XXXX-XXXX

1]Graduate School of Science, Kyoto University, Kitashirakawa Oiwakecho,
Sakyo, Kyoto, Kyoto, 606-8502, Japan 2]Scientific balloon, Institute of Space
and Astronautical Science, Japan Aerospace Exploration Agency, Yoshinodai
3-1-1, Chuou, Sagamihara, Kanagawa, 252-5210, Japan

# Development of Convolutional Neural Networks for an Electron-Tracking
Compton Camera

Tomonori Ikeda  Atsushi Takada  Mitsuru Abe  Kei Yoshikawa  Masaya Tsuda
Shingo Ogio  Shinya Sonoda  Yoshitaka Mizumura  Yura Yoshida  Toru Tanimori  [
<EMAIL_ADDRESS>[

###### Abstract

Electron-tracking Compton camera, which is a complete Compton camera with
tracking Compton scattering electron by a gas micro time projection chamber,
is expected to open up MeV gamma-ray astronomy. The technical challenge for
achieving several degrees of the point spread function is the precise
determination of the electron-recoil direction and the scattering position
from track images. We attempted to reconstruct these parameters using
convolutional neural networks. Two network models were designed to predict the
recoil direction and the scattering position. These models marked 41 degrees
of the angular resolution and 2.1 mm of the position resolution for 75 keV
electron simulation data in Argon-based gas at 2 atm pressure. In addition,
the point spread function of ETCC was improved to 15 degrees from 22 degrees
for experimental data of 662 keV gamma-ray source. These performances greatly
surpassed that using the traditional analysis.

MeV gamma-ray, MPGD, TPC, Deep learning

##  1 Introduction

Gamma-ray observations in the low-energy gamma-ray band from a hundred keV to
several MeV address various astrophysical phenomena; the nucleosynthesis and
the explosion mechanism in the supernovae Matz ; Diehl2003 ; Churazov2015 ,
the particle acceleration process in the active galactic nuclei or the gamma-
ray bursts Chiaberge2001 ; Ballo2002 ; Briggs1999 ; Preece2000  and electron-
positron annihilation line in the Galactic center region OSSE ; Siegert ; COSI
. However, the observation in this energy band remains poorly explored
compared to that of X-ray, GeV and TeV band, known as “MeV gap”. Even now,
COMPTEL COMPTEL , onboard Compton Gamma Ray Observatory launched in 1991, is
the most sensitive observation of the MeV sky. The causes of stagnant in MeV
observation are huge gamma-ray backgrounds from the atmosphere (albedo) and
generated in the instruments by cosmic-ray interactions. In addition,
conventional Compton cameras like COMPTEL have unclean images according to the
Compton circle due to the lack of the direction information of the recoil
electron. Such pseudo imaging is not capable of background rejection and does
not keep quantitative imaging TANIMORI2017 . Thus, it was pointed out that
conventional Compton cameras need the additional parameters of the recoil
direction of the Compton electron in order to reduce backgrounds
SCHONFELDER2004193 .

Electron-Tracking Compton Camera (ETCC) is a complete Compton camera with
tracking recoil electrons. It can record all information on Compton kinematics
for overcoming the problem of conventional Compton cameras. Hence ETCC
determines a unique incident gamma-ray direction TANIMORI2017 . A key device
of ETCC is the tracking detector to detect the recoil electron since the
point-spread function (PSF) is highly dependent on the determination accuracy
of the recoil direction and the scattering position Tanimori2015 . Thus, the
SMILE group (Sub-MeV gamma-ray Imaging Loaded-on-ballon Experiment) developed
the gas time projection chamber (TPC) based on $\mu$-PIC Takada , which is one
of the micro-pattern gas detectors. Although the gas TPC gives us the precise
track information for low energy electrons, these electrons are influenced by
the multiple scattering and make complex track images. The technical challenge
is the development of the analysis method with an exact determination of the
scattering position and the recoil direction from these track images.

The feature extraction from images is the field of computer vision. Neural
Networks (CNNs) have achieved great success in image classifications. Since
2010, many architectures like AlexNet AlexNet , VGG VGG , GoogLeNet GoogLeNet
, and ResNet ResNet  were proposed in the ImageNet Large Scale Visual
Recognition Challenge ImageNet  and achieved dramatic progress. In particle
physics and astrophysics experiments, modern machine learning techniques have
been actively applied and developed. There are several purposes of signal-
background recognition, particle identification, and event reconstruction.
Machine learning showed significant performance improvements than the
traditional way based on domain knowledge. In the gas TPC application, the CNN
was developed for the polarization extraction from photoelectron track images
taken with X-ray polarimeters and improved the polarization sensitivity by
10%–20% KITAGUCHI . The NEXT experiment, searching for neutrinoless double-
beta decay, utilized CNN to identify electron-positron pair production from
the topological signature. Then the signal efficiency was improved compared to
non-CNN-based analysis NEXT . In our case, CNN would also be a promising
approach.

In this paper, we describe the design of CNNs to predict the scattering
position and the electron-recoil direction from track images taken by the TPC.
In addition, we evaluate the imaging performance of the simulation and
experiment data, and compare with a traditional way.

##  2 Electron-Tracking Compton Camera

The dominant interaction process of MeV gamma-rays and materials is the
Compton scattering. The schematic view of the Compton kinematics is described
in Fig. 1. The conventional Compton camera obtains the information of
scattering gamma-ray energy $E_{\gamma}$, the absorption position of
scattering gamma-ray $P_{\rm abs}$, the scattering position of gamma-ray
$P_{\rm sct}$, and the electron-recoil energy $K_{\rm e}$ other than the
electron-recoil direction $\vec{e}$. Then, the scattering angle $\phi$ is
written as follows,

$\cos\phi=1-\frac{m_{\rm e}c^{2}}{E_{\gamma}+K_{e}}\frac{K_{\rm
e}}{E_{\gamma}}.$ (1)

The conventional Compton cameras are insufficient to resolve the kinematic
equation of Compton scattering due to the lack of the electron-recoil
direction. While the incident gamma direction $\vec{s}$ is only reconstructed
as a Compton circle, the overlapping of many Compton circles determines the
source location. In the conventional Compton camera, the PSF is only defined
by the angular resolution measure (ARM). The ARM means the angular distance
from the reconstructed Compton circle to the known source location or the
determination accuracy of Compton scattering angle $\phi$. The ARM depends on
the determination accuracies of the absorption point and the scattering point
of gamma-ray, and the energy resolution.

COMPEL was incredibly successful via Compton imaging techniques. However, the
achieved sensitivity was modest. One of the most important things we learned
from COMPTEL is that the huge gamma-ray backgrounds are generated in the
satellite itself by cosmic rays. Under such backgrounds, the sensitivity is
overestimated if we only use the ARM. Schönfelder reported the conceivable
background in COMPTEL and concluded that the most sensitive tool to increase
the sensitivity is to reduce the instrumental background rate
SCHONFELDER2004193 . He also proposed the measurement of recoil electrons in
order to suppress the background. From this effort, we developed ETCC as the
next generation Compton telescope.

The ETCC provides all parameters of Compton scattering including the electron-
recoil direction by the tracking detector. Figure 1(a) shows a schematic view
of the ETCC on the SMILE-2+ experiment, which is our second balloon experiment
to observe the celestial objects at the high altitudes Tanimori2020 . The ETCC
completely resolves the Compton kinematics and determines the unique incident
gamma-ray direction $\vec{s}$ using the following equation,

$\vec{s}=\frac{E_{\gamma}}{E_{\gamma}+K_{\rm
e}}\vec{\gamma}+\frac{\sqrt{K_{\rm e}(K_{\rm e}+2m_{\rm
e}c^{2})}}{E_{\gamma}+K}\vec{e}.$ (2)

The ETCC gives the complete PSF from the ARM and the Scattering Plane
Deviation (SPD) TANIMORI2017 , which is the accuracy of the determining the
scattering plane. The ETCC can achieve high background-rejection power by
localizing the arrival direction on the Compton circle thanks to the SPD.

Figure 1: (a) Schematic view of the ETCC on the SMILE-2+ experiment. The
incident gamma-ray coming from the source location $\vec{r}_{\gamma}$
undergoes Compton scattering into the TPC filled with argon-based gas. The
energy, scattering position $P_{\rm sct}$ and direction $\vec{r}_{e}$ of the
recoil electron are reconstructed by the TPC based on $\mu$-PIC. Finally, the
scattering gamma-ray $\vec{r}_{\gamma^{\prime}}$ is absorbed at $P_{\rm abs}$
with PSAs. (b) Schematic image of the TOT. (c) Track images of recoil electron
with 160 keV from $\mu$-PIC anode and cathode strips.

The ETCC of the SMILE-2+ experiment consists of pixel scintillator arrays
(PSAs) and a gas TPC based on a micro pattern gas detector $\mu$-PIC Takada .
The 36 PSAs with a thickness of 26 mm and the 18 $\times$ 4 PSAs with a
thickness of 13 mm are arranged bottom and side of TPC, respectively. Each PSA
is made of GSO (Gd2SiO5:Ce) scintillators of 8 $\times$ 8 pixels with a pixel
size of 6 $\times$ 6 mm2. PSAs are deployed as absorbers to detect the energy
$E_{\gamma}$ and the position $P_{\rm abs}$ of the scattering gamma-ray. The
TPC is filled with argon-based gas (95% Ar + 3% CF4 \+ 2% C4H10 at a pressure
of 2 atm) as a Compton-scattering target and has a drift length of 30 cm. The
electron track and energy information was detected by $\mu$-PIC, which has 768
$\times$ 768 strips with a pitch of 400 $\mu$m; the detection volume 30
$\times$ 30 $\times$ 30 cm3. In order to reduce the power consumption, every
two strips of $\mu$-PIC are grouped. Thus, the readout strip pitch is 800
$\mu$m in the SMILE-2+ experiment.

The anode and cathode strips of $\mu$-PIC are linearly arranged in the X and Y
directions, respectively. The Z direction is reconstructed by the measured
drift velocity and drift time. Thus two-dimension images (XZ and YZ) are
obtained. The strip signals are processed in FE2009bal CMOS ASIC chips
MIZUMOTO201540  including preamplifiers, shapers, and comparators. Each
amplified signal is compared to the threshold voltage and synchronized with
100 MHz (10 ns) clocks. Finally, the addresses of $\mu$-PIC strips with the
time-over-threshold (TOT) are recorded. Figure 1(b) shows the schematic image
of the TOT. The TOT is the time between rising and falling edges crossing the
threshold voltage, which is correlated with the convolution of the charge
information and the track-length along the z-axis on each strip. Since the TOT
is measured by the comparator with 100 MHz, it is digitized with 10 ns. Figure
1(c) shows a typical example of the electron track obtained by $\mu$-PIC. The
length of gray pixels along the Z-axis on each strip is corresponding to the
TOT multiplied by the drift velocity. The low density scatter medium and the
fine interval read-out detector $\mu$-PIC provides fine track images. The
scattering position $P_{\rm sct}$ and electron-recoil direction $\vec{e}$ are
reconstructed from these two images.

The traditional method to determine the scattering position is to utilize the
skewness of TOTs, which relies on the fact that the stopping power of electron
recoil depends on their residual energy, and the end-point of the track has
large TOTs. The skewness is written by the following equation,

$S=\frac{\mu_{3}}{\mu_{2}^{3/2}}=\frac{\langle(x-\langle
x\rangle)^{3}\rangle}{\langle(x-\langle x\rangle)^{2}\rangle^{3/2}}.$

It is represented as the dimensionless ratio between the third and second
moments of the TOTs. The skewness provides the information on whether the
endpoint or the start point is on the right or left side of the image.
Consequently, the maximum or minimum of track image is adopted as a scattering
position. After calculating the scattering position, the recoil direction is
determined by the linear fitting algorithm using TOTs within 4 mm of the
reconstructed scattering point to prevent multiple scattering effects. This
method is suitable for relatively straight tracks, while it is less accurate
for curved tracks by the multiple scattering.

##  3 Convolutional Neural Network

CNN is one of the artificial neural networks. It has an input layer, hidden
layers, and an output layer. The hidden layer is generally composed of a
combination of the convolutional layer, the pooling layer, and a fully
connected layer. The convolutional layer, which is the most important element
in the CNN, automatically extract features from the image by a weighting
matrix, called a filter or kernel. The pooling layer is used to down-sampling
the feature maps extracted in the convolutional layer by taking an average or
maximum. In a fully connected layer, all nodes are connected to all
activations in the previous layer. A deep hierarchical structure of the
convolutional layer and pooling layer has expressive capacity; a shallow layer
learns low-level features or local correlations, and a deep layer learns high-
level features or global correlations. Thus, general CNNs are made up of
stacks of the convolutional layer and pooling layer.

We implemented two independent network models to predict the scattering
position and the electron-recoil direction using the Keras/Tensorflow
framework Tensorflow . The common point in both network models is that the
input layer takes two-channel images of 256 $\times$ 512 pixels corresponding
to 204.8 $\times$ 194.56 mm2 in XZ and YZ dimensions. Thus, the size of the
input parameter ($\phi$, $\cos\theta$) is 2 $\times$ 256 $\times$ 512
($=$262144). In this section, we describe the characteristics of these network
models.

###  3.1 Network model to predict the electron-recoil direction

In 2014, the Visual Geometry Group (VGG VGG ) model was received high praise
in the classification category of the image recognition competition. This
architecture simply increases the number of layers by connecting the
convolution layer and the fully connected layer. The feature is that the
reduction of the filter size enables to implement a deeper network and learn
high-level features. We have designed a network model based on the VGG to
predict the electron-recoil direction.

The schematic view of the network model is shown in Fig. 2. This model has two
output layers to predict the cosine of zenith angle $\cos\theta$ and azimuth
angle $\phi$ of electron recoil direction. The prediction of each angle is
divided into 36 classes, each having a width of 0.055 and 10 degrees,
respectively. Thus, the size of the output parameter is 2 $\times$ 36\. Output
layers are activated by softmax functions to compute each of the probability
distribution. The hidden layer is a hierarchical representation of
convolutional layers and pooling layers. All convolutional layers have a
filter size of 3 $\times$ 3 and stride 1. All pooling layers takes the maximum
value with the filter size of 2 $\times$ 2 and stride 2. In addition, three
dropout layers, in which some number of layer outputs are randomly ignored,
are deployed to prevent over-fitting. In the final stage, two-dimensional
feature maps are converted to a one-dimensional vector in flatten layer and
connected to the fully connected layer with 512 nodes. All layers except for
pooling layers and output layers are activated by the rectified linear unit
(ReLU ReLU ).

Figure 2: Network model to predict the electron-recoil direction. White and
green boxes are input and output layer, respectively. Orange boxes represent
convolution layers with stride 1, which are denoted as (filter size) Conv,
(number of filters). Blue boxes are maximum pooling layer with the filter size
of 2 $\times$ 2 and stride 2. Purple boxes are dropout layers with the rate
set to 20%. Pink boxes represent full connected or flatten layers.

###  3.2 Network model to predict the scattering position

The accuracy of the scattering position is expected to be the pixel size
resolution. While the VGG model is able to learn high-level feature with the
deep layer, the detail position information is lost at the same time as the
size of feature maps are reduced. For spatially dense predictions like
semantic segmentation, which is the process of separating image into some
object of interest with pixel-level, Fully Convolutional Networks (FCN FCN ),
and U-Net U-Net  were developed. These architectures have an encoder part
followed by a decoder part. The encoder part is the typical classifier network
like the VGG. The decoder part expands the feature map and projects onto high-
resolution images. In order to predict the scattering position, we built the
network model upon the U-Net architecture.

The schematic view of the network model is depicted in Fig. 3. In order to
predict the scattering positions ($x$, $y$, $z$), this model has three output
layers, which have linear activation functions. The size of the output
parameter is 3 $\times$ 1\. The scattering position is predicted on a
regression problem. The hidden layer is comprised of the encoder and decoder
part. The first encoder section extracts features at different scale sizes
with down-sampling as well as the network to predict the electron-recoil
direction. Convolutional layers have 3 $\times$ 3 kernesl with stride 1, and
pooling layers are maxpooling operation and stride 2. In the second decoder
section, pooling layers are replaced with unpooling layers as up-sampling and
expands feature maps. We deploy the convolution transpose layer as unpooling
operation with the filter size of 2 $\times$ 2 and stride 2. In addition, up-
sampling images are connected by down-sampling images in concatenation layers
and recovered the high spatial resolution. The final up-sampling layer is
flattened and connected to output layers. All layers except for pooling layers
and output layers are activated by the ReLU.

Figure 3: Network model to predict the scattering position. White and green
boxes are input and output layer, respectively. Orange boxes represent
convolution layers with stride 1, which are denoted as (filter size) Conv,
(number of filters). Blue boxes are maximum pooling layers with a filter size
of 2 $\times$ 2 and stride 2. Purple boxes are dropout layers with the rate
set to 20%. Yellow boxes are unpooling layers. Pink box describes a flatten
layer. Black dotted arrows indicate concatenation operation to combine the
output of the convolution layers in the encoder part to the decoder part.

##  4 Training and validating network via a monte calro simulation

###  4.1 Preparing a training and validating data set

The training and validating data are prepared by the Monte Carlo (MC)
simulation package of the SMILE experiment based on the Geant4 Geant4 . In
order to reproduce experimental images, the following procedures are taken.

  * •

The full detector geometry is constructed in Geant4. Electron beams are
generated in TPC volume with the uniform position and direction. Then, the
interaction points and the deposit energies are recorded.

  * •

From the deposit energies, the number of ionized electrons is calculated
considering the fano factor. In addition, the collected positions of ionized
electrons on $\mu$-PIC strips are obtained according to the diffusion effect
simulated by MAGBOLTZ MAGBOLTZ .

  * •

Waveforms of $\mu$-PIC strips are represented by overlapping waveform
templates, which are calculated from the simulated $\mu$-PIC pulse signal
using Garfiled++ Takada_2013 . These waveforms are digitized and TOTs are
obtained by comparing to the threshold. Finally, calculated TOTs are encoded
to the experimental data format.

We analyze simulated data and make XZ and YZ images of 256 $\times$ 512 pixels
so that the tracks are in the center of the images like Fig. 1(b). The 630k
and 70k MC events for 5–200 keV, which were fully contained in the TPC volume,
were generated for the training and validating data sets, respectively. Also,
each class has almost the same number of samples; one class has about 17500
samples for the training data set.

###  4.2 Training and validating network

We trained the networks and optimized learnable parameters by minimizing the
loss function between the ground truth and the prediction. Training the model
to predict the electron-recoil direction, the ground truth is expressed by the
one-hot vector $\vec{p}$. For example, the $i$-th class constitutes angles
$\phi$ ranging from $\pi i/18-\pi$ to $\pi(i+1)/18-\pi$. Then, the $i$-th
label is one “1” and the rest is zero “0” in the one-hot vector. We employed
the cross-entropy loss, which is a typical loss function in the classification
task, represented by $H(p,q)=-\Sigma_{i}p_{i}\log q_{i}$, where $p_{i}$ is the
$i$-th label of the ground truth and $q_{i}$ is the predicted one. On the
other hand, we employed the mean square root as a loss function to train the
model to predict the scattering position. In both cases, an Adam optimizer was
used to minimize loss function with a learning rate of 0.0001. A batch size of
4 was chosen. We trained models for ten epochs where validation losses
saturated. The huge amount of calculations was done by NVIDIA Geforce GTX 2070
super graphics processing units (GPU).

Figure 4 shows confusion matrices at the $\phi$ and $\cos\theta$ class
predictions of several energies.In 5–10 keV of $\phi$, CNN almost
misclassifies and outputs two indexes in any true $\phi$. We confirmed that
such a high bias prediction appears when the highest predicted probability is
very low. Therefore, the angle $\phi$ is determined by random uniformly in
such a case. As the highest predicted probability exceeds a certain threshold,
the predicted angle $\phi$ is calculated as follows:

$\phi={\rm
atan2}\left(\sum_{i=0}^{N_{c}-1}p_{i}\sin\phi_{i},\sum_{i=0}^{N_{c}-1}p_{i}\cos\phi_{i}\right),$
(3)

where $p_{i}$ is the predicted probability in $i$-th class, $\phi_{i}$ is the
center angle of the $i$-th class and $N_{c}$ is the number of class. The
correlation matrix between the calculated angle $\phi$ and the true angle
$\phi$ are shown in Fig. 5. The high bias prediction in 5–10 keV was reduced
due to the uniform random. On the other hand, some bias prediction or the
nonuniform angular response remains in 10–200 keV. In order to remove the
nonuniform response, adding the nonuniform penalty term to the loss function
is valuable as discussed in Ref. KITAGUCHI . We will deploy such an advanced
method in our future work. The angle $\theta$ is calculated by the same way.
The confusion matrices and correlation matrices of $\cos\theta$ is described
in the bottom panels of Fig. 4 and Fig. 5, respectively.

Figure 4: Confusion matrices at the $\phi$ (top panel) and $\cos\theta$
(bottom panel) class predictions.  Figure 5: Top panel shows the correlation
matrices between the calculated angle $\phi$ and the true angle $\phi$. Bottom
panel is the correlation matrices between the calculated $\cos\theta$ and the
true $\cos\theta$

We define the two evaluated values. First one is the angular error of
$\cos\theta_{\rm err}=\vec{V}_{t}\cdot\vec{V}_{p}$, where $\vec{V}_{t}$ and
$\vec{V}_{p}$ are the true recoil vector and the predicted recoil vector,
respectively. Second one is the position error of
$R_{err}=|\vec{P}_{t}-\vec{P}_{p}|$, where $\vec{P}_{t}$ and $\vec{P}_{p}$ are
the true scattering position and the predicted scattering position,
respectively. We obtained these distributions by calculating for 70k
validating data. The angular and position resolutions are defined by these 50%
area value. In the traditional method, these values are calculated in the same
way.

The several reconstructed examples on images are depicted in Fig. 6. Using the
traditional method, the high-energy recoil events like the bottom figures
could not reconstruct with high precision due to the multiple scattering
effect. On the other hand, the CNN succeeded in resolving the scattering point
and recoil direction for such events.

Figures 7(a) and (b) show the energy dependence of the angular resolution and
the position resolution comparing to the traditional method. The CNN method is
better than the traditional method above 20 keV. In particular, it achieved
excellent performance above 50 keV. The angular resolution of 75 keV is 41
degrees. For predicting the scattering position, the CNN method indicates
better performance in all energy. The position resolution of 75 keV is 2.1 mm.

Figure 6: Four track images with the predicted scattering positions and recoil
directions. Red points and arrows are true scattering points and recoil
directions. Blue squares and arrows are predictions using the CNN. Figure 7:
(a) Angular resolution of the CNN method comparing to the traditional method
using the simulation data. Red and blue points indicate the CNN method and the
traditional method, respectively. Black dashed line is a principle limit of
multiple scatterings with 0.8 mm pitch readout. (b) Position resolution of the
CNN method comparing to the traditional method. Red and blue point are the CNN
method and the traditional method, respectively.

##  5 Performance of ETCC using the CNN method

We conducted the calibration measurement of ETCC for the SMILE-2+ instrument
on the ground. In order to investigate several energy response, calibration
data of 133Ba (mainly 356 keV), 22Na (511 keV), 137Cs (662 keV) and 60Co (1173
keV and 1332 keV) were taken. In addition, the 137Cs source was placed on
several zenith positions separate to 183 cm from the center of ETCC. We
reanalyzed these calibration data with the CNN method and evaluated the
performance of the experimental ETCC data.

###  5.1 Event selection

In the calibration measurement, there are not only direct incident gamma-rays
from the source but also backgrounds of ambient gamma-rays and cosmic-rays.
Therefore, we adopted the following event selection to extract direct incident
gamma-rays from the sources.

  * •

1 Hit event : We require one track in the TPC and one hit pixel signal in the
PSA, which removed most accidental coincidence events.

  * •

Fiducial volume : We require that an electron track is contained within the
fiducial volume of $25.72\times 25.72\times 29.8$ cm3. At the boundary of the
detection area, the electric field is distorted due to the supply voltage of
the PMT of the PSA around the TPC. Such events which do not retain the
original track information are eliminated.

  * •

Fully contain electron : The $dE/dx$ distribution or the correlation between
the track-length and the energy deposit of the track in the TPC distinguishes
the fully contained electrons, escaped electrons from the TPC, and cosmic-ray
muons. The $dE/dx$ of the fully contained electron is empirically proportional
to $E^{1.72}$ Sauli  of the recoil electron energy. We determined the fully
contained electron band by fitting with Gaussian function every energy bin
with a width of 10 keV, and the region within the $\pm 3\sigma$ quantiles was
chosen. More details can be found in Ref Tanimori2015 .

  * •

Compton kinematic test : The angle between the recoil electron direction and
the scattering gamma direction is defined as Compton $\alpha$ angle and
geometrically calculated by the following equation,

$\cos\alpha_{\rm geo}=\vec{g}\cdot\vec{e}.$ (4)

In addition, this angle is derived by the Compton scattering kinematics:

$\cos\alpha_{\rm
kin}=\left(1-\frac{m_{e}c^{2}}{E_{\gamma}}\right)\sqrt{\frac{K_{e}}{K_{e}+2m_{e}c^{2}}}.$
(5)

Therefore, $\Delta\cos\alpha$ ($=\cos\alpha_{\rm geo}-\cos\alpha_{\rm kin}$)
nearly zero ensures that the reconstructed events are true Compton-scattering
events. We limited as $|\Delta\cos\alpha|<0.5$ in order to extract Compton
scattering events.

  * •

Energy selection : Events, which reconstructed energy is within two times of
FWHM for the source energy, are used to extract the direct components.

After the event selection, the effective area of experiment and simulation
data for 662 keV were $2.3\times 10^{-1}$ cm2 and $2.1\times 10^{-1}$ cm2,
respectively.

###  5.2 Imaging performance

The incident gamma-ray direction is reconstructed by Eq. (2). To evaluate the
reconstruction accuracy, we defined the error angle $\cos\theta$ between the
reconstructed vector and the source vector. The PSF is the angular resolution
of $\theta$. Thus, we calculated the half area value of the $\cos\theta$
distribution as the PSF. Figure 8 shows the ARM, SPD, and $\cos\theta$
distribution using the CNN method with the traditional method as the 137Cs
calibration source was set at the zenith=0. The ARM using the CNN method is 10
degrees and is not different from the traditional method. We confirmed that
the supply voltage of PSAs around TPC distorted the electric field, and it
dominated the uncertainty of the scattering position. On the other hand, the
SPD was highly improved thanks to reducing the angular resolution of the
recoil direction. We obtained 59 degrees (FWHM) of SPD by fitting Gaussian
function. This benefit gave the sharp $\cos\theta$ distribution and improved
the PSF to 15 degrees.

Figure 8:  (a) ARM, (b) SPD and (c) $\cos\theta$ distributions of the
experiment data. Red and Blue histograms, which are normalized by the total
areas, indicate the CNN and the traditional method, respectively.

Figure 9(a) and (b) show the dependencies of PSF on gamma-ray energy and the
incident zenith-angle, respectively. For every gamma-ray energy and incident
zenith-angle, the CNN method marked better performance than the traditional
method.

Figure 9: (a) Gamma-ray energy dependence and (b) incident zenith-angle
dependence of PSF. Red and blue lines indicate the experiment data using the
CNN method and the traditional method, respectively. Gray dots and triangles
are the simulation data using the CNN method and the traditional method,
respectively.

Reconstructed source images of 137Cs for several zenith angles are shown in
Fig. 10 using equal-solid-angle projection, such as Lambert projection. The
bright points representing sources were observed, and the more focused images
were confirmed in the CNN method.

Figure 10: Reconstructed source images of 137Cs set at zenith angles of 0, 30,
60, and 90 degrees for the experiment data. The top and bottom columns are the
CNN and traditional methods, respectively.

##  6 Discussion

Since the scattering position resolution is smaller than the PSA position
uncertainty, further improvement in scattering points cannot be expected to
improve PSF. On the other hand, SPD remains improvement potential. Although
the recoil direction was dramatically improved using CNN, the physical limit
of the multiple scattering has not been reached yet. As the angular
resolutions of the azimuth and the zenith angles were calculated separately
for 50 keV electrons at the 4 mm distance from the scattering points, 29
degrees and 35 degrees were obtained, respectively. This indicates that the
zenith angular resolution mainly determines a 3-D angular resolution. The
Z-direction position resolution has a width corresponding to the TOT. From a
rough calculation, the drift speed of 38 mm/$\mu$s and TOT width of 80 ns
corresponding to 0.5 pC input charge give 3.0 mm position resolution. It is
3.8 times worse than the 800 $\mu$m read-out pitch of X and Y directions. This
is the cause of smearing the track information for low energy electrons, as
can be seen from the image of 42.3 keV in the upper left of Fig. 6.
Improvement can be expected by using a short shaping time amplifier. Also,
waveforms taken by the flush ADC have the Z-direction information DRIFT .
Therefore, utilizing the waveform images in CNNs would be a good approach.
Regarding the azimuth direction, it is adequate to make the read-out pitch
more acceptable and increase the track information using such as the three
projections read-out system HexaSauli ; DUNE . We confirmed that the three-
projections read-out system with 480 $\mu$m pitch achieved 50 degrees of the
3-D angular resolution from the simulation.

In this research, a network based on VGG was used as a demonstration. However,
residual network ResNet  (ResNet) is mainly used to realize a deeper layer in
recent research. In the future, we will incorporate this technology to improve
the neural network aspect.

##  7 Conclusion

We designed two CNNs based on the VGG and U-Net models to predict the
scattering position and the electron-recoil direction from track images taken
by ETCC. The angular resolution and scattering position resolution utilizing
CNNs are obtained as 41 degrees and 2.1 mm for 75 keV in simulation data. As
we adopted the CNN analysis to experimental calibration data, the resulting
PSF was improved by 32% compared with the traditional analysis and achieved 15
degrees. The CNN analysis surpassed the traditional analysis, and such
improved ETCC has the potential to reveal the MeV gamma-ray sky.

## Acknowledgment

This work was partly supported by JSPS (Japan Society for the promotion of
Science) KAKENHI (Grant-in-Aids for Scientific Research) (grant nos.
20K20428).

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