# Simulation Study on Super-Resolution for Coded Aperture Gamma Imaging
††thanks: The authors gratefully acknowledge the data storage service SDS@hd
supported by the Ministry of Science, Research and the Arts Baden-Württemberg
(MWK) and the German Research Foundation (DFG) through grant INST 35/1314-1
FUGG and INST 35/1503-1 FUGG. This paper was funded by the Zentrales
Innovationsprogramm Mittelstand (ZIM) under grant KK5044701BS0.

1st Tobias Meißner
3rd Jürgen Hesser  Mannheim Institute for Intelligent Systems in Medicine
Heidelberg University
Mannheim, Germany
<EMAIL_ADDRESS>Mannheim Institute for Intelligent
Systems in Medicine
Interdisciplinary Center for Scientific Computing (IWR)
Central Institute for Computer Engineering (ZITI)
CZS Heidelberg Center for Model-Based AI
Heidelberg University
Mannheim, Germany
<EMAIL_ADDRESS>2nd Werner Nahm
4th Nikolas Löw  Institute of Biomedical Engineering
Karlsruhe Institute of Technology (KIT)
Karlsruhe, Germany
<EMAIL_ADDRESS>Mannheim Institute for Intelligent Systems in Medicine
Heidelberg University
Mannheim, Germany
<EMAIL_ADDRESS>
###### Abstract

Coded Aperture Imaging (CAI) has been proposed as an alternative collimation
technique in nuclear imaging. To maximize spatial resolution small pinholes in
the coded aperture mask are required. However, a high-resolution detector is
needed to correctly sample the point spread function (PSF) to keep the
Nyquist-Shannon sampling theorem satisfied. The disadvantage of smaller
pixels, though, is the resulting higher Poisson noise. Thus, the aim of this
paper was to investigate if sufficiently accurate CAI reconstruction is
achievable with a detector which undersamples the PSF. With the Monte Carlo
simulation framework TOPAS a test image with multiple spheres of different
diameter was simulated based on the setup of an experimental gamma camera from
previous work. Additionally, measured phantom data were acquired. The captured
detector images were converted to low-resolution images of different pixel
sizes according to the super-resolution factor $k$. Multiple analytical
reconstruction methods and a Machine Learning approach were compared based on
the contrast-to-noise ratio (CNR). We show, that all reconstruction methods
are able to reconstruct both the test image and the measured phantom data for
$k\leq 7$. With a synthetic high-resolution PSF and upsampling the simulated
low-resolution detector image by bilinear interpolation the CNR can be kept
approximately constant. Results of this simulation study and additional
validation on measured phantom data indicate that an undersampling detector
can be combined with small aperture holes. However, further experiments need
to be conducted.

###### Index Terms:

coded aperture imaging, gamma imaging, image reconstruction, super-resolution,
nuclear medicine.

##  I Introduction

Figure 1: The basic principle of planar Coded Aperture Imaging: A mask with
pinholes projects the source image onto the detector, where multiple
overlapping projections emerge. Decoding or image reconstruction is necessary
to obtain the original source image. Figure modified from [1].

Accurate localization and visualization of radioactive sources is an essential
task in nuclear medicine [2], high-energy astrophysics [3] and in monitoring
of nuclear waste [4, 5]. Recently, small handheld gamma cameras for localizing
sentinel lymph nodes in breast cancer patients are under investigation [6, 7,
8, 9]. Due to the high-energy photons involved refractive lenses cannot be
used for producing an image of the scene and instead parallel or pinhole
collimators are employed to capture the necessary spatial information [2].
However, the size of the opening is usually subject to a balanced trade-off
between the number of captured photons (photon efficiency) and the spatial
resolution as the former increases and the latter decreases with the size of
the pinhole. A high photon efficiency is desired since the guiding principle
in the medical domain is to reduce the exposed radiation to As Low As
Reasonable Achievable (ALARA). Thus, photon flux is limited and achieving high
quantum yield is of major importance.
To improve the mentioned trade-off Coded Aperture Imaging (CAI) has been
introduced [10, 11]: A mask between object and detector consisting of a
radiopaque material with pinholes encodes the directional information of
incoming gamma rays. As Figure 1 shows, each pinhole in the mask generates a
projection of the source image on the detector resulting in a multitude of
overlapping projections. Therefore, image reconstruction (also referred to as
decoding) becomes necessary. If the distance between source and collimator is
large and the extension in depth is small relative to the distance, CAI can be
considered as an image-to-image mapping and is denoted as planar Coded
Aperture Imaging [1]. In this paper, the captured detector image is denoted as
$p(x,y)$, the original source image and its reconstruction as $f(x,y)$ and
$\hat{f}(x,y)$, respectively.

The term super-resolution refers to the process of combining several ”low
resolution, noisy, slightly shifted observations” [12] to reconstruct an image
of the underlying high resolution scene, as Figure 2 illustrates.

Figure 2: Super-resolution is referred to the process to combine multiple low
quality images to reconstruct an image of the underlying high-resolution
scene. Figure modified from [12].

Because the spatial resolution in CAI is mainly influenced by the mask’s
pinhole diameter [13], increasing the MURA rank and thus the amount of
pinholes while reducing its diameter would increase the spatial resolution. So
far, the pinhole diameter has been chosen such that the utilized detector can
properly sample the resulting PSF [8, 9].
To the best of the authors’ knowledge, no research group has investigated the
combination of small pinholes and a low-resolution detector. Therefore, the
investigated hypothesis is as follows: Existing CAI reconstruction methods are
capable of reconstructing point sources from an undersampling detector, and
thus achieving super-resolution, at reasonable quality even though the
detector cannot resolve the higher spatial resolution of the aperture. This is
due to the shifted but overlapping projections caused by the coded aperture.

##  II Methods

###  II-A Simulating a coded aperture test image

For simulating a test image the Monte Carlo simulation toolkit TOPAS [14], a
wrapper library around Geant4 [15], is deployed. Unlike ray-casting
simulations, TOPAS accounts for photon-mass interactions like scattering and
mask penetration, and is therefore considered to be the gold standard in gamma
imaging [16].
The geometrical components and dimensions were simulated according to the
experimental gamma camera from Rozhkov et al. [16]. The main characteristics
can be summarized as follows: A 2$\times$2 mosaicked, 1 mm thick Tungsten not-
two-holes-touching (NTHT) MURA mask of rank 31 with pinholes of 0.34 mm in
diameter (denoted as $d$) was placed 42 mm ($a$) in front of a 2 mm thick
256$\times$256 pixelated CdTe semiconductor detector coupled to a Timepix©
readout circuit. The detector has a side length of 14.1 mm and hence a single
pixel size $s=0.0551$ mm. The virtual object plane is 172 mm ($b$) in front of
the mask plane, resulting in a field of view (FoV) of 57.75$\times$57.75 mm.
The test image consists of three spherical sources with diameters $d_{1}$,
$d_{2}$ and $d_{3}$ of 1, 2 and 3 mm distributed within the FoV as Figure 8(a)
shows. $10^{9}$ gamma photons with a photon energy of 140.5 keV (corresponding
to the photon peak of ${}^{99\text{m}}\text{Tc}$ the most commonly used
radiotracer in nuclear medicine [2]) were distributed to the three sources
according to their area. Every photon hitting the detector was collected and
stored in a so called phase space file. In addition to the coded aperture a
single pinhole collimator with the same diameter was simulated to serve as a
reference for the reconstructed images. The captured pinhole image was
smoothed by Gaussian blurring with a $\sigma$ of 2 pixels.
The ground truth image was generated from the geometrical model and remains
binary: 1 for where a source is located and 0 everywhere else.

###  II-B Measured data from the experimental gamma-camera

Captured images from an experimental gamma-camera also used in previous work
[16, 17] was used to validate the effect of super-resolution on real-world
measurement data. The camera set-up was the same as for the simulation of the
test image. The phantom has the basic form of a cylinder with a height of 80
mm and 50 mm in diameter, where tubes along the vertical axis were filled with
${}^{99\text{m}}\text{Tc}$. These three tubes have a diameter of 1.1 mm, and
two of them are 15 mm long while the central one is 20 mm long. The total
activity at the beginning of the measurements was 83 MBq. A depiction of the
geometric computer model can be seen in Figure 3. The phantom was exposed to
the gamma-camera for 2 min and afterwards rotated by 3°. This way, a total of
120 images were captured. Outlier replacement as described in [17] was applied
afterwards.

Figure 3: The utilized phantom with its three tubes (red) filled with
${}^{99\text{m}}\text{Tc}$ of which 120 images where captured.

###  II-C Generating low-resolution detector images

To analyze the effect of different pixel sizes low-resolution images of
different resolutions were produced as follows: The captured photons from the
phase space file and from the measured phantom data respectively were binned
into images of different low resolution. The actual detector served as
reference with a resolution of 256$\times$256 pixels, which corresponds to the
resolution of the final reconstructed image. Therefore, the super-resolution
factor $k$ is introduced. It means that $k\times k$ high-resolution pixels are
reconstructed from a single low-resolution pixel. Note that the absolute
detector size remains the same: 14.1$\times$14.1 mm. The single pixel size $s$
changes proportional to $k$: $s=k\cdot 14.1\,$mm$/256$. Finally, all low-
resolution images were upsampled by bilinear interpolation to 256$\times$256
pixels in order to fit the synthetic high-resolution PSF and to fit into the
CED-IN respectively. The process of generating low-resolution images is shown
in Figure 4 for $k=8$.

Figure 4: Pixels of the high-resolution detector image from the TOPAS
simulation are accumulated (here with $k=8$ into 32$\times$32 pixels) to form
the low-resolution detector image. Afterwards this image is upsampled by
bilinear interpolation to the high-resolution of 256$\times$256 pixels.

###  II-D Analytical reconstruction methods

Figure 5: Left: The measured point spread function (PSF) of the experimental
gamma camera, where the pixel intensity represents photon counts. Center: The
two-holes-touching (THT) version of the used rank 31 MURA mask $h(x,y)$
fundamental for MURA Decoding and MLEM. Right: The respective decoding pattern
$g(x,y)$ for MURA Decoding. Note the additional positive square at the center
of $g(x,y)$ [18]. Both patterns were resized to 256$\times$256 pixels by
nearest neighbor interpolation to maintain the original shape.

Three different methods for super-resolution reconstruction are analyzed and
compared in this paper: MURA decoding [19], a convolutional Maximum Likelihood
Expectation Maximization algorithm (MLEM) [20] and a convolutional encoder-
decoder network (CED) from previous work [17].
MURA Decoding is the most commonly used reconstruction method. It consists of
a single circular convolution of the detector image $p(x,y)$ with the decoding
pattern $g(x,y)$:

$\hat{f}(x,y)=p(x,y)\circledast g(x,y)$ (1)

with ”$\circledast$” denoting the circular convolution operator. All circular
operations in this paper are carried out by periodically padding the second
operand to twice its size, i.e. to 512$\times$512 pixels, and cropping the
result to its central 256$\times$256 pixels.
The decoding pattern $g(x,y)$ is based on $h(x,y)$ and its definition can be
found in [18]: It is equivalent to changing all $0$ to -$1$ and adding a
positive pixel to the center of the PSF [18].
The MLEM algorithm works in iterations and is derived from a random Poisson
process. It consists of a combination of forward and backward projections,
where ten iterations were deployed in this paper:

$\hat{f}^{k+1}(x,y)=\hat{f}^{k}(x,y)\odot\left[\frac{p(x,y)}{\hat{f}^{k}(x,y)\circledast
h(x,y)}\otimes h(x,y)\right]$ (2)

where ”$\odot$” denotes point-wise multiplication and ”$\otimes$” is circular
cross-correlation.
Instead of the real measured PSF with round pinhole projections, the two-
holes-touching (THT) version of the PSF without gaps between neighboring
pinholes is used for reconstruction, since it suppresses periodical noise
[17]. Both the THT-PSF and its corresponding decoding pattern are of
rectangular structure and a square of 8 bright pixels represents the position
and bounding box of each projected pinhole. They also define the reconstructed
resolution of 256$\times$256 pixels. Figure 5 depicts the measured PSF, the
THT-PSF and the decoding pattern.
Additionally, MURA Decoding with low-resolution THT-PSF was implemented, where
the synthetic high-resolution THT-PSF was not used, but the down-sampled THT-
PSFs emulating a low-resolution detector. Since the reconstructions come in
low-resolution, the reconstructed images were upsampled to 256$\times$256
pixels by bilinear interpolation.

###  II-E Reconstruction by Machine Learning

A Convolutional Encoder-Decoder (CED) is a widely used form of Convolutional
Neural Networks (CNN). A CED consists of trainable parameters that transform
an input into an output image. However, this transformation is not derived
from a mathematical description but by providing a sufficient amount of paired
training images. First experiments were conducted on the application of CNNs
to CAI reconstruction, but its validation exclusively relied on simulated and
low-resolution images [21] or were only visually compared on few images [13].
While the two analytical reconstruction methods solely rely on the PSF of the
gamma camera, which acts as a linear approximation of the imaging system, the
CED-IN is in theory capable of more complex mappings [22]. Recent advances in
Machine Learning in the field of image reconstruction [23, 24, 25] underline
the potential of CEDs for CAI reconstruction. The CED used in this paper is
denoted as CED-IN because it was trained with a convolutional simulation based
on natural photographs from the ImageNet database [26]. Its architecture is
presented in Figure 6 and for more information on the training process and
data simulation the reader is referred to [17].

Figure 6: The convolutional encoder-decoder network architecture deployed in
this paper. The top row represents the number of filters per layer and the
bottom row the feature map size in pixels.

After reconstructing all images, the contrast-to-noise ratio (CNR) is
calculated based on the reconstructed and the ground truth image. The binary
ground truth image enables a separation of the reconstruction into the signal
part $S$ and background part $B$. The following definition of CNR is employed
[27]:

$\text{CNR}=\frac{\left|\bar{S}-\bar{B}\right|}{\sigma_{B}}\,,$ (3)

where $\bar{S}$ denotes the mean intensity of the signal, $\bar{B}$ the mean
intensity and $\sigma_{B}$ the standard deviation of the background.

(a) Simulated test image: The black dotted vertical lines mark the critical
super-resolution factors $\tilde{k}_{\text{THT-PSF}}=4$ and
$\tilde{k}_{1}=11$.The red dotted line represents the CNR of the smoothed
image captured by a pinhole collimator and serves as reference.

(b) Measured phantom data: Error bars represent the standard deviation at each
data point. The vertical dotted lines denote the critical super-resolution
factors $\tilde{k}_{\text{THT-PSF}}=4$ and
$\tilde{k}_{\text{measurement}}=12$.

Figure 7: The contrast-to-noise ratio (CNR) for reconstructions of different
reconstruction methods depending on the super-resolution factor $k$.

(a) Simulated test image: For reference, the coded aperture simulation in
256$\times$256 pixels, the smoothed pinhole collimator simulation in the same
resolution and the ground truth is shown on the right-hand side.

(b) Measured phantom data: On the right-hand side the coded aperture detector
and the ground truth (GT) images are shown.

Figure 8: Exemplary reconstructions of the analyzed reconstruction methods at
different super-resolution factors $k$. The CNR is printed in the top left
corner of each image.

###  II-F Nyquist-Shannon sampling theorem

The Nyquist-Shannon sampling theorem states that the sampling frequency of a
pixelated representation must be larger than twice the maximum frequency of
the periodic image [28]. Thus, when the smallest occurring structure is
sampled by two pixels or less, an image is not represented unambiguously which
leads to aliasing and hence signal degradation [28]. Since the aforementioned
analytical reconstruction methods MURA Decoding and MLEM consist of one or
more convolutions of two discretized signals, a reconstruction without
aliasing artefacts is only possible when both images were sampled by enough
pixels. Thus, critical super-resolution factors $\tilde{k}$ were determined
both for the coded aperture test image and the THT-PSF $h(x,y)$. The smallest
point source of the test image is 1 mm wide and therefore much larger than the
pinhole diameter: $d_{1}\gg d$. Hence, the smallest structure on the detector
caused by the small point source can be approximated by $t=d_{1}\cdot m=1.244$
mm. For $h(x,y)$ the smallest structure $t$ is 8 pixels wide, i.e. $t=8\cdot
s$ (Figure 5). For the given gamma camera with its magnification factor
$m=(1+a/b)=1.244$, the smallest depicted structure $t$ and the single pixel
side length of $s=0.0551$ mm $\tilde{k}$ can be defined as follows:

$\tilde{k}=\left\lfloor\frac{1}{2}\cdot\frac{t}{s}\right\rfloor$ (4)

where $\left\lfloor\cdot\right\rfloor$ denotes rounding off to the next
smallest integer value.

##  III Results

###  III-A Critical super-resolution factors

The following critical super-resolution factors $\tilde{k}$ were obtained from
the Nyquist-Shannon sampling theorem: The THT-PSF $h(x,y)$ must be sampled by
at least 64$\times$64 pixels leading to the following critical super-
resolution factor: $\tilde{k}_{\text{THT-PSF}}=4$. This means that the
synthetic high-resolution THT-PSF in this paper is 4-times oversampled.
However, the test image with the 1 mm point source results in a higher
critical super-resolution factor of $\tilde{k}_{1}=\left\lfloor
11.29\right\rfloor=11$. The tubes of the phantom captured by the experimental
gamma-camera have a diameter of 1.1 mm, that are magnified to approximately
1.37 mm and thus 24.83 pixels. This results in a critical super-resolution
factor for the measured data of $\tilde{k}_{\text{measurement}}=\left\lfloor
12.42\right\rfloor=12$.

###  III-B Results on the test image

Figure 7(a) shows the CNR of the four reconstruction methods over the super-
resolution factor $k$. The red dotted line at $\text{CNR}=7.63$ denotes the
smoothed image captured with a pinhole collimator where no reconstruction was
required. It is depicted central on the right-hand side together with the
ground truth image and the coded aperture test image in Figure 8(a). The left-
hand side shows exemplary reconstructions for $k=1,3,6,11$ and $16$.
Clearly visible, the CNRs of the reconstruction methods with the synthetic
high-resolution THT-PSF increase until $k=3$ and steadily decline afterwards.
The CED-IN is an exception, where the CNR increases further until falling
below its baseline at $k=13$. For all reconstruction methods using the
synthetic high-resolution THT-PSF the smallest point source starts to
disappear for $k>11$ and is hardly visible for $k=16$.
MURA Decoding with the respective low-resolution THT-PSF does not exceed its
baseline CNR and falls beneath the pinhole reference at $k=5$ and again for
all $k>9$. The reconstructions for $k\geq 11$ fail entirely and resemble no
similarity to the ground truth anymore.

###  III-C Results on the measurement data

Analogously to the test image Figure 7(b) shows the CNR for the presented
reconstruction method at different super-resolution factors $k$. However,
because 120 reconstructions were analyzed the marker represents the average
CNR and additional error bars represent the standard deviation for each
reconstruction method and super-resolution factor $k$.
In general, compared to the test image, lower CNRs can be observed. Similarly,
all methods except for the MURA Decoding with low-resolution THT-PSF, slightly
rise for small $k$ and then fall after approximately $k=7$. This behavior can
also be seen in the exemplary images in Figure 8(b). The maximum median CNR is
reached by the CED-IN with 3.42 at super-resolution factor $k=6$. Visually, a
higher background noise is present compared to the test image and the three
line sources are prominent until they start to disappear for larger $k$.

##  IV Discussion

###  IV-A Simulated test image

For the given setup the Nyquist-Shannon sampling theorem states, that for
super-resolution factors above $k=4$ the PSF is not sufficiently sampled
anymore. If the THT-PSF is undersampled it loses its characteristic to
properly function for CAI reconstruction. The simulation study of this paper
shows this behavior where CNRs of MURA Decoding with low-resolution THT-PSF
drop notably for $k\geq\tilde{k}_{\text{THT-PSF}}$ and the reconstruction of
$k=11$ show major artefacts rendering the three sources unrecognizably. But,
when upsampling the low-resolution detector image to a high-resolution of
256$\times$256 pixels and using a synthetic high-resolution THT-PSF for
reconstructions, the CNRs drop slower, as MURA Decoding and MLEM demonstrate.
For $k\leq 6$ the CNRs even stay approximately constant.
Additionally, the reconstructed images are visually closer to the expected
output. For $k>\tilde{k}_{1}$, as predicted by the Nyquist-Shannon sampling
theorem, the smallest point source cannot be reconstructed properly and starts
to dissolve.
In this simulation study super-resolution with small $k$ and upsampling by
bilinear interpolation even had a positive impact on the CNR and the
reconstructions look smoother.

###  IV-B Measured phantom data

Similar behavior was observed for the measured phantom data. For all
reconstruction methods, the CNR does not decrease until approximately $k=7$,
even though undersampling of the THT-PSF starts at $k=4$. However, the gain in
CNR for the measured phantom data is far less for small $k$ compared to the
simulated test image.
Interestingly, the CED-IN, even though not trained on processing upsampled
low-resolution images, performs better than all other reconstruction methods
for super-resolution factors of $k\geq 4$. For the measured phantom data it is
the best reconstruction method for all $k$. This indicates that the CED-IN
generalized from the training domain of natural photographs to discrete
sources on a dark background, even when the low-resolution input image was
upsampled by bilinear interpolation. Especially the background is
reconstructed more uniform compared to all other methods.
This implies that the CED-IN was taught to compress the inputted detector
image into a robust representation of the image, suppressing noise and dead
pixels, like in the top right corner of the coded aperture image in Figure
8(b).
The test image generated by Monte Carlo simulation shows in theory that super-
resolution in CAI is possible and the simulated low-resolution detector images
based on phantom data captured by an experimental gamma-camera strengthen this
hypothesis. However, the question remains as to how a real low-resolution
detector would affect the reconstruction. Erroneous pixels on a low-resolution
detector will have a higher impact since it is supposed to capture a larger
fraction of the coded aperture pattern. Another point not investigated in this
paper are other types of gamma sources. Especially extended sources are known
to cause problems in CAI [29].

##  V Conclusion

The conducted simulation study indicates that super-resolution reconstruction
for planar CAI is possible even if the detector is not capable of sampling the
PSF with a sufficient amount of pixels. Instead, a synthetic high-resolution
THT-PSF is combined with upsampling the captured low-resolution detector image
by bilinear interpolation. This way, established reconstruction methods were
able to reconstruct the simulated test image. However, for large super-
resolution factors, the smallest point source could not be reconstructed as
the Nyquist-Shannon sampling theorem predicted. Applying the same technique to
simulated low-resolution detector images from data of a hot-rod phantom
captured with an experimental gamma-camera strengthen these findings. For
future research, though, further experiments with a more realistic
undersampling detector including erroneous pixels are needed.

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