Preamble

MuTEA: Muon Trajectory Estimate Algorithm for Muon Tomography
Ting Wang¹², Yu Wang¹²†, Shubin Liu¹²³, Zhihang Yao¹³, Gefeng Zhang¹², Zhiyong Zhang¹², and Changqing Feng¹²
¹State Key Laboratory of Particle Detection and Electronics, University of Science and Technology of China, Hefei 230026, China
²Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
³School of Nuclear Science and Technology, University of Science and Technology of China, Hefei 230026, China

Muon scattering tomography utilizes the penetrating ability of cosmic ray muons for non-destructive detection of materials inside shielding layers, with applications in geophysical exploration and homeland security. Trajectory reconstruction is essential for muon scattering tomography, as high-precision reconstruction of muon scattering trajectories enhances imaging quality and material discrimination. This study presents a muon trajectory estimate algorithm that reconstructs 3D scattering trajectories based on Coulomb scattering principles and Bayesian inference. This approach estimates the most probable trajectory of muons in materials, deriving accurate lateral displacements and scattering angles to improve imaging quality. Geant4 simulation validates the algorithm's effectiveness, indicating that trajectory reconstruction uncertainty is roughly positively correlated with the ratio of object thickness to radiation length. Furthermore, imaging experiments were conducted using a muon tomography instrument to image low-, medium-, and high-Z materials shielded by stainless steel. Two no-reference image quality assessment methods—gradient magnitude and contrast metric—were adopted for quantitative evaluation. Imaging results show that the trajectory-optimized PoCA algorithm can image and distinguish materials such as tungsten blocks, a hollow copper canister, and sand.

Keywords: muon trajectory estimate algorithm, muon tomography, image quality assessment

Introduction

The muon is a fundamental particle within the Standard Model of particle physics and possesses distinctive physical properties. It is a lepton with a rest mass of 105.7 MeV/c², approximately 207 times that of the electron. Although muons decay after approximately 2.2 µs, relativistic effects prolong the lifetime of high-energy muons, allowing them to reach the ground. At sea level, cosmic ray muons have an average energy of approximately 3 to 4 GeV \cite{reference1}. With their highly penetrating capability and non-destructive properties, muons serve as natural particle probes employed in material identification and imaging.

According to imaging principles, there are two main methods of muon imaging: transmission radiography and scattering tomography (MST). In transmission radiography, object reconstruction is achieved by measuring muon flux attenuation through materials. In 1955, Australian engineer E.P. George conducted transmission radiography experiments in a tunnel \cite{reference2}. By comparing the muon flux at the entrance to the tunnel with that inside, the density of the rock above the tunnel was successfully estimated. This technique has been applied in volcano monitoring \cite{reference3}, geological structure detection \cite{reference4}, and archaeological research \cite{reference5}.

Rather than relying on flux attenuation, MST reconstructs the structure of the material by measuring the scattering information between incident and outgoing trajectories. The Los Alamos National Laboratory (LANL) first introduced this method in 2003 and applied it to imaging of tungsten objects and iron scaffolds \cite{reference6}. Decision Sciences Corporation combined the response of muons and electrons to develop the Multi-Mode Passive Detection System (MMPDS) for customs and container security inspection \cite{reference7}. After the Fukushima accident, LANL and Toshiba used Monte Carlo simulation software to build a reactor model and reconstruct the reactor's internal cavities \cite{reference8}. Domestic research institutions have also made significant achievements in muon imaging. Tsinghua University pioneered muon imaging using a large-area MRPC (Multi-Gap Resistive Plate Chamber) system to image lead blocks \cite{reference9, reference10}. Lanzhou University developed a plastic scintillator-based muon imaging system and conducted research on high-Z material imaging and archaeogeophysics \cite{reference11, reference12}. The University of Science and Technology of China employed a high-resolution muon imaging system based on Micromegas, achieving imaging of centimeter-sized tungsten blocks and transmission imaging of ancient volcanoes \cite{reference13, reference14}.

The effectiveness of MST relies on imaging algorithms to reconstruct the internal structure of objects from muon scattering data. In recent years, research teams worldwide have made significant achievements in muon scattering tomography algorithms. For example, LANL proposed several imaging algorithms including the Point of Closest Approach (PoCA) \cite{reference15}, Maximum Likelihood Scattering and Displacement (MLSD) \cite{reference16}, Maximum A Posteriori (MAP) \cite{reference17}, and Gaussian Scale Mixture (GSM) \cite{reference18}. In addition, there are clustering algorithms for rapid imaging \cite{reference19}, two-step deflection proportion algorithms \cite{reference20}, deep learning optimization algorithms \cite{reference21}, and methods using muons and their secondary particles to aid imaging \cite{reference22}. Currently, the primary algorithms used for imaging are PoCA and MLSD, or improvements based on these two methods, which have demonstrated the feasibility of muon scattering tomography. However, most algorithms approximate muon trajectories within the object using piecewise linear segments during imaging. This approximation may cause bias in the reconstruction of scattering points. For example, when the PoCA algorithm reconstructs scattering points using piecewise linear segments, these points may be reconstructed outside the imaging region, resulting in the loss of valid events \cite{reference23}. The MLSD algorithm requires PoCA results as initial values \cite{reference24} and also relies on the same piecewise linear approximation, which can propagate reconstruction errors from PoCA. Estimating muon scattering trajectories can solve these issues and improve imaging accuracy.

To enhance imaging accuracy, this paper proposes a muon trajectory estimation algorithm (MuTEA) based on multiple Coulomb scattering. The algorithm uses prior knowledge of the muon incident and outgoing trajectories and establishes a probability distribution of the trajectories based on Bayesian inference to calculate the 3D most probable scattering trajectory. The estimated scattering trajectories can improve the calculation of equivalent scattering points and penetration paths within fine voxels to calculate the scattering density for image reconstruction. The effectiveness of the algorithm in reconstructing scattering trajectories was verified using Monte Carlo simulations. Then, a MuTEA-optimized PoCA algorithm was proposed and applied to a muon tomography instrument to image low-, medium-, and high-Z materials under stainless steel shielding.

II. Muon Scattering Tomography Algorithm

A. Principle of Muon Scattering

When cosmic ray muons pass through material, multiple Coulomb scattering occurs, resulting in an angular deflection between incident and outgoing trajectories. When projecting the scattering angle onto two orthogonal planes, the distribution of the projected scattering angle in each plane can be approximated as a normal distribution with a mean of 0 and a standard deviation of σ(θ) \cite{reference25}, where σ(θ) is given by Eq. (1):

$$
\sigma(\theta) = \frac{13.6 \text{ MeV}}{p\beta c} \sqrt{\frac{L}{L_{\text{rad}}}} \left[1 + 0.038 \ln\left(\frac{L}{L_{\text{rad}}}\right)\right]
$$

where $p$ is the muon momentum, $\beta c$ is the velocity of the muon, $L$ is the thickness of the material, and $L_{\text{rad}}$ is the radiation length of the material. $L_{\text{rad}}$ \cite{reference26} is related to the atomic number and mass number of a material, and can be expressed as Eq. (2):

$$
L_{\text{rad}} = 716.4 \frac{A}{Z(Z+1)} \ln\left(\frac{287}{\sqrt{Z}}\right) \text{ g·cm}^{-2}
$$

where $A$ is the atomic mass number and $Z$ is the atomic number. Eq. (2) can be approximated as $L_{\text{rad}} \propto Z^{-2}$. Thus, materials with a larger atomic number have a shorter radiation length. Therefore, under the same mass thickness condition, the standard deviation of the scattering angle distribution is approximately proportional to the atomic number of the target material. To better describe the relationship between the material and the scattering angle distribution, a quantity called scattering density $\lambda$ is defined \cite{reference27}:

$$
\lambda = \frac{\sigma(\theta)^2}{L} = \left(\frac{13.6 \text{ MeV}}{p\beta c}\right)^2 \frac{1}{L_{\text{rad}}}
$$

The scattering density quantifies the amount of muon scattering in a material and serves as the basis for image reconstruction and material identification in MST.

The PoCA algorithm is the most basic and widely adopted method in MST, offering the advantages of simple imaging principles and fast calculation speed. It simplifies the complex multiple Coulomb scattering interactions between muons and material into a single effective scattering event. The muon scattering trajectory is simplified as the extension of the incident trajectory to the scattering point and the extension from the scattering point toward the outgoing trajectory. Therefore, the algorithm has high computational efficiency but poor accuracy. The MLSD algorithm improves density inversion accuracy by statistically estimating the scattering-parameter distribution through Monte Carlo iterations. Nevertheless, it still employs piecewise-linear trajectory approximations that may accumulate errors and degrade imaging resolution. Most algorithms require scattering trajectories for density reconstruction, yet they rely on the simplified piecewise-linear trajectories provided by PoCA. Accurate reconstruction of muon scattering trajectories can upgrade these algorithms and enhance imaging resolution.

B. Muon Trajectory Estimate Algorithm

1. Theoretical Derivation

The overall scattering effect of muons in material is described by Eq. (1), and this relation remains valid when the material is divided into multiple layers. Therefore, when a muon enters an object's surface at a given direction, its exit direction and position are random variables that follow this distribution. With the measured exit direction and position as posterior information, Bayesian estimation is employed to estimate the probability distribution of the muon's trajectory within the imaging region. Based on this distribution, the most probable trajectory can be obtained.

To calculate the trajectory, a dedicated XOZ coordinate system is defined, where the muon incident direction is the positive direction of the Z-axis and the incident point is the origin, as shown in Fig. 1 [FIGURE:1]. The gray rectangles at the top and bottom of Fig. 1 represent detectors used to measure the muon trajectory. $MO(z_{mo}, x_{mo}, \theta_{mo})$ denotes the outgoing point coordinates, where $z_{mo}$ is the total thickness of the penetration, $x_{mo}$ is the total transverse displacement, and $\theta_{mo}$ is the cumulative deflection angle. $MD(z_{md}, x_{md}, \theta_{md})$ represents the coordinates of the middle point of the trajectory, describing the muon's state at the penetration depth $z_{md}$.

Let $P(MI(z_{mi}, x_{mi}, \theta_{mi}) \rightarrow MD(z_{md}, x_{md}, \theta_{md}))$ be the probability that the muon is incident from $MI$ and outgoing at $MD$ with angle $\theta_{md}$, hereafter simplified as $P(MI \rightarrow MD)$. Similarly, let $P(MD(z_{md}, x_{md}, \theta_{md}) \rightarrow MO(z_{mo}, x_{mo}, \theta_{mo}))$ be the probability that the muon is incident from $MD$ and outgoing at $MO$ with angle $\theta_{mo}$, hereafter simplified as $P(MD \rightarrow MO)$. The joint probability of the muon passing through $MD$, given that it is incident from $MI$ and outgoing from $MO$ with angle $\theta_{mo}$, can be expressed as $P(MI \rightarrow MD) \cdot P(MD \rightarrow MO)$.

According to the definition of conditional probability, the probability that the muon is incident from $MI$ and outgoing from $MO$ with angle $\theta_{mo}$, given that it has passed through $MD$, can be expressed as $P(MI \rightarrow MO | MD)$. On the other hand, the probability that the muon is incident from $MI$ and outgoing from $MO$ with angle $\theta_{mo}$ is $P(MI \rightarrow MO)$. Let $P_A = P(MI \rightarrow MD)$, $P_B = P(MI \rightarrow MO)$, and $P_{B|A} = P(MD \rightarrow MO)$. Then, the most probable muon scattering trajectory can be derived from the posterior probability according to Bayesian inference, as shown in Eq. (4):

$$
P(MI \rightarrow MO | MD) = \frac{P(MI \rightarrow MD) \cdot P(MD \rightarrow MO)}{P(MI \rightarrow MO)}
$$

For scattering trajectory reconstruction, the Gaussian approximation of the binary scattering distribution, as derived by Fermi and Rossi \cite{reference28}, is adopted as the probability density function (Eq. 5):

$$
F(z, x, \theta_{xoz}) = \frac{z^2}{\pi^{3/2}} \exp\left[-\omega^2\left(\frac{3x^2}{z^3} - \frac{3x\theta_{xoz}}{z^2} + \frac{\theta_{xoz}^2}{z}\right)\right]
$$

Here, $\omega = \frac{2p\beta}{E_s}$, where $E_s = 21.2$ MeV/c and $p$ is the muon momentum (MeV/c). The normalized penetration depth and lateral displacement are defined as $z = Z/X_0$ and $x = X/X_0$, respectively, where $Z$ is the material thickness, $X$ is the lateral displacement, and $X_0$ is the radiation length. For homogeneous materials, $z$ and $x$ are normalized by a constant $X_0$; for inhomogeneous cases, a radiation-length-weighted average is used. $\theta_{xoz}$ is the angle of deflection of the muon.

The trajectory probability distribution is then obtained by combining the probability density function with the above coordinate information:

$$
P(x_{md}, \theta_{md}) = \frac{z_{mo}^2}{\pi^{3/2}(z_{mo} - z_{md})^3 z_{md}^3} \exp\left(-\frac{a\theta_{md}^2 + b\theta_{md} + c}{z_{mo} - z_{md}}\right)
$$

where $a$, $b$, and $c$ are defined as follows:

$$
\begin{cases}
a = \omega^2(F + G) \
b = \omega^2(b_0 + b_1 x_{md}) = \omega^2\left(\theta_{mo}F - \frac{3x_{mo}F}{z_{mo}^2}\right) + \omega^2\left(\frac{3F}{z_{mo}^2} - \frac{3G}{z_{md}^2}\right)x_{md} \
c = \omega^2(c_0 + c_1 x_{md} + c_2 x_{md}^2) = \omega^2\left(\theta_{mo}F - \frac{3x_{mo}\theta_{mo}F}{z_{mo}^2} + \frac{3x_{mo}^2F}{z_{mo}^3}\right) + \omega^2\left(\frac{3\theta_{mo}F}{z_{mo}^2} - \frac{6x_{mo}F}{z_{mo}^3}\right)x_{md} + \omega^2\left(\frac{3F}{z_{mo}^3} + \frac{3G}{z_{md}^3}\right)x_{md}^2
\end{cases}
$$

where $F$, $G$, and $H$ are defined as:

$$
F = \frac{z_{mo} - z_{md}}{z_{mo}z_{md}}, \quad G = \frac{z_{md}}{z_{mo}(z_{mo} - z_{md})}, \quad H = \frac{z_{mo}}{z_{md}(z_{mo} - z_{md})}
$$

The optimal trajectory parameters are determined by maximizing the trajectory probability distribution function $P(x_{md}, \theta_{md})$. This involves calculating partial derivatives and then numerically solving the equations $\frac{\partial P}{\partial x_{md}} = 0$ and $\frac{\partial P}{\partial \theta_{md}} = 0$ to obtain the maximum likelihood estimates. The most probable scattering parameters (lateral displacement $x_{md}$ and angular deflection $\theta_{md}$) at different penetration depths $z_{md}$ are derived as follows:

$$
x_{md} = \frac{2ac_1 - b_0b_1}{1 - 4ac_2}, \quad \theta_{md} = \frac{2b_0c_2 - b_1c_0}{1 - 4ac_2}
$$

The most probable scattering trajectory in the XOZ plane is reconstructed by fitting $z_{md}$ to the corresponding $x_{md}$. Since muon transport in the XOZ and YOZ planes is statistically independent, the lateral displacement $y_{md}(z_{md})$ in the YOZ plane is solved in the same way. The 3D trajectory is then obtained by combining the results from both planes.

2. Scattering Density Reconstruction

The MuTEA accurately estimates 3D scattering trajectories that optimally connect the incident and outgoing trajectories. The reconstructed trajectories are smoother and more closely follow the expected Coulomb scattering curvature. By precisely determining the spatial positions and directions of muons, these improved scattering data can be integrated into imaging algorithms to reconstruct high-resolution images.

In this study, the reconstructed trajectory is employed to optimize the PoCA algorithm, improving both the precision of scattering point localization and the accuracy of scattering density reconstruction. In the absence of scattering, the muon follows the same trajectory as its direction of incidence. When scattering occurs, the muon is most likely deflected along the MuTEA-predicted trajectory before exiting along the observed outgoing direction. Consequently, the scattering point $\mathbf{r}s$ is determined as the position on $T$, formalized as:}}$ that minimizes the geometric distance to both $T_{\text{in}}$ and $T_{\text{out}

$$
\mathbf{r}s = \arg\min))} \in T_{\text{MuTEA}}} (d(\mathbf{r}, T_{\text{in}}) + d(\mathbf{r}, T_{\text{out}
$$

For voxel-based density reconstruction, the imaging region is divided into discrete voxels. The voxel containing the scattering point contributes to the scattering effect. Then, the scattering density $\lambda_j$ for the $j$-th voxel is calculated as:

$$
\lambda_j = \frac{1}{N_j} \sum_{i=1}^{N_j} \frac{\theta_i^2}{h_{i,j}}
$$

where $N_j$ is the total number of muon events passing through the $j$-th voxel, $\theta_i$ is the scattering angle of the $i$-th event within the voxel, and $h_{i,j}$ is the penetration depth of the muon in the voxel. For each labeled voxel, the penetration depth $h_{i,j}$ is calculated via line integration along the MuTEA-predicted trajectory:

$$
h_{i,j} = \int \sqrt{1 + \left(\frac{dx}{dz}\right)^2 + \left(\frac{dy}{dz}\right)^2}\,dz
$$

The scattering density is then calculated using Eq. (11) to complete the image reconstruction.

III. Performance Evaluation

A. Trajectory Uncertainty Analysis

To quantify the uncertainty inherent in estimated muon scattering trajectories, an error analysis is performed via the inverse of the local Fisher information matrix \cite{reference29}. First, the parameter estimates of the scattering trajectories are derived by maximizing the likelihood function, and the covariance matrix is calculated as the inverse of the Fisher information matrix. Subsequently, the diagonal elements of the covariance matrix are extracted to calculate standard errors, thereby obtaining the distribution width of the scattering trajectory errors $\sigma_{x_{md}}$, as shown in Eq. (15):

$$
\hat{I}(\hat{x}{md}, \hat{\theta}}) = \begin{pmatrix} 2c_2 & b_1 \ b_1 & 2a \end{pmatrix}, \quad \text{Cov}(\hat{x{md}, \hat{\theta}}) = \hat{I}^{-1}(\hat{x{md}, \hat{\theta})
$$

$$
\sigma_{x_{md}} = \sqrt{\text{Cov}(\hat{x}{md}, \hat{x}})} = \sqrt{\frac{2(z_{mo} - z_{md})^3 z_{md}^3}{3\omega^2 z_{mo}^3}
$$

The spatial uncertainty $\sigma_{x_{md}}$ in muon trajectory estimation depends on the normalized penetration depth $z_{md} = Z/X_0$, which combines two critical factors: the material thickness $Z$ and its radiation length $X_0$. Thicker material and smaller radiation length (materials that scatter muons more easily) both lead to greater uncertainty in the estimated scattering trajectory. Notably, the distribution width reaches its maximum when the muon normalized penetration depth satisfies $z_{md} = z_{mo}/2$ (the center of the object's thickness).

B. Simulation Setup

To evaluate the performance of MuTEA in reconstructing muon scattering trajectories, a comparison is made between muon trajectories simulated by Geant4 \cite{reference30} and those estimated by MuTEA. The Geant4 simulated muon detection system consists of upstream and downstream trajectory detection modules with an image reconstruction area in the middle, as shown in Fig. 2 [FIGURE:2]. The trajectory detection module consists of three ideal detectors, each measuring 15 cm × 15 cm with 5 cm spacing, using 5 mm thick argon as the sensitive layer. Given that both scattering trajectory reconstruction and its uncertainty analysis are independent of the scattering object area, the simulation focuses on the effects of object thickness and type. To obtain simulated muon scattering trajectories, the object is sliced along the Z-axis with 1 mm thickness and set as sensitive layers, recording muon positions as it passes through each layer.

In the simulation process, the ECOMUG generator is used to generate muons \cite{reference31}. For this study, a hemispherical surface with a radius of 20 cm serves as the muon generation plane. The generated muons have energies ranging from 0.1 to 1000 GeV. The QGSP_BERT physics model is adopted to accurately simulate high-energy muon interactions.

C. Muon Scattering Trajectory Reconstruction

In Geant4 simulation, a 15 cm × 15 cm × 5 cm tungsten block was used for trajectory acquisition. Fig. 3 [FIGURE:3] shows the MuTEA-reconstructed trajectories for coplanar (incident and outgoing trajectories in the same plane) and non-coplanar (trajectories in different planes) scenarios, respectively. The results verify that MuTEA can effectively connect the incident and outgoing trajectories in both scenarios, and the reconstructed curvature relatively matches the simulation.

Simulation results show that at the object's central thickness, the deviations between MuTEA-estimated lateral displacements and simulated data approximately follow Gaussian distributions, with standard deviations of 0.0473 mm in the XOZ plane and 0.0470 mm in the YOZ plane, as shown in Fig. 4 [FIGURE:4]. Hereafter, the standard deviation of the lateral displacement reconstruction bias is used to quantify the trajectory reconstruction error. Fig. 5 [FIGURE:5] presents trajectory reconstruction errors at different penetration depths, revealing a maximum width at the object's center, consistent with theoretical error analysis.

Impact of Object Thickness and Radiation Length on Trajectory Reconstruction

The MuTEA's reconstruction performance was evaluated through two critical parameters. First, simulations revealed that trajectory reconstruction error increases with tungsten thickness, as shown in Fig. 6 FIGURE:6. This is attributed to thicker objects leading to both longer scattering paths and larger cumulative multiple Coulomb scattering angles, resulting in greater uncertainty in the estimated scattering trajectory.

Secondly, comparative simulations of equal-thickness samples revealed that materials with smaller radiation lengths produce significantly broader trajectory distributions in the XOZ plane, as shown in Fig. 6 FIGURE:6. This confirms that scattering strength depends fundamentally on the material's radiation length. These results demonstrate that MuTEA's reconstruction accuracy follows predictable physical trends, with trajectory uncertainty roughly positively correlated with $Z/X_0$. The algorithm's performance is thus physically consistent, with reconstruction precision governed by both thickness and material (radiation length) properties of the target object.

Impact of Positional Uncertainty on Trajectory Reconstruction

The effectiveness of MuTEA in reconstructing scattering trajectories has been verified under ideal conditions. However, in practical experiments, factors such as the inherent spatial resolution of the detector and installation misalignment can lead to errors in the fitting of incident and outgoing muon trajectories. These errors can affect the accuracy of scattering trajectory reconstruction. To study the impact of positional uncertainty on trajectory reconstruction, different position measurement uncertainties are added to the obtained detector hit positions. Specifically, the hit position is modified as $x_{\text{hit}} = x_{\text{real}} + \Delta x$, where $x_{\text{real}}$ is the true hit position and $\Delta x$ is random error generated from a Gaussian distribution with a specified variance.

The trajectory reconstruction performance under different positional uncertainty conditions is shown in Fig. 7 [FIGURE:7]. As spatial resolution deteriorates, reconstruction error increases, with positional uncertainty becoming the dominant error source beyond 300 µm. Notably, the maximum reconstruction error shifts from the geometric center ($z_{md} = z_{mo}/2$) toward boundaries, while the error is minimal at the center. These results indicate that high spatial resolution systems (better than 300 µm) are essential for accurate scattering trajectory reconstruction, which can further improve the clarity of the reconstructed image.

IV. Scattering Tomography Experiment

A. Imaging Quality Evaluation Methods

To quantitatively evaluate the reconstructed imaging quality, two no-reference image quality assessment (NR-IQA) methods are adopted: gradient magnitude and contrast metric \cite{reference32}. Gradient magnitude describes the structural features of object boundaries, while the contrast metric enables material identification by comparing intensity (scattering density) differences between objects.

1. Gradient Magnitude

Gradient highlights edges and structural features by quantifying the rate of data variation, providing a basis for target identification. By applying the finite difference method, the gradient components are computed as partial derivatives along each dimension of the multidimensional array:

$$
\nabla f(x, y) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right), \quad G = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2}
$$

The gradient magnitude ($G$) directly reflects the strength of local changes—the higher the magnitude, the sharper the boundary. In this study, gradient magnitude is employed for edge detection to delineate targets within imaging regions. This method is particularly effective for detecting boundaries between different materials, as it highlights regions with significant intensity changes.

2. Contrast Metric

Contrast metric $\kappa$ quantifies the statistical differences between two regions in an image, reflecting their distinguishability. It is defined as the ratio of the difference between the mean intensities of the two regions to their combined standard deviation:

$$
\kappa = \frac{|\mu_1 - \mu_2|}{\sqrt{\sigma_1^2 + \sigma_2^2}}
$$

where $\mu_1, \mu_2$ and $\sigma_1, \sigma_2$ are the mean and standard deviation of the two regions, respectively. This indicator compares the signal difference (numerator) with the noise fluctuation (denominator). When $\kappa > 1.0$, the two regions are distinct and can be clearly differentiated; when $0.5 \leq \kappa \leq 1.0$, there is partial separability but reduced sensitivity to boundary details; when $\kappa < 0.5$, the two regions are similar and difficult to effectively distinguish.

B. Experimental Setup

The muon scattering tomography instrument developed in the authors' laboratory is shown in Fig. 8 [FIGURE:8]. The system consists of six layers of Micromegas detectors, six front-end cards (FECs), a data acquisition (DAQ) board, a host computer, and high-voltage modules.

The Micromegas detector used in this paper was designed in the authors' laboratory and manufactured using the thermal bonding method (TBM) \cite{reference34}. It is a dual-chamber detector that primarily consists of anode readout strips, two resistive electrodes, two mesh electrodes, two drift electrodes, and two gas chambers, as shown in Fig. 9 [FIGURE:9]. The detector is read out in both X and Y directions, where the X and Y strip layers are placed in the second and second-to-last layer of the PCB, respectively, providing better spatial consistency in two directions. The sensitive area of the detector is 600 mm × 600 mm. In the test, the mesh voltage and drift voltage are set to -580 V and -760 V, respectively. With this configuration, the gain of detectors is greater than $10^4$.

The FEC is designed based on the Asic for General Electronics for Time Projection Chambers (AGET) \cite{reference35} and applied to read out Micromegas detectors in muon imaging \cite{reference36}. The DAQ board is a generic back-end board designed in our laboratory for various applications \cite{reference37}. The entire system operates in self-trigger mode. A separate pre-trigger signal is generated for each detector layer FEE and transmitted to the DAQ. The DAQ makes a logical judgment on all the pre-trigger signals received to determine the valid muon trajectory. In this MST instrument, a global trigger is generated when a valid pre-trigger signal is reported in both the upstream and downstream trajectory modules at the same time.

In practical applications such as container inspection by scattering tomography, metallic shielding is common. To address this, this study evaluates imaging performance for low-, medium-, and high-density materials under stainless steel shielding (sizes: 40 cm × 40 cm × 0.5 cm). The three test specimens are described as follows:

• High-Z Material: Two objects—a 4-cm-thick Chinese character "zhong" composed of 2 cm tungsten cubes with two 4 cm × 4 cm gaps on both sides, and a 6 cm × 8 cm × 8 cm tungsten block constructed by stacking 2 cm tungsten cubes.
• Medium-Z Material: A hollow copper canister, 14 cm diameter × 14 cm height, 3 cm wall thickness, featuring an 8 cm diameter × 8 cm cylindrical cavity.
• Low-Z Material: A T-shaped structure composed of three sand-filled plastic canisters—two identical 20 cm × 12 cm × 12 cm side containers and one 23 cm × 12 cm × 12 cm central container.

The geometric configurations of these materials are detailed in Fig. 10 [FIGURE:10]. During the experiments, imaging tests were first performed on the tungsten cubes assembled into the shape of the Chinese character "zhong". Next, the larger tungsten block and the hollow copper canister were placed together on the detection platform. The sand-filled T-structure was tested independently. The location of the materials is shown in Fig. 11 [FIGURE:11].

C. Imaging Results

Data Filtering

In analyzing multiple Coulomb scattering, the overall angular distribution of scattering angles is commonly approximated by a Gaussian model. However, large-angle scattering events caused by single hard scatters exhibit a long-tail distribution that deviates from the Gaussian assumption \cite{reference38}. Additionally, non-Gaussian large-angle scattering events cannot be estimated by the trajectory calculation shown in Eq. (5). To reduce the effect of large-angle scattering events on scattering density estimation, data filtering is applied based on the plane scattering angle distribution. An angular threshold is set by generating a statistical histogram of plane scattering angles and using this as a reference. Data with scattering angles above this threshold are filtered out to improve accuracy.

1. Chinese Character "zhong" Composed of Tungsten Cubes

After accumulating 40 hours of experimental data, imaging results were reconstructed using the MuTEA-optimized PoCA algorithm. Fig. 12 [FIGURE:12] shows the results for the Chinese character "zhong" composed of tungsten cubes. The 2D gray image reveals distinct features corresponding to the structure of "zhong". The square shape of the stainless-steel shielding plane is also visible in Fig. 12(a). The XY-plane cross-section is taken at the center of the character and processed with threshold filtering and smooth interpolation. As shown in Fig. 12(b), the character "zhong" is clearly visible, with the two sides of the character appearing as rectangular regions. The size of the light-colored rectangular areas on both sides of the character is approximately 4 cm × 4 cm, which matches the actual size of the gaps in the character.

2. Hollow Copper Canister and Large Tungsten Block

After 40 hours of experimental data acquisition, Fig. 13 [FIGURE:13] presents the imaging results of the hollow copper canister and the 6 cm × 8 cm × 8 cm tungsten block. The 2D gray image shows circular and rectangular regions corresponding to the copper canister and tungsten block, respectively. To examine the hollow structure of the copper canister, an XY-plane cross-sectional view was taken at its center. Through threshold filtering and smoothed interpolation processing of the 2D XY cross-section, background noise was effectively reduced. As shown in Fig. 13(b), the copper canister displays a ring-shaped structure, with color scale values in the central region lower than those in the outer area. The measured hollow region has diameters of approximately 6 cm in both the X and Y directions.

The large tungsten block appears as a rectangular area in the image, with edge lengths slightly larger than the actual physical dimensions. The sizes of object regions in the color scale map match the actual objects. The minor deviations may be attributed to factors such as algorithmic and systematic errors, voxel discretization (square-shaped voxels), and positioning during imaging.

Fig. 13(c) presents the gradient magnitude map, which aligns with visual observations. The results demonstrate clear identification of the hollow region in the copper canister. Additionally, the $\kappa$ value is employed to quantify the separability between different regions: the large tungsten block area ($R_1$), the solid region of the copper canister ($R_2$), and the hollow region of the canister ($R_3$). The contrast results for each region are shown in Fig. 14 [FIGURE:14]. The $\kappa$ value between the large tungsten block ($R_1$) and the solid region of the copper canister ($R_2$) is measured at 2.59, indicating a significant density contrast that matches theoretical expectations. Meanwhile, the $\kappa$ value of 1.13 between the solid ($R_2$) and hollow ($R_3$) regions of the copper canister confirms successful differentiation between these regions.

3. T-shaped Sand-Filled Structure

After 64 hours of data acquisition, the T-shaped sand-filled structure was successfully reconstructed, as shown in Fig. 15 [FIGURE:15]. The 2D gray image results describe the T-shaped geometry, with an XY-plane cross-section taken at its center. After threshold filtering and smooth interpolation, the 2D XY section (Fig. 15(b)) clearly reveals the T-shaped structure. Notably, two square regions appear on either side of the top of the vertical line of the T-shape in the XY cross-section. The reconstructed structure was found to coincide approximately with the vertical aluminum columns that support the stainless-steel panels. Moreover, the sizes of the reconstructed T-shape closely match those of the physical object.

As shown in Fig. 15(c), the gradient magnitude map clearly outlines the edges of the T-shaped sand. Fig. 16 FIGURE:16 presents the contrast between the T-shaped sand ($R_1$) and the background ($R_2$), with a $\kappa$ value of 1.17 indicating successful differentiation. However, the $\kappa$ value between the T-shaped sand ($R_1$) and the aluminum profile columns ($R_3$) is measured at 0.34, as shown in Fig. 16(b). Because the radiation lengths of sand and aluminum are similar, and the aluminum area is not a solid structure, the results are reasonable. These results demonstrate the effectiveness of the MuTEA-optimized PoCA algorithm for discerning low-Z materials in metal-shielded scenarios.

V. Conclusion and Discussion

This paper presents a muon scattering trajectory estimation algorithm based on multiple Coulomb scattering. By using prior knowledge of the muon incident and outgoing trajectories, the algorithm establishes a probability distribution model based on Bayesian inference to calculate the most probable 3D scattering trajectory. The estimated scattering trajectories enable precise localization of voxel penetration trajectories and their corresponding lengths, while calculating new equivalent scattering points.

Geant4 simulations verify the validity of MuTEA for reconstructing muon trajectories. Simulation results show that reconstruction uncertainty is approximately proportional to the ratio of object thickness to radiation length. Meanwhile, an MST instrument was built to perform imaging experiments under stainless-steel shielding for low-, medium-, and high-Z materials. The imaging results demonstrate that the MuTEA-optimized PoCA algorithm accurately reconstructs the sand, the hollow copper canister, and the tungsten block, and further reveals the internal cavity inside the canister. Subsequently, two reference-free image quality assessment methods—gradient magnitude and contrast metric—were introduced to evaluate the reconstructed image quality. The metrics were used to describe material structure and to perform material differentiation, further demonstrating the algorithm's ability to reconstruct high-resolution images.

In future work, the 3D scattering trajectories estimated by MuTEA will be combined with other algorithms, such as the MLSD algorithm, to enhance image reconstruction. Furthermore, the present trajectory estimation algorithm is derived under the assumption of uniform materials. Future studies will improve this approximation to achieve more accurate trajectory estimation in non-uniform materials.

Acknowledgement

The authors would like to thank Anshun Zhou, Yong Zhou, Haibin Fei, Sicheng Wen, Xiaoqi Zhang, Han Han, Zhenhua Ye, and Meizi Jingluo at Jianwei Scientific Instruments (Anhui) Technology Corporation for their help with the manufacture of the Micromegas detector.

Author Contributions

All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Ting Wang, Yu Wang, Shubin Liu, Zhihang Yao, Gefeng Zhang, Zhiyong Zhang, and Changqing Feng. The first draft of the manuscript was written by Ting Wang, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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