organization=Department of Nuclear Engineering and Radiological Sciences,
University of Michigan, addressline=2355 Bonisteel Blvd, city=Ann Arbor,
postcode=MI 48109, country=United States

# Reconstruction of Fast Neutron Direction in
Segmented Organic Detectors using Deep Learning

Jun Woo Bae, Tingshiuan C. Wu and Igor Jovanovic

(August 2022)

###### Abstract

A method for reconstructing the direction of a fast neutron source using a
segmented organic scintillator-based detector and deep learning model is
proposed and analyzed. The model is based on recurrent neural network, which
can be trained by a sequence of data obtained from an event recorded in the
detector and suitably pre-processed. The performance of deep learning-based
model is compared with the conventional double-scatter detection algorithm in
reconstructing the direction of a fast neutron source. With the deep learning
model, the uncertainty in source direction of 0.301 rad is achieved with 100
neutron detection events in a segmented cubic organic scintillator detector
with a side length of 46 mm. To reconstruct the source direction with the same
angular resolution as the double-scatter algorithm, the deep learning method
requires 75% fewer events. Application of this method could augment the
operation of segmented detectors operated in the neutron scatter camera
configuration for applications such as special nuclear material detection.

##  1 Introduction

Directional detection of fast neutrons is a technique that has seen
significant attention in the search for special nuclear material (SNM) for
nuclear security and safeguards applications. For directional detection of
fast neutrons, a neutron scattering camera technique that employs segmented
detectors in two independent detection volumes [1, 2, 3], a single-volume
detector [4, 5, 6], or a time-projection chamber [7, 8] has been devised. The
method of double-scatter backprojection relies on the kinematics by recording
the energy and position of particles before and after scattering [9]. In an
organic scintillator, even if the light output-energy relationship is
precisely calibrated for recoil protons, the deposited energy does not always
directly relate to the recorded scintillation light output because of the
sequence of multiple scatters through which the energy can be deposited
combined with detector nonlinearity, along with the relatively poor energy
resolution [10]. For this reason, when a particle scatters in two separate
detection planes, a method of calculating the speed, that is, energy of a
neutron, uses the known distance and measured time-of-flight (ToF) between
those two planes [11]. In addition to the fact that the double-scatter events
are relatively rare because scatters must occur in both detection planes for
the same incident neutron, the reconstruction accuracy can be limited by the
achievable time resolution. In order to reconstruct the source direction, many
detector interactions are required, as most events typically produce only
single scatters in a compact detector. This motivates the development of a
directional detection method that could employ the information provided by a
greater fraction of all neutron interactions with the detector and not only
those that result in double scatter. This paper explores the application of
machine learning to take advantage of the information provided by the entire
range of neutron interactions and topologies recorded in a segmented detector
to reconstruct the incident neutron direction.

Machine learning is now used to solve a variety of problems in physical
sciences [12]. In particle physics, where massive amounts of data need to be
interpreted, the machine learning has already been recognized as a powerful
tool and is increasingly adopted [13]. Beyond traditional machine learning
models that utilize statistical correlation of parameters, deep learning
models that use neural networks have become more versatile and can be trained
using high-dimensional data or time-dependent sequential data [14]. In the
field of ionizing radiation measurement and high-energy physics, deep learning
has been used for nuclide identification [15] and particle classification
[16]. In these applications, artificial neural networks are used for
spectroscopic analysis, to detect rare events, and to examine the
correspondence between the experimental and simulated data by utilizing
generative adversarial networks.

In this study, we present the design and performance of a deep learning model
in reconstructing the direction of an isolated source of fast neutrons. The
model uses a series of detector signals that comprise events within a detector
volume that provides position sensitivity through detector segmentation. The
method arranges event information in the form of a sequence data and uses this
data to train a recurrent neural network-based deep learning model. The
training and test data are generated by simulation, and the performance of the
method is compared to the double-scatter backprojection method that uses the
same data.

##  2 Materials and Methods

###  2.1 Simulation Overview

A single or a series of interactions in a detector induced by a single
incident neutron constitutes an event. Each event carries a certain degree of
information related to the direction of the incident fast neutron. In order to
reconstruct the angle of incidence of fast neutrons, a series of signals can
be used that originate from recoil protons produced by neutron scattering in a
hydrogen-rich material such as an organic scintillator. The energy transfer to
recoil proton is modeled as a Monte Carlo particle transport simulation and
converted to light output such that it is proportional to $E^{3/2}$ [17].
Detector segmentation in 3 dimensions is used to achieve position resolution;
as a result, the following parameters can be associated with the scintillation
detector signal: light output, time, and position (which is discretized to a
detector segment). The detected signal is stored as sequential data according
to the chronological order in which the energy deposition occurs. The sequence
data have a length of 16 and contain 5 values: light output, time, and a
discrete position of the detector segment ($i,j,k)$. There are several cases
of interactions that produce ambiguous input as sequence data but occur
relatively infrequently. In the first case, multiple interactions occur in the
same detector segment. The duration of an event was set to 600 ns, which is of
the order of a typical digitized waveform for an organic scintillator combined
with a silicon photomultiplier. If interactions take place within this
interval, it is assumed that they cannot be distinguished in real-time data
processing. Therefore, the light output produced by the multiple interactions
in this case are added. The interaction beyond this time interval is rejected
and not stored in the sequence data. In the second case, interactions are
recorded to occur at the same time because of limited time resolution. In this
case, the interaction occurring closer to the face of the detector is stored
as the first in a sequence.

Figure 1: Illustration of the process of fast neutron direction reconstruction
using deep learning. The proton recoil represents a detector signal. The
signal is stored into a dataset, which is split into test and training
datasets. The model is pre-trained by the training dataset, and it is
validated with the test dataset.

Figure 1 shows the reconstruction process of fast neutron using deep learning.
A dataset is prepared using a standard radiation particle transport simulation
code, the GEANT4 (GEometry ANd Tracking) simulation toolkit [18]. We employed
a dataset with $1.3\times 10^{5}$ detection events resulting from $10^{6}$
incident fast neutrons; 80$\%$ of dataset is used for training of a deep
learning model, while the remainder is used as a test dataset. The full solid
angle seen by the detector is divided into 10 equal polar and 10 azimuthal
segments, such that each angular segment subtends the same azimuthal angle
($\phi$) and the same cosine of polar angle ($\cos{\theta}$). The performance
of the trained model is evaluated with randomly sampled events from the test
dataset.

###  2.2 Detector Geometry and Source Definition

Figure 2 shows the detector and source geometry used for the simulation. The
simulated detector is a 8 $\times$ 8 $\times$ 8 3-D segmented detector plastic
scintillator with a density of 1.023 g/cm3 and H:C ratio of 1.107 ; this data
is obtained from the GEANT4/NIST material database [19]. The detector material
and geometry are based on that used in Ref. [20], and the physical
segmentation is extended from 2 to 3 dimensions. Each cell of detector is a
cube with a side length of 5.4 mm, and the spacing among cells is 0.4 mm; the
entire detector assembly is a cube with side length of 46 mm. For this study,
a simple monoenergetic 2.45-MeV neutron spectrum is chosen. The source is
positioned as shown in Figure 2. The initial direction of the neutrons is
randomly sampled, such that the azimuthal and polar angles follow a uniform
distribution from $\phi\in[-\pi/2,\pi/2]$ and $\cos{\theta}\in[-1,1]$ First,
the source direction is set towards the center of the detector. The source
position is then shifted parallel to the set source direction, which avoids
directing the source only to the center of the detector at a specific angle.
Otherwise, the neutrons emitted at a specific angle would would consistently
interact with the same detector segment, leading to a bias between the source
direction and the position of interaction in the detector. The range of shift
is $\pm\ \sqrt{3}/2\cdot 46$ mm such that the entire field-of-view of the
cubic detector is covered. This corresponds to the condition usually referred
to as far-field detection.

Figure 2: Detector and source geometry used for the simulation. The direction
of a neutron is randomly sampled such that it is directed towards the center
of the detector, and its position is shifted parallel to this direction to
emulate far field detection.

###  2.3 Deep Learning Model and Training

Figure 2 shows the detector and source geometry used for the simulation. The
neutron incident direction is discretized such that directional reconstruction
can be treated as a classification problem. Earlier studies that attempt to
reconstruct the neutron direction using the backprojection algorithm show a
typical angular resolution of $\sim$0.14 rad [21, 22, 23], which corresponds
to 22 bins over a $1-\pi$ angular range. However, this makes the number of
events assigned to each bin small, increasing the computational cost. For this
reason, discretization is performed into 10 bins for $\cos(\theta)$ and 10
bins for $\phi$, so that there is a total of 100 possible incident directions.
In the test dataset, the source direction is reconstructed by classifying it
into one of the discrete angular segments.

From the dataset, we find that a 2.45-MeV neutron had an average of $1.30\pm
0.58$ and a maximum of 7 interactions with the simulated detector. The length
of the sequence frame was therefore set to 16 to ensure a full set of
interactions can be stored while limiting the computational cost. Since the
input data contain 5 parameters, as described in Section 2.1, the dimension of
input data is 16 $\times$ 5.

The recurrent neural network (RNN) is a type of model that can receive a
sequential input and learn the dependencies between values in a sequence. This
characteristic makes it suitable for capturing the relevant physics contained
in the correlations between individual interactions of the interaction
sequence comprising an event. Figure 3 shows a configuration of the model
designed for this study. The input layer is followed by two long short-term
memory (LSTM) layers and a fully connected recurrent neural network (FCRNN)
layer [24]. The fully connected layer is followed by a series of RNNs. The
number of output nodes of the FCRNN layer is identical to the total number of
discrete angular segments (S).

Figure 3: Design schematic of the RNN-based neural network model. The model
takes sequence data with a length of 16. The model consists of two LSTM
layers, one FCRNN, and fully connected layers.

The model calculates the normalized probability of classification in the
corresponding angular segment, such that the sum of the probabilities over all
segments is unity. The source direction $(\phi,\cos\theta)$ is reconstructed
by selecting the maximum probable angular segment.

$\phi_{r}=\phi_{i},\;\;\theta_{r}=\cos^{-1}\left(\cos\theta_{i}\right),\;\;\\{i|i\in
P^{(\phi,\theta)}(i)=max(P^{(\phi,\theta)}(i))\\}.$ (1)

Here, $P^{(\phi,\theta)}(i)$ is the probability of reconstructing the $i$-th
direction segment, $\phi_{r}$ and $\theta_{r}$ are the reconstructed azimuthal
and polar angles, and $\phi_{i}$ and $\cos\theta_{i}$ are the azimuthal angle
and cosine polar angle that correspond to the $i$-th angular segment. The
angular accuracy ($\psi$) is defined as the angle between two unit vectors
representing the true source direction ($\vec{v}_{t}$) and the reconstructed
source direction ($\vec{v}_{r}$):

$\psi=\cos^{-1}(1-d^{2}/2),$ (2) $d^{2}=|\vec{v}_{t}-\vec{v}_{r}|^{2}.$ (3)

###  2.4 Event Updating and Comparison with the Double-Scatter Method

The deep learning-based method is compared with the double-scatter
backprojection method that uses the ToF and distance between scattering events
[6]. The same detector segmented design and simulated event data are used for
the comparison. The position of an interaction is assumed to be in the center
of a scintillator where it occurs. Bayesian inference is used to update the
probability:

$P^{(n+1)}(i)=\frac{P^{(*)}(i)\cdot P^{(n)}(i)}{\sum_{i}^{S}(P^{(*)}(i)\cdot
P^{(n)}(i))},$ (4)

where $P^{(n+1)}(i)$, $P_{i}^{(n)}(i)$ and $P_{i}^{(*)}$ represent
probabilities of i-th angular segment for the $(n+1)$-th, $n$-th event update,
and the probability of reconstructing the $i$-th angular segment after the
$n$-th event. In the double-scatter method, the angular cones are calculated
as‘[11]

$\theta_{DS}=\tan^{-1}\sqrt{E_{p}/E_{n}},$ (5)
$E_{n}=\frac{m}{2}\left(\frac{\sqrt{\Delta x^{2}+\Delta y^{2}+\Delta
z^{2}}\cdot l}{\tau}\right)^{2}.$ (6)

Here, $\theta_{D}S$ is the angle between incident neutron and scattered
neutron that defines the angular cone, $E_{p}$ and $E_{n}$ are energies of
recoil proton and scattered neutron, $m$ is neutron mass, $\Delta x$, $\Delta
y$, $\Delta z$ are number of segments between two scattered detector cell in
direction of $x$, $y$, and $z$, $l$ is a side length of a detector cell, and
$\tau$ is the time of flight between the detector cells in which double
scatter takes place. The spatial distance between the two scatter signals is
calculated as described in Eq. (6), assuming that the signal is generated at
the center of the detector cell. The discrete spatial information leads to
over- or under-estimation of the energy of scattered neutron as well as error
of the vector between two scatter signals. This is an intrinsic limitation in
the application of the double scatter method in a detector that has a
segmented volume.

In the RNN method, each event is used to update the reconstructed source
direction, whereas only double-scatter events lead to updating the source
direction in the case of double-scatter backprojection method. Event-time
smearing corresponds to the time resolution of 0, 0.2, and 0.5 ns and is
applied and light output signals are cut off with threshold of 0, 0.1, 0.2
MeVee to make the simulation more realistic.

##  3 Results and Discussion

Figure 4 shows an example of reconstructed source direction using the deep
learning model. The probability for each angular segment follows a set color
scale, while the red box represents the true source direction that corresponds
to $-0.6<\cos\theta<-0.8$ and $1.26<\phi<1.57$ rad, respectively. As the
number of events increases from 1 to 48, the maximum probability increases
from 0.02 to 0.8. The probability distribution is relatively broad for few
events and shrinks with accumulated statistics.

Figure 4: Reconstructed direction of the source neutron using the deep
learning model for different numbers of events. The red box indicates the
angular segment corresponding to the true direction.

Figure 5 demonstrates the performance of the RNN model for several incident
neutron directions based on 100 events from the test dataset. The red box in
each figure represents the true direction, and the corresponding $\cos\theta$
and $\phi$ are presented at the top of each figure. The reconstructed
direction matches the true direction.

Figure 5: Reconstructed direction using deep learning model with different
source directions based on 100 events. The red box and the angular coordinates
provided at the top of each panel correspond to the true source direction.

Figure 6 shows the distribution of reconstructed directions for two cases: (1)
isotropic case, where the source has no preferred direction (background), and
(2) a source placed at a single direction. In the case of an isotropic source,
there is no clear preference for reconstructed direction. In contrast, the
reconstruction is consistent in the case of the source emitting from a single
direction (segment 25). Small peaks are also shown at segments 15 and 35,
which are adjacent to the polar angle of the true segment. The Gaussian fit of
the peak shows a mean of 24.8 and a full-width half maximum of 1.16. This
demonstrates that the trained deep learning model can clearly distinguish
between a point source and an isotropic source (background).

Figure 6: Distribution of reconstructed direction for 1000 trials. The source
angle is randomly chosen to emulate background (black), while it is fixed at a
single direction for the signal (segment 25 – red). Each trial uses 100
events.

Figure 7 shows the comparison of reconstructed source direction using the deep
learning model and the double-scatter method, including the evolution of
angular accuracy with event statistics. In this example, the true source
direction of $0<\cos{\theta}<0.2$ and $0.628<\phi<0.942$ rad, which is
displayed as the red box. The angular accuracy in Figure 7 (c) changes
relatively frequently for the deep learning model because it is updated with
every event. In contrast, the double-scatter backprojection method leads to
step-wise accuracy updates because only a subset of the detected events
include double scatters. In the method using the deep learning model, all
events can be updated without waiting for the double scatter event, therefore
the detection efficiency can be improved, which can improve the sensitivity
when used for the detection of special nuclear materials or radioactive
isotopes. It can also be seen that the change in angular accuracy occurs at
the same time because both methods update the same event at the same time.
When there is a double-scatter event, both the accuracies of the double-
scatter method and the deep learning model experience a large change. While a
single-scattering event makes a small, incremental change to the reconstructed
angle in the deep learning model, a double-scatter event has a much greater
impact.

Figure 7: Reconstructed source direction (a) employing the deep learning model
and (b) the double scatter backprojection method. The red box represents the
true direction. (c) Evolution of angular accuracy for different numbers of
events. Input data were randomly sampled from all events in the test dataset.

Figure 8 compares the angular accuracy obtained from the deep learning model
and the double-scatter backprojection method. Figure 8 and (b) shows a
distribution of angular accuracy for 100 events for the deep learning model
and the double-scatter backprojection algorithm, respectively. The deep
learning model shows a better (less dispersed) angular accuracy. In Figure 8
(c), the angular accuracy is averaged over all angular segments. The angular
accuracy is different for different angular segments, which is why the average
was calculated to evaluate the overall trend for each method. The error bar
represents one standard error among angular segments. The mean angular
accuracies for the deep learning model and double-scatter backprojection
method for 100 events are 0.301 $\pm$ 0.327 and 0.557 $\pm$ 0.571 rad,
respectively. The deep learning model reaches the accuracy of the double-
scatter backprojection method with $\sim$20 events.

The miniTimeCube-based neutron scatter camera [22] reports the angular
difference of ($\theta$, $\phi$)(0.1 and 0.13) rad with 10,093 neutron events
by averaging direction vectors of first two recoils. The iSANDD neutron
scatter camera [23] reports $\cos\theta=0.99$, which corresponds to the
angular difference of 0.14 rad with 5$\times$106 2.45 MeV neutrons by
reconstructing angular cone using the double-scatter backprojection method.
Because the number of events is different, quantitative comparison is not
possible, but the results show that the use of the deep learning model can
lead to good accuracy with relatively few ($\sim$100) events.

Figure 8: Mean angular accuracy over all angular segments when using (a) deep
learning model and (b) double-scatter backprojection method. (c) The evolution
of mean angular accuracy is averaged over all angular segments for the deep
learning model and the double-scatter backprojection method.

Figure 9 shows the effect on the time resolution and the light output
threshold on the evolution of angular accuracy with event statistics. The true
source is $0<\cos{\theta}<0.2$ and $0.628<\phi<0.942$ rad which is same
direction used in Figure 7. The solid lines represent a set of 100 events; the
same events are used for all depicted cases. The dashed lines represent a mean
of 10 repetitions; 100 events are randomly sampled for each repetition. In the
case of the deep learning model, 9 independent models are trained using the
data that is processed for each condition. Therefore, the evolution of angular
accuracy seems to vary a lot depending on the conditions. In case of the
double-scatter backprojection method, the time resolution affects the $\tau$,
which affects the estimation of $\theta_{DS}$. On the other hand, the light
output threshold can affect the event, which leads to decrease the number of
double scatter events or misclassification of double-scatter event by cutting
off signals with small light output. Therefore the time resolution primarily
affects the angular accuracy derived from a single event, while the light
output threshold affects the evolution of the angular accuracy by reducing the
number of events from which the information can be extracted. According to the
average trend, in the double-scatter backprojection method, when a higher
light output threshold and worse time resolution are applied, the angular
accuracy is worse. However, in case of deep learning model, this is not always
the case, and there are instances where the reconstruction is better even with
high light output threshold and time resolution. This may be explained by the
fact that deep learning models are trained by the datasets to which these
specific conditions apply.

Figure 9: Evolution of angular accuracy using the deep learning model and the
double-scatter backprojection method for several different time resolutions.
All reconstructions use the same event dataset.

##  4 Conclusion

In this paper, we investigated the performance of an RNN-based deep learning
method in reconstructing the direction of fast neutrons incident onto a
segmented scintillation detector. The performance was studied using event data
generated through GEANT4 simulation. Data included a series of recoil proton
signals generated by an incident fast neutron source, and they were stored as
a sequence. The light output thresholds and time resolution was applied to
make the data more realistic. To train the model and perform reconstruction
from the sequential data,a RNN-based neural network architecture was designed
as a deep learning model. A double-scatter method for estimating direction of
fast neutron was utilized and compared with the proposed deep learning-based
method as a benchmark. The deep learning-based method shows a better angular
accuracy, or equivalently requires fewer events to reach the set accuracy.
While the single and multiple scatters are discarded in the double-scatter
backprojection method, they can provide useful information in case of the deep
learning-based method. It is expected that the model can reconstruct the
direction of the source even if some of signals are removed by the neutron-
gamma discrimination algorithm or by a pulse-height event threshold. This
study is still limited in that it uses only simulated data; significantly more
insight will be provided when the model is trained and evaluated with real
experimental data in the future.

##  5 Acknowledgments

This work was supported by the Department of Energy National Nuclear Security
Administration, Consortium for Monitoring, Verification and Technology (DE-
NE000863), Nuclear Global Fellowship Program through the Korea Nuclear
International Cooperation Foundation(KONICOF) funded by the Ministry of
Science and ICT, and partially supported by the Department of Energy, Nuclear
Energy University Program Fellowship.

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