# The Influence of Photon Attenuation on Tumor-to-Background and Signal-to-Noise Ratios for SPECT Imaging.

**ABSTRACT** Expanding on the work of Nuyts et. al [1], Bai et. al. [2], and Bai and Shao [3], who all studied the effects of attenuation and attenuation correction on tumor-to-background ratios and signal detection, we have derived a general expression for the tumor-to-background ratio (TBR) for SPECT attenuated data that have been reconstructed with a linear, non-iterative reconstruction operator O. A special case of this is when O represents discrete filtered back-projection (FBP). The TBR of the reconstructed, uncorrected attenuated data (TBR(no-AC)) can be written as a weighted sum of the TBR of the FBP-reconstructed unattenuated data (TBR(FBP)) and the TBR of the FBP-reconstructed "difference" projection data (TBR(diff)). We evaluated the expression for TBR(no-AC) for a variety of objects and attenuation conditions. The ideal observer signal-to-noise ratio (SNR(ideal)) was also computed in projection space, in order to obtain an upper bound on signal detectability for a signal-known-exactly/background-known-exactly (SKE/BKE) detection task. The results generally show that SNR(ideal) is lower for tumors located deeper within the attenuating medium and increases for tumors nearer the edge of the object. In addition, larger values for the uniform attenuation coefficient μ lead to lower values for SNR(ideal). The TBR for FBP-reconstructed, uncorrected attenuated data can both under- and over-estimate the true TBR, depending on several properties of the attenuating medium, including the shape of the attenuator, the uniformity of the attenuator, and the degree to which the data are attenuated.

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The Influence of Photon Attenuation on Tumor-to-Background

and Signal-to-Noise Ratios for SPECT Imaging

Edward J. Soares, Ph.D.*,†, Michael A. King, Ph.D.†, Charles L. Byrne, Ph.D.‡, Howard C.

Gifford†, and Andre Lehovich, Ph.D.†

* Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610

† Department of Radiology, University of Massachusetts Medical School, Worcester, MA 01655

‡ Department of Mathematical Sciences, University of Massachusetts at Lowell, Lowell, MA 01854

Abstract

Expanding on the work of Nuyts et. al [1], Bai et. al. [2], and Bai and Shao [3], who all studied the

effects of attenuation and attenuation correction on tumor-to-background ratios and signal detection,

we have derived a general expression for the tumor-to-background ratio (TBR) for SPECT attenuated

data that have been reconstructed with a linear, non-iterative reconstruction operator O. A special

case of this is when O represents discrete filtered back-projection (FBP). The TBR of the

reconstructed, uncorrected attenuated data (TBRno-AC) can be written as a weighted sum of the TBR

of the FBP-reconstructed unattenuated data (TBRFBP) and the TBR of the FBP-reconstructed

“difference” projection data (TBRdiff). We evaluated the expression for TBRno-AC for a variety of

objects and attenuation conditions. The ideal observer signal-to-noise ratio (SNRideal) was also

computed in projection space, in order to obtain an upper bound on signal detectability for a signal-

known-exactly/background-known-exactly (SKE/BKE) detection task. The results generally show

that SNRideal is lower for tumors located deeper within the attenuating medium and increases for

tumors nearer the edge of the object. In addition, larger values for the uniform attenuation coefficient

μ lead to lower values for SNRideal. The TBR for FBP-reconstructed, uncorrected attenuated data

can both under- and over-estimate the true TBR, depending on several properties of the attenuating

medium, including the shape of the attenuator, the uniformity of the attenuator, and the degree to

which the data are attenuated.

I. Introduction

Over the last decade, several authors have investigated the usefulness of attenuation correction

on lesion detection and localization in single-photon emission computed tomography (SPECT)

and positron emission tomography (PET). In 2000, Wells et. al. [4] compared ordered-subset

expectation-maximization (OS-EM) reconstruction with filtered back-projection (FBP) to

determine if attenuation correction helped for detection and localization of thoracic gallium-

labeled lesions in SPECT. Using psychophysical studies, they found no difference in observer

performance using images derived from FBP reconstruction, with and without attenuation

compensation (AC). While OS-EM with AC did improve observer performance relative to the

other methods, OS-EM without AC yielded comparable detection to images reconstructed with

E.J. Soares is with the Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610 USA

(telephone: 508-793-3368, e-mail: esoares@holycross.edu).

M.A. King, H.C. Gifford, and A. Lehovich are with the Department of Radiology, Division of Nuclear Medicine, University of

Massachusetts Medical School, Worcester, MA 01655 USA (telephone: 508-856-4255, e-mail: Michael.King@umassmed.edu).

C.L. Byrne is with the Department of Mathematical Sciences, University of Massachusetts at Lowell, Lowell, MA 01854 (telephone:

978-934-2447, e-mail: Charles Byrne@uml.edu).

NIH Public Access

Author Manuscript

IEEE Nucl Sci Symp Conf Rec (1997). Author manuscript; available in PMC 2009 January 23.

Published in final edited form as:

IEEE Nucl Sci Symp Conf Rec (1997). 2007 ; 5: 3609–3615. doi:10.1109/NSSMIC.2007.4436905.

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FBP and no AC. Narayanan et. al. [5] studied compensation strategies for 99mTc myocardial

perfusion imaging. They compared FBP with no AC, with OS-EM reconstruction with AC,

with AC and scatter correction, and with AC, scatter correction, and resolution compensation.

The results of their psychophysical studies suggested that accurate modeling and compensation

for these degrading effects improved detection of coronary artery disease. FBP exhibited the

lowest observer performance, while performance increased as image-degrading factors were

included in the OS-EM reconstruction process.

Nuyts et. al [1] examined the effect of the attenuation artifact on the tumor-to-background ratio

(TBR) and tumor detection for PET. They showed the attenuation artifact had scaling and

shifting components. The scaling component was the result of loss of counts due to attenuation,

while the shifting component is a smooth negative contribution that causes an artificially high

TBR in some circumstances. Bai et. al. [2] also investigated the effects of attenuation in PET.

They noted that images without AC could have locally enhanced contrast in regions of uniform

attenuation. In regions of nonuniform attenuation such as the thorax, high count foci can

disappear in images without AC, or even appear photopenic. This effect is dependent on the

size, location, and density of the foci. Bai and Shao [3] extended their analysis to SPECT

oncological imaging. They found similar behavior in tumor detectability as compared with

their analysis of PET imaging. Chang’s attenuation correction method was shown to not

improve estimation of TBR, relative to no-AC.

Motivated by this previous work, we have attempted to analytically quantify the effect of

photon attenuation in the projection data and the lack of compensation for the degradation in

SPECT reconstructed images. We have derived an analytic expression that relates the TBR of

FBP-reconstructed, uncorrected attenuated data to that of FBP-reconstructed, unattenuated

data and FBP-reconstructed, “difference” projection data. Using simulation studies, we

demonstrate the effect of object size and shape, non-uniformity of attenuator, and degree of

attenuation on the TBR and ideal observer signal-to-noise ratio (SNR).

II. Theory

A. Imaging Model

In our analysis, we consider a simple signal-known-exactly/background-known-exactly (SKE/

BKE) detection task. Let f0 denote the discretized signal-absent object and f1 denote the

discretized signal-present object. Thus, the background and tumor (signal) are given,

respectively, by

(1)

The discrete Radon transform and the discrete attenuated Radon transform can be expressed

as matrices, H = (hij) and Ha = (aij hij), respectively, where hij is the weight for determining

the contribution of object pixel j to measurement i and aij is the associated attenuation factor.

Next, we define the difference transform ΔH, where ΔH = H − Ha = ((1 −aij)hij). The

importance of this operator becomes clear in our subsequent derivations.

Both signal-absent and signal-present objects are then projected incorporating the effects of

photon attenuation. Thus, we may write the mean attenuated projections of the signal-absent

and signal-present objects as

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(2)

The object-dependent noise n is defined as the deviation between a measurement and its mean

(3)

which, when incorporated with (2), yields the imaging model

(4)

Next, the data are reconstructed with a linear, non-iterative operator O. An example of O would

be discrete filtered back-projection (FBP). Thus, we may write the reconstructed signal-absent

and signal-present images as

(5)

with mean reconstructed signal-absent and signal-present images given by

(6)

since the noise in (3) is defined tacitly as having zero mean. Finally, using equations (1) and

(6), we see that the mean reconstructed background and tumor images may be written as

(7)

and

(8)

respectively.

B. Signal-to-Noise Ratio (SNR)

In order to objectively assess the effect of photon attenuation on tumor detectability, we

computed the ideal observer signal-to-noise ratio (SNRideal) using the projection data. This is

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slightly unconventional, as it is the reconstructed images that are typically used for diagnostic

purposes and thus image quality tends to be assessed in reconstruction space. But evaluating

SNRideal in projection space serves two purposes. First, it sets an upper bound on detectability,

as SNRideal is invariant to invertible reconstruction operators O and may only decrease after

image reconstruction. One explanation for such a decrease would be if the reconstruction

operator O contains null components. Second, it will allow us to determine which signal

locations within the background produce data with higher detectability measures. In some

sense, it’s a measure of favorability of tumor location under our SKE/BKE paradigm.

The ideal observer uses the full joint probability distributions of both signal-present and signal-

absent data to compute a test statistic based on maximizing the likelihood ratio. For simplicity,

we used the normal approximation to the Poisson-distributed data

(9)

where the signal-absent and signal-present class covariance matrices are diagonal and given

by

(10)

Assuming the probabilities of occurrence of each class are P0 and P1, respectively, then

SNRideal can be calculated as

(11)

where ḡ ḡT are the mean attenuated projections of the tumor. The reader will note that the inverse

of the average class covariance matrix (P0Σg0 + P1Σg1)is easily computed, as this matrix is

diagonal.

C. Tumor-to-Background Ratio (TBR)

The tumor-to-background ratio (TBR) is a local measure of contrast between the tumor activity

and that of the surrounding background. Intuitively, we might hypothesize that a higher TBR

would correlate with higher detectability, as the tumor “stands out” more against the local

background. However, tumor detection also depends on the noise properties of image,

particularly the noise correlations. It is well known that FBP reconstruction introduces noise

correlations that inhibit human observer performance in SKE/BKE detection tasks. Thus, we

wished to compare this contrast measure with our detectability figure-of-merit SNRideal to

better understand any relationships that may exist.

In the following, we use the mean reconstructed background and tumor (eqns. (7) and (8))

rather than those containing the noise terms. This is due to the fact that the inclusion of the

noise terms would introduce an additional level of complexity in our derivation. Also, the

average activity in both the reconstructed background and tumor is computed within a region-

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of-interest (ROI), and this averaging process would most likely minimize the influence of the

noise terms on the overall TBR.

To estimate the average tumor activity and average background activity, we must first define

a ROI-observer template w that is non-zero for pixels within the ROI and zero otherwise. Often,

the template values are designed to sum to unity. We can now compute the estimated average

tumor activity (T̂) in the ROI as

(12)

as well as the estimated average background activity (B̂) in the ROI as

(13)

Finally, we define the TBR of the reconstructed, uncorrected attenuated data as

(14)

The quotient above can be simplified using component-operation long division as discussed

in [6] to yield

(15)

We note that

(16)

Dividing both sides by wtOHafB (assuming it is non-zero), we yield the identity

(17)

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Substituting (17) into (15), we are able to write

(18)

Let us now assume that the attenuated data are reconstructed with FBP. Thus, O is the discrete

FBP matrix and so (18) becomes

(19)

where TBRFBP is the TBR of the FBP-reconstructed un-attenuated data and TBRdiff is the TBR

of the images derived from the FBP-reconstructed difference projection data (difference

between the un-attenuated and attenuated data).

The reader’s attention should be directed to the weighting factor in (19)

(20)

The numerator of α is simply the average activity within the ROI of the FBP-reconstructed

background from unattenuated data. We would assume that this value would be close to that

in the original object. However, the denominator plays a crucial role in determining the overall

TBRno-AC. As discussed in [1]–[3], the FBP-reconstructed background from uncorrected,

attenuated data may contain regions of zero or negative activity. If the average background

activity in the ROI is small, this will produce an extremely large weighting factor and thus a

high TBRno-AC.

III. Methods

We studied uniformly emitting objects and uniform and nonuniform attenuators in our

experiments. Examples of these can be seen in Figure 1. Each object was of dimension 64× 64

pixels. The emitters included both circular (radius = 24 pixels) and elliptical (major axis = 28

pixels, minor axis = 20 pixels) objects, so that we could see the effect of object shape on both

SNRideal and TBRno-AC. We simulated a background activity level of 100 counts per object

pixel in our uniform emitters. Tumors were modeled as 10% positive contrast uniform disks

(radius = 3 pixels). The attenuators also included both circular and elliptical shapes and were

simulated to model attenuating mediums for photon energies 73 keV (201Tl), 140 keV (99mTc),

and 247 keV (111In), in order to see the effect of degree of attenuation on both SNRideal and

TBRno-AC. The uniform attenuation coefficients for each of these energies are given,

respectively, by μ = .129 cm−1, μ = .15 cm−1, and μ = .192 cm−1. Finally we studied a non-

uniformly attenuating medium using the MCAT simulation modeling 99mTc emission and

attenuation.

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For each object and attenuator pair, we created 2D maps of SNRideal and TBRno-AC as a

function of tumor location within the object. For each object pixel, a tumor was placed within

the object, centered at that object pixel. If the tumor fell completely within the object’s support,

we then computed SNRideal (using (11)) and TBRno-AC (using (19)) and set the value of the

SNR and TBR maps at that object pixel accordingly. Otherwise, these values were set to zero.

As a result, brighter pixel values in these maps correspond to higher SNRideal and TBRno-AC.

In this way, one can clearly determine which locations within the object correspond to higher

detectability and local contrast. We used the support of the tumor as our ROI template w,

although it is clear that the way with which one defines the ROI will affect the value of

TBRno-AC.

IV. Results

A. Circular Objects

In Figure 2, we show the results for the circularly-shaped emitter and attenuator. The top row

contains a 1D plot of SNRideal vs. radial tumor location for the cases of no attenuation (μ = 0)

and the aforementioned levels of uniform photon attenuation. The bottom row contains a 1D

plot of TBRno-AC vs. radial tumor location for the same cases. The reader will note that due to

the isotropic nature of the object, computing 2D SNR and TBR maps was unnecessary.

It can be seen that tumors placed closer to the center of the emitter yield lower values for

SNRideal. As the tumor is moved closer to the edge, SNRideal increases. This is due to the fact

that the effect of attenuation decreases as we move toward the edge and corrupts the data less.

Also, we can see that there is a clear ordering to the SNRs, with SNRideal consistently lower

for μ = .192 cm−1, followed by μ = .15 cm−1, and then finally μ = .129 cm−1. Again, lower

attenuation coefficients lead to less corrupt data and thus higher SNRs.

Finally, we note that SNRideal for μ = 0 is mostly flat throughout the object, but with a slight

increase at the edge. This is not an artifact of the reconstruction process. This slight increase

exists because when the tumor is placed near the edge of the object, the noise variance in the

support of the projected tumor is proportional to the projected background. The value of this

projected background decreases at the edge of the projection because of the object shape. This

yields lower values for the noise variance, which slightly increases SNRideal.

Examining the TBRno-AC plots, we see that when no attenuation is present (μ = 0), our contrast

measure is invariant to tumor location with a value of approximately 7.4%. The reader will

note that this is less than our simulated value of 10% contrast. This is due to the fact that the

ROI template was defined as the support of the tumor and since some of the tumor activity is

represented as partial volumes, averaging all of the tumor activity would lead to a slight

decrease in the simulated contrast.

As we might predict, TBRno-AC is lowest in the center of the object because photon attenuation

decreases local contrast, and if not corrected, this decrease will be retained in the FBP-

reconstructed image. However, as the tumor is moved closer to the edge, TBRno-AC increases,

as attenuation corrupts the data less. Throughout most of the background, the values of

TBRno-AC for the three levels of attenuation exhibited the same ordering as was seen with

SNRideal. Again, higher levels of photon attenuation lead to decreased contrast, relative to

lower levels of attenuation. However, as we move toward the edge, TBRno-AC exceeds the

correct value and this ordering

The data contained in Figure 3 provides the explanation for this effect. In this figure, we show

FBP reconstructions of the uncorrected, attenuated projections of the uniform background for

the three levels of attenuation. As we can see from all three images, there is a hot ring at the

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periphery of each of the reconstructions, relative to the cooler interior. However, the ring is

hotter for lower attenuation coefficients, which can be seen by examining the maximum values

on the respective colorbars. Since the average activity in these images within the support of

the tumor is used to compute the denominator of the scaling factor α (eqn. (20)), this value will

be larger for lower attenuation coefficients. Thus, TBRno-AC (eqn. (19)) will be higher for

larger attenuation coefficients (μ = 0.192 cm−1) vs. smaller ones (μ = 0.129 cm−1).

B. Elliptical Objects

In Figure 4, we show the results for the elliptically-shaped emitter and attenuator. The top row

contains 2D images of SNRideal (left) and TBRno-AC (right), both vs. radial tumor location for

the cases of no attenuation (μ = 0). The middle row contains analogous images for the case

where μ = 0.129 cm−1, while the bottom row contains analogous images for the case where μ

= 0.15 cm−1. The reader will note that due to the anisotropic nature of the object, computing

2D SNR and TBR maps was necessary.

When no attenuation is present, both SNRideal and TBRno-AC are approximately invariant to

tumor location, as was also seen in our study of circular objects and attenuators.

When attenuation is present, it can be seen that tumors placed closer to the center of the emitter

yield lower values for SNRideal. As the tumor is moved closer to the edge, SNRideal increases.

Again, this is due to the fact that the effect of attenuation decreases as we move toward the

edge and corrupts the data less. Also, we note that SNRideal is higher at the edge along the

minor axis of the ellipse, relative to the edge along the major axis. Tumors located along the

minor axis, especially those near the edge, undergo less attenuation producing data that are

less corrupted and thus have higher SNRs.

The 2D maps of TBRno-AC exhibit an unusual behavior. At the center of the object, we see

artificially high values for the TBR. These values then decrease as we move outward radially,

and then increase again as we move outward in a direction close to the major axis of the ellipse.

The peak is larger for the case where μ = 0.15 cm−1 vs. μ = 0.129 cm−1, suggesting that the

average activity in the FBP reconstructions of the uncorrected, attenuated projections of the

uniform elliptical background is smaller for the case where μ = 0.15 cm−1. This would also be

the case for those regions along the direction of the major axis which have artificially high

TBRs.

The data contained in Figure 5 provides the explanation for this effect. In this figure, we show

FBP reconstructions of the uncorrected, attenuated projections of the uniform elliptical

background for levels of attenuation μ = 0.129 cm−1 and μ = 0.15 cm−1. As we can see from

the two images, there is a hot ring at the periphery of each of the reconstructions, relative to

the cooler interior. However, the ring is hotter for lower attenuation coefficients, which can be

seen by examining the maximum values on the respective colorbars. It is more difficult to see

which regions are cooler for the different cases. However, we hypothesize that artificially high

TBRs would correlate with regions of low activity in these images, while low TBRs would

correlate with regions of high activity. Again, this is due to the fact that the average activity in

these images within the support of the tumor is used to compute the denominator of the scaling

factor α (eqn. (20)).

C. MCAT

In Figure 6, we show results for our study of the effect of uniform emission and non-uniform

attenuation on SNRideal and TBRno-AC using the MCAT simulation. In this case, our tumor

was a 10% contrast disk of radius 6 pixels placed within the uniform MCAT object. The

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background activity level remained at 100 counts per pixel. The MCAT attenuation map

reflected that for 99mTc attenuation.

First, because of the larger tumor size (radius = 6 pixels instead of radius = 3 pixels), we note

that the value of SNRideal with no attenuation present has increased. The larger tumor leads to

greater tumor conspicuity from the perspective of the ideal observer, and thus a larger overall

SNRideal. Further, when no attenuation is present, both SNRideal and TBRno-AC are

approximately invariant to tumor location, as was also seen in our previous studies.

When no attenuation is present, both SNRideal and TBRno-AC become location dependent. The

SNRideal map is consistent with the observation that tumors located deeper within the

attenuating medium lead to data that is more corrupt and thus produce lower values for

SNRideal. The TBRno-AC map has a striking similarity to a blurred version of the attenuation

map (see Fig. 1, bottom right image). The lung areas have lower values for TBRno-AC, while

the heart and mediastinum areas have higher values.

The data contained in Figure 7 provides the explanation for this effect. In this figure, we show

the FBP reconstruction of the uncorrected, non-uniformly attenuated projections of the uniform

elliptical background. The reader will note that the lung areas possess higher counts, while the

heart and mediastinum areas exhibit lower counts. Again, since the average activity in this

image within the support of the tumor is used to compute the denominator of the scaling factor

α (eqn. (20)), this value will be larger for areas with a lower attenuation coefficient. Thus,

TBRno-AC (eqn. (19)) will be higher for areas with higher levels of attenuation vs. lower ones.

V. Conclusion

We have derived a general expression for the TBR of SPECT attenuated data that have been

reconstructed with a linear, non-iterative reconstruction operator O. A special case of this is

when O represents discrete FBP. We showed that the TBR of the reconstructed, uncorrected

attenuated data can be written as a weighted sum of the TBR of the FBP-reconstructed

unattenuated data and the TBR of the FBP-reconstructed difference projection data. We

evaluated our expression for a variety of objects and attenuation conditions. The ideal observer

SNR was also computed to obtain an upper bound on signal detectability for a signal-known-

exactly/background-known-exactly detection task. The results generally showed that the ideal

SNR is lower for tumors located deeper within the attenuating medium and increases for tumors

nearer the edge of the object. In addition, larger values for the uniform attenuation coefficient

μ lead to lower values for the ideal SNR. The TBR for FBP-reconstructed, uncorrected

attenuated data can both under- and over-estimate the true TBR, depending on several

properties of the attenuating medium, including the shape of the attenuator, the uniformity of

the attenuator, and the degree to which the data are attenuated. We intend to extend this work

by incorporating the contributions of object-dependent noise, in order to more accurately assess

the effects of these imaging factors on SNR and TBR.

Acknowledgements

The authors would like to thank P. Hendrik Pretorius, Ph.D. of the University of Massachusetts Medical School for

his help in creating the MCAT simulations and Johan Nuyts, Ph.D. of Katholieke Universiteit Leuven for insights

regarding attenuation and tumor-to-background ratios. This work was supported in part by the National Institute for

Biomedical Imaging and Bioengineering under grant R01-EB002798. The contents are solely the responsibility of the

authors and do not necessarily represent the official views of the National Institutes of Health.

This work was supported in part by the National Institute for Biomedical Imaging and Bioengineering under grant

R01-EB002798. The contents are solely the responsibility of the authors and do not necessarily represent the official

views of the National Institutes of Health.

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References

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Oncology. 2006 IEEE Medical Imaging Conference Proceedings;

4. Wells R, King M, Simkin P, Judy P, Brill A, Gifford H, Licho R, Pretorius P, Schneider P, Seldin D.

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5. Narayanan M, King M, Pretorius P, Dahlberg S, Spencer F, Simon E, Ewald E, Healy E, MacNaught

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Fig. 1.

Simulated emitters (top row) and attenuators (bottom row) used in our study. Top row: uniform

disk (left), uniform ellipse (center), uniform MCAT (right). Bottom row: uniform disk (left),

uniform ellipse (center), non-uniform MCAT (right).

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Fig. 2.

Plots of SNRideal (top) and TBRno-AC (bottom) as a function of radial tumor location for a

circularly-shaped uniform emitter and uniform attenuator.

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Fig. 3.

FBP reconstruction of the uncorrected, attenuated projections of the uniform disk background:

μ = 0.129 cm−1 (left), μ = 0.15 cm−1 (center), μ = 0.192 cm−1 (right).

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Fig. 4.

Plots of SNRideal (left) and TBRno-AC (right) as a function of radial tumor location for an

elliptically-shaped uniform emitter and uniform attenuator. Top row: No attenuation; Middle

row: μ = 0.129 cm−1; Bottom row: μ = 0.15 cm−1.

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Fig. 5.

FBP reconstruction of the uncorrected, attenuated projections of the uniform elliptical

background: μ = 0.129 cm−1 (left), μ = 0.15 cm−1 (right).

Soares et al. Page 15

IEEE Nucl Sci Symp Conf Rec (1997). Author manuscript; available in PMC 2009 January 23.

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