (three-dimensional) simulations are shown in Figure7. Figures 7(a)–7(f) show slices from the
reconstructed images from systems A through F, respectively, at the stage in the MR algorithm at which they are closest to the original phantom (Figure 7(g)). The number of iterations used for the systems is 9 (system
A); 11 (system B); 11 (system C); 12 (system D); 20 (system E); 15 (system F). Although most systems use a fixed number of
iterations in MR reconstruction, it is not unreasonable to use a different
number of iterations for each system.
The normalized error predictions for the three-dimensional simulations
and the normalized true error are shown in Figure 8. The reconstructed images
from the final (three-dimensional) simulations are evaluated both in terms of
noise and in terms of lesion detection (an indirect measure of spatial
resolution). In order to evaluate the
comparative performance of the systems, a composite error is used. This error composite is defined as the sum of the mean squared error of the entire image,
(20)ξ1=∑i=1m3(x−A−1BTp)j2
and the square root of the noise power along the profile of the three
small lesions in the bottom half of the phantom,
(21)ξ2=∑j(x−A−1BTp)j2
as shown in Figure 9. Sensitivity
is measured in the noise parameter, ξ1, and resolution is measured
in the small-lesion profile, ξ2.
Figure 10 compares profiles along the line shown in Figure 9. Using this error composite, the reconstructed images can be evaluated in terms of noise
and lesion detection, or sensitivity and resolution.Figure 7
Comparison of reconstructed images from three-dimensional simulations.Figure 8
Predicted and measured errors for three-dimensional simulations.Figure 9
Illustration of line used for profile comparison.Comparison
of system profiles for (a) original phantom, (b) system A, (c) system C, and
(d) system E.
(a)(b)(c)(d)


## 3.3. Analysis

The two-dimensional simulations show
the condition number and singular values to be useful in determining relative
uncertainty in reconstructed images.
Both error estimates (17) and (18)) perform well in estimating error,
but the error estimate which involves the entire singular value spectrum—the noise amplification factor—more accurately predicts system
performance. However, the
characteristics of any small two-dimensional system are much closer to ideal
than those of a real-world system, and are easier to model.The composite error for the
three-dimensional system, as previously defined, was created in order to
measure both system sensitivity and spatial resolution. For example, systems A and B are able to
resolve the three lesions in the bottom half of the phantom, but contain
substantial amounts of noise, as evidenced by the noisy reconstruction of the
large bright circle at the top half of the phantom (Figures7(a) and 7(b)). Systems D and E contain relatively low amounts of noise (Figures 7(d) and 7(e)), but the three small lesions at
the bottom are almost indistinguishable.
System F (Figure 7(f)) has the worst reconstructed image; although the
background is mostly homogeneous, the bright spot at the top is not well defined,
and there is a large artifact in the center of the image. This artifact is most likely due to poor
placement of pinholes in the aperture.
System C is a good compromise between the high-noise problems of the
one- and two-pinhole apertures (systems A and B) and the poor resolution of the
nine- and ten-pinhole apertures (systems E and F). It also has the lowest error prediction. Note that systems C and D contain the same
number of holes, yet have observably different performance, as reflected in the
error predictions, the actual composite error, and the reconstructed images.Note that
although the condition-number-based error predictor described in (14) and the
composite error measurement described in (20) and (21) show some correlation in
this simulation, the error predictor cannot predict exact performance for a
particular phantom. The error predictor
is derived from the projection and backprojection equations, but has no
relation to the phantom in question. It
can therefore be used to predict performance generally, but cannot predict
performance exactly for a specific phantom.
On the other hand, the composite error measurement used above is
significant only for this particular phantom, as it relies partially on a profile whose location was specifically
selected to match the location of the lesions to be detected. It should be taken as most significant, then,
that the systems which performed well in simulation were generally likely to
also have low error predictions.


## 4. Conclusion

The objective of
this research was to create an error estimate that could predict the relative
performance of pinhole-based SPECT systems with a reasonable degree of
accuracy. To achieve this, the singular
value decomposition of the system’s projection-backprojection matrix was
analyzed.  The singular value
decomposition allows for a frequency-based analysis, similar to a Fourier
analysis. It was based on a function termed thenoise amplification factor, which is
based on the entire singular value spectrum and the photon count of the system. Because of the large amount of computation
required to calculate the entire singular value spectrum for a real-world
system, a second error predictor was created, based on the condition number and
the photon count of the system. However,
because the condition number does not contain information from the entire
singular value spectrum, it cannot account for the more subtle differences
between systems, and is therefore less reliable than the noise amplification
factor.These error
predictors were shown to be useful in the prediction of system
performance. Six systems with varying
numbers and arrangements of pinholes were used to compare predicted and actual
errors. The predictions were shown to be
useful in determining a preferred system configuration.The design of a
pinhole-based SPECT system is a problem of system design with many
variables. The number of pinholes,
arrangement of pinholes, detector size and distance from the aperture,
acceptance angle, and many other variables all affects the efficacy of an SPECT
system in ways that are interrelated.
Thus, system optimization cannot be reduced to a combination of
single-variable optimizations.
Simulation of each possible system configuration is also unfeasible,
because of the nearly infinite number of configurations available, and because
results will vary depending on the phantom used. For this reason, an unbiased error predictor,
based only on the system configuration and not on any empirical data, will provide
great benefits to system designers.A drawback of the
SVD-based analysis is the case of an overspecified system. In such a case, the condition number is
infinity because the singular values corresponding to high frequencies are
zero. In this case, the system
resolution must be decreased to a point that all systems under consideration
can be analyzed.The most obvious
use for an unbiased error predictor, such as the one described in this paper,
is in system optimization. It is
therefore the most important of the extensions of this research. However, in order to move to the goal of
system optimization, research in this preliminary stage of performance
prediction must be expanded.In the
mathematical derivations presented in this paper, Poisson noise was added at
the detector. In the frequency-based
analysis of the system, this noise was modeled as having equal power at all
frequencies. An analysis of the Poisson
noise in terms of the singular-value-based frequency spectrum, and
incorporation of this knowledge into the error estimate, would add another
degree of accuracy to the present error predictions.When using the MR
algorithm for image reconstruction, iteration of the algorithm is terminated
after a certain number of iterations.
Because of this, high-frequency information is attenuated in the
reconstructed image. In order to reflect
this in the error predictor, the singular value spectrum must be truncated, as
shown in (10)–(12).
To do so accurately would require a stronger knowledge of the
relationship between the number of iterations performed in the MR algorithm and
its effect on the singular value spectrum.
It is possible that this relationship can be explained as simply as a
high-pass-type transfer function which is applied at each iteration of the
algorithm, but it is most likely that the relationship is more complex.Because
calculation of the entire set of singular values for a real-world system is
computationally expensive, a function of the condition number was used in this
paper to predict system performance.
However, it is very unlikely that this is the optimal predictor, even if
only using the condition number of the system.
The present system could be vastly improved and a detailed system
analysis could be made much simpler if a method could be devised to create a
rough estimate of the singular value spectrum, or if a better estimate of the
noise amplification factor could be derived.
If not, a more refined estimation of the noise amplification factor,
based on the condition number, would still improve the error estimate somewhat.



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*Source: 769195-2008-12-21.xml*