Preamble

Geo-EM-ML: Geometrically Constrained EM-ML Algorithm for Multi-Source Localization in Pipeline Systems

Zijia Kuang¹, Wei Wang¹, Yuanjie Bi²,†, Rui Qiu³,⁴,‡, Chuangao Wang¹, and Senlin Liu¹

¹Institute of Nuclear Safety and Environmental Engineering Technology, China Institute of Atomic Energy, Beijing 102413, China
²School of Science, Shenzhen Campus of Sun Yat-sen University, Shenzhen 518107, China
³Department of Engineering Physics, Tsinghua University, Beijing 100084, China
⁴Key Laboratory of Particle and Radiation Imaging of Ministry of Education, Beijing 100084, China

In nuclear facility maintenance, accurately locating multiple radiation sources within complex pipeline systems is paramount for ensuring the safety of maintenance personnel and optimizing work routes to minimize radiation exposure. However, existing localization methods often rely on two-dimensional grid spaces or are limited to a few sources, and they frequently neglect the attenuation effects caused by pipeline materials, leading to reduced accuracy and increased computational complexity. This study introduces the Geo-EM-ML algorithm, a novel hybrid approach that integrates the Expectation-Maximization (EM) algorithm and Maximum Likelihood Estimation (MLE), enhanced by geometric constraints tailored to the three-dimensional continuous space of pipeline systems. Additionally, a ray tracing program is employed to model the attenuation effects accurately, ensuring reliable detector response calculations. Experimental results demonstrate that Geo-EM-ML achieves a high success rate exceeding 94% in scenarios with up to six radiation hotspots, maintaining average position error and relative activity error below 8.58 mm and 4.31%, respectively. The algorithm exhibits robustness across varying pipeline shielding materials, wall thicknesses, source intensities, and detector configurations.

The Geo-EM-ML algorithm represents a significant advancement in multi-source localization, offering a scalable and precise solution for complex pipeline environments in nuclear facilities, thereby mitigating safety risks and optimizing maintenance workflows.

Keywords: Multiple sources localization, Maximum Likelihood Estimation, Expectation-Maximization Algorithm, Shielding scenario

Introduction

In nuclear facilities, pipeline systems transporting radioactive liquids often develop radiation hotspots where residual deposits accumulate in valves, bends, and other complex geometries due to factors like adsorption, corrosion, and surface roughness. These areas can remain persistently radioactive even after standard flushing. Such unidentified or poorly mapped hotspots present a significant threat to maintenance and experimental personnel, as a lack of precise knowledge of their locations and activity levels prevents effective work planning, resulting in increased radiation exposure. Therefore, localizing and characterizing these radiation sources is crucial for ensuring occupational safety and optimizing radiation protection measures.

Currently, there are three methods for obtaining radiation hotspot information: (1) estimation based on staff experience, (2) direct measurement near the pipeline or sampling from the pipeline, and (3) measurement using directional detectors (e.g., gamma cameras, Compton cameras) \cite{1,2,3,4,5}. The first method can lead to significant deviations due to the subjective nature of estimates. The second method, while more direct, exposes personnel to potentially high radiation doses. Additionally, due to spatial constraints, many locations may be inaccessible to close-range detectors. The third method using directional detectors has the disadvantages of long imaging time and limited field of view. Therefore, developing a multi-source localization algorithm based on non-directional detectors helps workers locate radiation sources quickly and accurately.

Multi-source localization is an inverse problem that aims to determine the location and activity of sources based on the count data collected by a group of detectors positioned around the sources. In contrast to the search for a lost single source in nuclear security scenarios \cite{6,7,8,9,10,11,12,13,14,15,16,17,18}, multi-source localization algorithms are inherently more complex and challenging to develop.

Currently, several algorithms are used to solve the multi-source localization problem, mainly including least squares estimation, maximum likelihood estimation/expectation maximization, Bayesian estimation, and neural network.

a. Least Squares Estimation: Chen et al. proposed a multi-source localization algorithm based on the least squares method, which can locate up to two radiation sources in a three-dimensional environment. They designed a screening mechanism to determine whether the detector participates in the calculation through geometric relationships, which improves the localization accuracy and algorithm robustness in the presence of shielding \cite{19}.

b. Maximum Likelihood Estimation/Expectation Maximization: Deb et al. implemented the mixed use of directional and non-directional detectors under the expectation-maximization framework so that the radiation source localization work can consider both cost and efficiency \cite{20}. Deb modeled the multi-source localization problem as an optimization problem of a high-dimensional function and solved the maximum likelihood estimation (MLE) through Fisher score iteration. Since the multi-source localization problem is a non-convex optimization problem, finding a reasonable initial estimate is an inherent need. Deb proposed an expectation-maximization-based algorithm to find the approximate distribution of the source intensity in space. Then, the local maxima of the spatial distribution are identified as the initial estimates for the maximum likelihood estimation \cite{21}. Hellfeld et al. improved the expectation-maximization algorithm and proposed the Point Source Localization (PSL) algorithm. The expectation-maximization algorithm usually divides the space into grids and estimates the radiation source activity at each grid point. The Point Source Localization (PSL) algorithm treats the position and radiation source activity as continuous variables. The PSL algorithm is suitable for the localization of a single radiation source. To solve the multi-source localization problem, Hellfeld et al. proposed the Additive Point Source Localization (APSL) algorithm. The Additive Point Source Localization (APSL) algorithm estimates the number of radiation sources by gradually adding new radiation sources and uses the Bayesian information criterion as the stopping criterion, thereby improving the robustness of the algorithm \cite{22,23}. Bandstra et al. improved the Point Source Localization (PSL) algorithm to enable it to estimate the position of radiation sources accurately in the presence of shielding. They used LiDAR to build a voxel model of the scene and calculated the shielding effect through a ray tracing algorithm. They regarded the entire scene as composed of the same material and added the linear attenuation coefficient of this material as an estimated parameter to the PSL algorithm, thereby achieving better localization accuracy in practical tests than the original PSL algorithm \cite{24}. Abdelhakim proposed heuristic techniques to enhance the efficiency of maximum likelihood estimation for localizing radioactive sources \cite{25}.

c. Bayesian Estimation: Morelande et al. proposed a Bayesian estimation framework that transformed the task of estimating radiation source parameters into a model selection problem and utilized partial Bayes factors (PBF) to compare the evidence of different models for determining the number of radiation sources. Their work utilized the sequential Monte Carlo method, especially by introducing the progressively corrected importance sampling (PCIS) technique, effectively approximating the posterior distribution of radiation count data collected by detectors when multiple radiation sources exist \cite{26,27,28,29}. Chin et al. proposed an efficient Bayesian estimation framework combining the particle filter and mean shift technique. Their proposed Bayesian framework ensures constant computational complexity, accounts for the influence of shielding, and enables parallel processing, making it applicable to large-scale sensor networks \cite{30}. Yee et al. compared two Bayesian inferential approaches for detecting and estimating radioactive point sources: one using importance sampling with progressive correction and another using reversible-jump Markov chain Monte Carlo sampling. The methods differ in their handling of measurement models and background radiation, with experimental data demonstrating the effectiveness of both approaches \cite{31}. Anderson et al. proposed a new method for radiation measurement using mobile robots, which can accurately locate and characterize radioactive sources in real scenarios containing multiple radiation sources and complex environments through recursive Bayesian estimation combined with attenuation modeling. In addition, their system implemented autonomous isotope identification and measurement point optimization based on Fisher information, significantly improving measurement efficiency and accuracy \cite{32}. Lazna and Zalud developed a method for localizing multiple radiation sources using a particle filter. Their approach employs an autonomous mobile robot for information-driven data collection that switches between minimizing Shannon entropy and reducing measurement variance in unexplored areas, reducing exploration time by approximately 40% while maintaining safe clearance from obstacles \cite{33}.

d. Neural Network: The multi-source localization methods based on neural networks are mainly divided into recurrent and feedforward neural networks. Wacholder et al. modeled the multi-source localization problem as a combinatorial optimization problem and solved it using the Hopfield recurrent neural network. The Hopfield recurrent neural network does not require pre-training of network parameters but updates the state of neurons iteratively to minimize the energy function and obtain the optimal solution \cite{34}. Mendes et al. transformed the multi-source localization problem in two-dimensional space into an object detection problem and detected the possible positions of radiation sources from the radiation distribution map using a pre-trained convolutional neural network \cite{35}. Hu et al. developed a method for multi-unknown gamma radiation source localization using convolutional neural networks (CNN) and a self-avoiding walk (SAW) algorithm. Their approach simulated energy deposition distributions of multiple gamma radiation sources using Geant4, collected energy deposition values in a region using the SAW algorithm to build training datasets, and then trained a CNN to determine radiation source positions. This method can locate up to three radiation sources with at least 86% accuracy, even in complex environments with obstacles creating shielding effects \cite{36}. Abdelhakim proposed a feature extraction technique to convert the position information and count data of the detectors into feature vectors and then used the decision tree regression algorithm to learn how to predict the activity and position of the radiation sources from the feature vectors \cite{37}. Hao et al. designed a Source Distribution Inversion Convolutional Neural Network (SDICNN) to obtain the distribution information of complex source terms from radiation parameters in space. The SDICNN comprises a fully connected network (FCN) and a convolutional neural network (CNN). The FCN obtains low-resolution source distribution parameters from a single sampling point in the radiation field, and the CNN uses a structure similar to the super-resolution CNN (SRCNN) to complete the delicate reconstruction of the source distribution \cite{38}. Okabe et al. designed a detector pixel layout inspired by Tetris and increased the response contrast between pixels by adding inter-pixel padding materials between pixels. They used a neural network composed of filtering layers and a deep U-net to predict the direction of radiation sources. Based on the predicted direction and the incident radiation intensity, they used posterior probability maximization to estimate the position of the radiation source \cite{39}.

e. Other Methods: Hautot et al. applied Visual Simultaneous Localization and Mapping (VSLAM) methods to indoor 3D topographical and radiological mapping. Their system combines robotics and computer vision sensors to allow fast real-time localization of radiological measurements in space with near real-time source identification and characterization, improving operators' dosimetry evaluation and intervention scenarios in potentially contaminated areas \cite{40}. Khan et al. designed a detection system based on three CsI(TI) detectors for searching lost radioactive sources. Their system determines source position by analyzing dose rate differences between the three detectors. The central detector (B) determines the distance between the source and system along the axis, while the side detectors (A and C) determine the deviation from the axis. Through experimental and simulated analyses of dose rates at various detector-source distances and detector separations, they demonstrated that their system can locate single or multiple gamma-ray sources with small deviation angles. The approach is particularly useful for quickly searching for radioactive sources in radiation environments, requiring only dose rate measurements and distance differences to determine source locations \cite{41}.

However, existing research encountered challenges when applied to the localization of multiple radiation sources in complex pipeline systems. Firstly, most previous methods focused on multi-source localization in two-dimensional planes or discrete grid spaces, whereas radiation sources in complex pipeline systems are continuously distributed in three-dimensional space \cite{20,21,25,26,27,28,29,30,31,33,34,35,36,37}. Besides, most previous studies primarily considered attenuation from the air, whereas radiation sources in complex pipeline systems are shielded by the pipeline wall, leading to errors in estimating the radiation source position \cite{19,20,21,22,23,24,25,26,27,28,29,31,32,33,34,35,37,39}. Moreover, previous studies primarily focused on localizing only a few radiation sources (fewer than four), whereas complex pipeline systems typically involve a much larger number of radiation sources, exacerbating the ill-posedness of the inverse problem \cite{19,20,24,25,26,27,28,29,32,33,34,35,36,37,38,39,40,41}.

This paper proposes a geometrically constrained EM-ML algorithm (Geo-EM-ML) to solve the multi-source localization problem in complex pipeline systems. At the initial stage, the expectation-maximization (EM) algorithm is used to estimate the position and activity of the radiation sources quickly. Maximum likelihood estimation (MLE) is used to optimize the initial estimation provided by the EM algorithm. During the optimization process, geometric constraints are incorporated into MLE to reduce the algorithm's search space and enhance convergence efficiency. Meanwhile, a parameter filter is designed to estimate the number of radiation sources progressively. Finally, the algorithm continuously optimizes and adjusts the position, number, and activity of the radiation sources until the estimated number of radiation sources no longer changes.

The primary contributions of the paper are as follows:
- The proposed hybrid algorithm can rapidly provide an approximate orientation of the radiation sources within the pipeline system and efficiently compute the number, positions, and activity of the radiation sources in ∼3min. Tests with up to twelve hotspots were conducted, yielding excellent results.
- Our algorithm introduces geometric constraints, which narrows the search space of radiation sources and improves the robustness and convergence speed of the algorithm.
- We designed a filter strategy to progressively optimize the estimation of the number of radiation sources, improving the stability and accuracy of the algorithm while reducing the computational burden in the later stages.
- We used ray tracing to accurately model the radiation shielding effect, ensuring the accuracy and stability of the algorithm in complex pipeline systems. The stainless steel pipelines with a wall thickness of up to 15 cm were tested in this study, achieving a success rate of around 95%.

The rest of this paper is organized as follows: Sec. II introduces the details of the Geo-EM-ML algorithm; Sec. III tests the algorithm under different experimental settings to demonstrate the feasibility of the algorithm, and conducts ablation experiments on various sub-modules of the algorithm to illustrate their usefulness; Sec. IV summarizes the work of this paper.

II. Method

A. Preliminaries

The problem to be solved by our algorithm is that there are multiple point sources in the pipeline system, which are composed of the same radioactive nuclide and emit gamma rays, as shown in Figure 1 [FIGURE:1]. The total number of these point sources $M$, the position $\mathbf{r}_m$ and the activity $\lambda_m$ of each point source are unknown. Therefore, the parameter distribution to be estimated is represented as shown in Eq. (1).

$$
\theta = {\mathbf{r}m, \lambda_m}^M
$$

$N$ detectors are deployed in the free space outside the pipeline system, and their positions $\mathbf{r}_n$ are known. It should be noted that, in practical applications, deploying a large number of detectors simultaneously may be constrained by time, manpower, and economic costs. To address this issue, a single detector can be mounted on a mobile robot or robotic arm to perform repeated measurements at different positions. Alternatively, personnel can carry handheld detectors to conduct sequential measurements across multiple locations. For radiation fields that can be considered static during the total measurement period, this approach yields data equivalent to those obtained from simultaneous measurements by multiple detectors, effectively forming a virtual detector array. This measurement strategy demonstrates strong feasibility and practicality in the context of nuclear facility maintenance.

During the detection time $t$, the photon peak counts received by detector $n$ are also known and denoted as $C_n^t$, and the corresponding count rate is $c_n$.

A well-studied forward problem is formulated as determining the count rate $c_n(\theta)$ of the photon peak at the position $\mathbf{r}_n$ of detector $n$ based on the distribution of radiation sources $\theta$ and the geometry and material of the shielding in the environment. In the case where the radiation source distribution $\theta$ is known, the count rate of the photon peak on the $n$-th detector contributed by the $m$-th radiation source is as shown in Eq. (2).

$$
c_{n,m}(\theta) = \alpha_{n,m}\lambda_m = \frac{A\mu e^{-\bar{\rho}{n,m}d\lambda_m}}}{4\pi d_{n,m}^2
$$

Where $\alpha_{n,m}$ is the attenuation coefficient from the $m$-th radiation source to the $n$-th detector, $A$ is the detector area, $\mu$ is the detector detection efficiency, $\bar{\rho}{n,m}$ is the average linear attenuation coefficient from the detector to the radiation source, and $d_m|$ is the distance between the detector and the radiation source.} = |\mathbf{r}_n - \mathbf{r

It should be noted that Eq. (2) neglects the contribution of scattered photons, which is justified by the use of full-energy peak counting in the detection process. The detectors employed in this algorithm are designed to measure photopeak counts within a narrow energy window centered on the characteristic gamma-ray energy of the source.

The detectors used in this study have consistent detector area and detection efficiency, and are isotropic detectors. Since the influence of each radiation source on the detector is independent, the count rate of the photon peak on the detector is as shown in Eq. (3).

$$
c_n(\theta) = \sum_{m=1}^M c_{n,m}(\theta) = \sum_{m=1}^M \alpha_{n,m}\lambda_m
$$

The problem studied in this paper is the inverse problem of the above forward problem, i.e., determining the distribution $\theta$ of radiation sources based on the positions of the detectors and the count of the photon peaks ${\mathbf{r}n, C_n^t}^N$ and the geometry and material of the shielding in the environment. The inverse problem is generally solved using optimization methods, i.e., finding the optimal parameters $\theta^$ in the solution space such that the difference between the estimated count rate ${c_n(\theta^)}{n=1}^N$ and the actual count rate ${c_n}^N$, determining the direction of the next sampling based on the estimated count rate and the actual count rate of the photon peaks detected, until the optimal parameters $\theta^*$ are found.}^N$ is minimized. The entire solution process essentially involves continuously sampling the parameters $\theta$ in the solution space, calculating the corresponding count rate ${c_n(\theta)}_{n=1

When solving the inverse problem using the least squares estimation (LSE), the algorithm calculates the square difference between the estimated count rate and the actual count rate ${c_n}_{n=1}^N$ as the loss function $L(\theta)$, and selects the parameter $\theta^*$ that minimizes $L(\theta)$ as the distribution of the radiation sources, as shown in Eq. (4).

$$
\theta^* = \arg\min L_{LSE}(\theta) = \arg\min \sum_{n=1}^N (c_n(\theta) - c_n)^2
$$

When solving the inverse problem using the maximum likelihood estimation (MLE), the algorithm calculates the probability of the photon peak counts received by each detector as the likelihood function, and selects the parameter $\theta^*$ that maximizes the likelihood function as the distribution of the radiation sources, as shown in Eq. (5).

$$
\theta^* = \arg\max L_{MLE}(\theta) = \arg\max \prod_{n=1}^N P(C_n^t | \theta)
$$

Where $P(C_n^t | \theta)$ is the probability of the photon peak counts received by detector $n$ when the source distribution is $\theta$, which follows the Poisson distribution, as shown in Eq. (6).

$$
P(C_n^t | \theta) = \frac{(c_n(\theta)t)^{C_n^t} e^{-c_n(\theta)t}}{C_n^t!}
$$

B. Geo-EM-ML Algorithm Structure

We propose a hybrid algorithm, Geo-EM-ML, which combines the expectation-maximization algorithm and the maximum likelihood estimation method, incorporating the geometric information of complex pipeline systems to address the multi-source localization problem. The algorithm pipeline is shown in Fig. 2 [FIGURE:2]. To address the trade-off between accuracy and computational efficiency in traditional discrete grid methods, the proposed algorithm employs a novel two-stage strategy: coarse localization in the discrete domain followed by fine optimization in the continuous domain. This strategy is progressively realized through three phases: Initial Estimation, Pre-optimization, and Joint Optimization. The approach first utilizes the global search capability of discrete-domain methods to obtain a robust initial estimate. It then transitions to the continuous domain for high-precision refinement. This design overcomes the accuracy bottlenecks inherent in purely discrete methods and avoids the tendency of purely continuous methods to become trapped in local optima.

Initial Estimation: This stage performs coarse localization in a discrete domain. The entire pipeline system is first discretized into $M_{em}$ uniform units, forming a search space of potential sources with center positions $\mathbf{r}m^{em}$. At this stage, the only parameters to be estimated are the source activities at these discrete locations, expressed as $\theta}} = {\lambda_m^{em{m=1}^{M.}}$. The activity values are iteratively updated using the Expectation-Maximization (EM) algorithm. Given the activities $\lambda_m^{em,k}$ at iteration $k$, the updates for the next iteration, $\lambda_m^{em,k+1}$, are computed according to Eq. (7) \cite{21

$$
\lambda_m^{em,k+1} = \lambda_m^{em,k} \frac{\sum_{n=1}^N \alpha_{n,m} t \Psi(C_n^t, t\alpha_{n,m}\lambda_m^{em,k})}{\sum_{n=1}^N \sum_{j=1}^{M_{em}} \alpha_{n,j} t \Psi(C_n^t, t\alpha_{n,j}\lambda_m^{em,k})}
$$

Where $\Psi(C, \lambda) = e^{-\lambda}\lambda^C/C!$ is the probability mass function of the Poisson distribution. Without any prior information, the initial activities for the iteration are set to a uniform distribution. The EM algorithm can rapidly (approx. 2s) generate a global overview of the activity distribution, providing a high-quality initial input for identifying potential source regions in subsequent stages.

Pre-optimization: This stage refines and reduces the dimensionality of the initial activity distribution while keeping the source positions fixed. The optimization follows the Maximum Likelihood Estimation (MLE) as defined in Eq. (5), but its optimization variables are restricted to the activities at discrete grid points. First, Maximum Likelihood Estimation (MLE-1) is applied to the full activity set obtained from the EM algorithm to yield more accurate estimates. Subsequently, a local maximum search is performed to identify peaks in the activity distribution, effectively filtering out numerous low-activity pseudo-sources and significantly reducing the number of candidates. Finally, Maximum Likelihood Estimation (MLE-2) is used again to optimize the activities of this greatly reduced set of candidate sources, providing a manageable and more accurate input for the final joint optimization.

Joint Optimization: This final stage transitions from the discrete domain to the continuous domain for fine-grained localization. Unlike the pre-optimization stage, it jointly estimates both source positions $\mathbf{r}m$ and activities $\lambda_m$, i.e., the parameters to be estimated are $\theta = {\mathbf{r}_m, \lambda_m}^M$. By including positions in the optimization, the algorithm breaks free from the constraints of the discrete grid, thereby achieving higher, sub-grid-level localization accuracy. Each iteration alternates between two MLE-based strategies: first, an unconstrained Maximum Likelihood Estimation (MLE-3) is used to freely optimize source positions and activities, allowing for broad parameter adjustments and corrections. Then, a geometrically constrained Maximum Likelihood Estimation (MLE-4) is applied, incorporating the pipeline geometry as a probabilistic prior to ensure the physical plausibility of the source locations. This alternating optimization balances flexibility in adjustment with physical realism. After each iteration, a source fusion step refines the number of sources, and the process repeats until convergence to the final localization result.

C. Key Techniques

1. Geometric Constraints

In the Geo-EM-ML algorithm, geometric constraints are used to provide prior information on the position of the radiation sources, thereby improving the convergence speed and accuracy of the algorithm. Geometric constraints are used in two modules of the Geo-EM-ML algorithm. The initial estimation module uses the Expectation-Maximization algorithm to estimate the activity of the radiation source, requiring geometric constraints to restrict the sampling positions of the radiation source. In the joint optimization module, geometric constraints are added to the likelihood function in the form of probabilities to constrain the position of the radiation sources.

In the initial estimation module, we propose two sampling methods for the position of the radiation sources, denoted as PIPE and SPACE. The PIPE sampling method uses the geometric model of the pipeline system as prior information, while the SPACE sampling method does not use any prior information. Fig. 3 FIGURE:3 shows that when using the PIPE sampling method, the radiation sources are uniformly distributed within the pipeline system. Fig. 3(b) shows that when using the SPACE sampling method, the radiation sources are uniformly distributed within the free space around the pipeline system.

In the joint optimization module, geometric constraints are incorporated into the likelihood function through probabilistic modeling. For each radiation source position $\mathbf{r}_m$, we calculate its minimum distance to the pipeline midline $D(\mathbf{r}_m)$. This distance's probability density distribution is approximated using a Gaussian function:

$$
f(D(\mathbf{r}m)) = \frac{1}{\sqrt{2\pi d}(S)}} e^{-\frac{D(\mathbf{rm)^2}{2d}(S)}
$$

Where $d_{min}(S)$ represents the minimum possible distance between radiation sources. This constraint term combines with the detector response probability to form the constrained likelihood function:

$$
L_{MLE}(\theta) = \prod_{n=1}^N P(C_n^t|\theta) \cdot \prod_{m=1}^M f(D(\mathbf{r}_m))
$$

This probabilistic formulation allows geometric constraints and detector responses to be optimized on the same scale, ensuring both geometric compliance and numerical stability. The Gaussian variance setting ($d_{min}(S)$) balances constraint strength with optimization smoothness.

2. Source Fusion and Filtering

In the Geo-EM-ML algorithm, source fusion and filtering are employed to reduce the number of estimated radiation sources, thereby improving the algorithm's convergence speed and accuracy while correctly determining the number of radiation sources. These techniques are applied in two main stages of the algorithm: the pre-optimization stage and the joint optimization stage.

In this algorithm, the source fusion distance $d_{min}(S)$ and the activity filtering threshold are two key hyperparameters set by personnel based on prior knowledge of the site. Notably, the activity filtering threshold is not a fixed absolute value but is indirectly defined by setting a ratio of maximum to minimum activity (e.g., 10:1). Whenever activity filtering is required, the algorithm first identifies the maximum activity among all current candidate sources and then dynamically determines a specific activity threshold based on the preset ratio. Sources with activities below this threshold are then eliminated.

In the pre-optimization stage, specifically in the MLE-1 and MLE-2 modules, local maxima search and activity threshold filtering are primarily used to reduce the number of candidate radiation sources. The local maxima search identifies peaks in the activity distribution to eliminate pseudo-sources with lower activity values, while activity threshold filtering directly removes candidate sources below a preset threshold.

In the joint optimization stage, involving the MLE-3 and MLE-4 modules, a dynamic fusion strategy is implemented. After each iteration, the algorithm performs local maxima clustering, merging radiation sources with spatial distances less than the minimum possible distance $d_{min}(S)$ into a single source. The position of the merged source is determined through activity-weighted averaging, while its activity is the sum of the activities of the merged sources.

Additionally, the algorithm employs an adaptive activity filtering strategy. In the early stages of iteration, activity threshold filtering is not applied to retain all potential radiation sources. As iterations progress and the fusion strategy becomes less effective, activity threshold filtering is gradually introduced to eliminate potential pseudo-sources. The algorithm is considered to have converged when the number of radiation sources remains unchanged between consecutive iterations.

This multi-stage source fusion and filtering strategy significantly enhances the algorithm's performance. It not only reduces the number of candidate sources from $\mathcal{O}(10^2)$ in the initial EM stage to $\mathcal{O}(10^0)$ in the final stage but also improves localization accuracy by eliminating duplicate estimates and reducing the impact of measurement noise. Furthermore, the dynamic adjustment strategy effectively prevents the algorithm from converging to local optima, ensuring stable convergence.

3. Shielding Effects

The key to accurately calculating the detector response for environments with various materials is determining the distance radiation travels through each material from the source to the detector. Specifically, for complex pipeline systems, the task involves calculating the distances traveled by the line segment between the $n$-th detector and the $m$-th radiation source through the pipeline material and air, $d_{n,m,pipe}$ and $d_{n,m,air}$. Fig. 4 [FIGURE:4] shows the key steps of the calculation, including listing the intersection points and classifying the line segments.

The pipeline system is first modeled as a water-tight triangle mesh to ensure no holes or cracks on its surface and that the internal and external spaces are clearly defined. Then, the set of line segments to be solved, denoted as $(\mathbf{r}_n, \mathbf{r}_m)$, $n = 1, \ldots, N$, $m = 1, \ldots, M$, is determined based on the positions of the detector and the radiation source.

Next, the position of all intersection points of each line segment with the pipeline model is obtained using the list_intersections function in the Open3D library, as denoted in Eq. (10) and shown in Fig. 4(a) \cite{42}.

$$
\mathcal{R}{n,m} = {\mathbf{r}}, \ldots, \mathbf{r{n,m,I}}
$$

Where $I_{n,m}$ represents the number of intersection points of the line segment with the pipeline model, and $\mathbf{r}{n,m,i}$ represents the position of the $i$-th intersection point. The intersection points in the set are sorted in ascending order of distance, that is, $|\mathbf{r}} - \mathbf{rm| > |\mathbf{r}} - \mathbf{rm|$, $\forall i \in {1, \ldots, I - 1}$.

Since the detector is positioned outside the pipeline model, the segment of the path from the final intersection point, $\mathbf{r}{n,m,I}}$, to the detector position, $\mathbf{rn$, must traverse through the air. Additionally, the segment between the last two intersection points, $\mathbf{r}}-1}$ and $\mathbf{r{n,m,I$, allowing us to classify each segment as occurring in air or within the pipeline model, as shown in Eq. (11) and illustrated in Fig. 4(b).}}$, lies within the pipeline model. This alternation between air and pipeline can be determined by the parity of the total number of intersection points, $I_{n,m

$$
\rho(\mathbf{r}{n,m,i}, \mathbf{r}}) = \begin{cases
\rho_{pipe}, & \text{if } i \bmod 2 \neq I_{n,m} \bmod 2 \
\rho_{air}, & \text{if } i \bmod 2 = I_{n,m} \bmod 2
\end{cases}
$$

Where $i \in {0, \ldots, I_{n,m}}$, $\rho_{pipe}$ and $\rho_{air}$ represent the attenuation coefficients of the pipeline model and air, respectively.

The starting and ending points of the line segment are defined as $\mathbf{r}{n,m,0} = \mathbf{r}_m$ and $\mathbf{r}}+1} = \mathbf{rn$. Finally, the average attenuation coefficient of the line segment, $\bar{\rho}$, can be calculated as shown in Eq. (12).

$$
\bar{\rho}{n,m} = \frac{\sum}^{I_{n,m}} \rho(\mathbf{r{n,m,i}, \mathbf{r}}) |\mathbf{r{n,m,i+1} - \mathbf{r}}|}{|\mathbf{r}_n - \mathbf{r}_m|
$$

III. Results and Discussion

A. Experimental Settings

In all experiments, the simulated radiation sources are Cs-137 isotopes emitting gamma rays at 662 keV. The detector responses are calculated based on the full-energy peak counts at this characteristic energy, ensuring that the measurements specifically reflect unscattered primary gamma photons.

The overall size of the pipeline system is $3.5655\,\text{m} \times 0.495\,\text{m} \times 1.3628\,\text{m}$, with different shielding materials or pipe wall thicknesses designed according to the specific model. Detectors are distributed throughout the entire pipeline model space using the GRID or RANDOM method, as shown in Fig. 5 [FIGURE:5]. The GRID distribution method divides the overall space of the pipeline model into multiple equally sized small cubes, with the center point of each small cube as the position of a detector. The RANDOM distribution method randomly generates the position of a detector within each small cube to ensure that the detectors are evenly distributed throughout the space.

One hundred sets of radiation source positions and activity parameters were randomly generated for each experimental setting. The detector responses for each set of source parameters were generated using two different methods.

Point Kernel Integration (PKI) method: This deterministic approach consists of the following steps: First, a water-tight triangular mesh file of the pipeline model is generated. Second, detector positions, source distributions, detection times, and other detector parameters are specified. Third, the ray-tracing algorithm described in Sec. II C 3 is used to calculate the average linear attenuation coefficient along the line connecting each source to each detector. Finally, the detector response is calculated according to Equations (2) and (3). This method provides fast and deterministic calculations, offering a theoretical upper bound for algorithm performance.

Monte Carlo Simulation method: This statistical approach uses Geant4 software with the following workflow: First, the pipeline model is constructed in Geant4 using boolean entities or by importing triangular mesh files. Second, isotropic particle guns are positioned at radiation source locations, with emission probabilities determined by the activity ratios of the radiation sources. Third, particles are randomly emitted and their transport through the pipeline model is simulated, including various physical processes such as scattering and absorption. Finally, detectors count particles at the full-energy peak to obtain the detector response. This method introduces statistical noise that closely resembles actual detector response characteristics, providing a more realistic assessment of algorithm performance in practical applications.

The point kernel integration method is more suitable for theoretically illustrating the best possible algorithm's performance. In contrast, the detector response data generated by the Monte Carlo method, which introduces random noise, is closer to the actual detector response and is more suitable for illustrating the performance of the Geo-EM-ML algorithm in practical applications.

Fig. 6 [FIGURE:6] shows the detector response obtained under the same source distribution parameters and pipeline model using different generating methods, along with the relative deviation between the two data sets. To better visualize the trend, the detector indices are reordered according to the relative error from smallest to largest. Despite the statistical errors the Monte Carlo results introduced, the overall trends are consistent. In addition, all high-count detectors have a relative deviation of less than 5%. Due to the more significant impact of high-count detectors on the algorithm, the fact that all high-count detectors have low deviation helps narrow the gap between point kernel integration and Geant4. The result also demonstrates the effectiveness of the detector response calculation for pipeline models with shielding materials proposed in Sec. II C 3.

The final results produced by the Geo-EM-ML algorithm are denoted as $\theta_{opt} = {(\mathbf{r}m^{opt}, \lambda_m^{opt}), m = 1, 2, \ldots, M}}$, and the true source parameters are denoted as $\theta_{gt} = {(\mathbf{rm^{gt}, \lambda_m^{gt}), m = 1, 2, \ldots, M}}$. The position error is defined as $\Delta\mathbf{rm = |\mathbf{r}_m^{gt} - \mathbf{r}_m^{opt}|$. The standard distance is defined as the distance from each radiation source to the nearest detector, that is, $d} = \min_n d_{n,m}$. The normalized position error is defined as the ratio of the position error to the standard distance, that is, $\delta r_m = \frac{|\mathbf{rm^{gt} - \mathbf{r}_m^{opt}|}{d$.}}$. The relative activity error is defined as the ratio of the absolute activity error to the proper activity, that is, $\delta\lambda_m = \frac{|\Delta\lambda_m|}{\lambda_m^{gt}}$. The absolute activity error is defined as $\Delta\lambda_m = \lambda_m^{gt} - \lambda_m^{opt

In all experiments, the criteria for evaluating the algorithm's success are defined by the following three conditions. First of all, the estimated number of radiation sources should be correct, i.e., $M_{opt} = M_{gt}$. Secondly, the maximum normalized position error of the estimated radiation sources should be less than 50%, i.e., $\max_m \delta r_m < 50\%$. Finally, the maximum relative activity error of the estimated radiation sources should be less than 50%, i.e., $\max_m \delta\lambda_m < 50\%$.

The success rate $p$ is defined as the ratio of successful experiments to the total number of experiments, where an experiment is considered successful if it meets the three conditions mentioned above.

Fig. 7 [FIGURE:7] provides a comparison of the ground truth, the initial estimation provided by the EM sub-module, and the final result in one set of experiments. Note that, aside from the identified hotspot locations, the rest of the pipeline system is assumed to be non-radioactive or to have activity levels below the detection threshold.

To quantitatively evaluate the algorithm's localization performance, the normalized position error is chosen as the primary metric instead of the absolute error. The absolute error, while straightforward, fails to capture the proportional significance of the error in relation to the problem's scale. For instance, a 1-meter error is substantial in a small-scale setup but may be negligible in a large one. By normalizing the absolute error, the relative error offers a scale-invariant metric that enables fair and meaningful comparisons across diverse experimental configurations. Furthermore, the choice of the normalization factor in the relative error calculation is critical. We adopt a problem-specific characteristic length, $d_{standard}$, rather than the norm of the ground truth position vector, $|\mathbf{r}^{gt}|$, since the latter depends on the arbitrary choice of the coordinate system's origin. A different origin would yield a different relative error for the same physical localization result, making the metric unreliable. In contrast, $d_{standard}$ is defined based on the intrinsic geometry of the setup (such as the distance between two sources being localized), representing a physically meaningful scale. This ensures the evaluation metric is robust, reproducible, and justly reflects the algorithm's accuracy relative to the inherent challenge of each specific scenario.

B. Experimental Results and Analysis

Implementation and Basic Performance: The parameters of the baseline experiment were set as follows: the shielding material of the pipeline model was stainless steel, the wall thickness of the pipe was 5 mm, the number of radiation sources was 6, the ratio of the minimum to the maximum activity of the radiation sources was 1:2, the number of detectors was 164, and the distribution of detectors was GRID.

As shown in Fig. 7(b), the EM sub-module roughly identifies the area where the hotspot is located, helping users identify the possible location of the radiation source. Comparison of Fig. 7(c) and Fig. 7(a) shows that the final result accurately reflects the radiation sources' number, position, and activity.

The baseline experiment was run on an Intel(R) Core(TM) i5-12600 CPU @ 4.80 GHz with 128 GB of memory. The time distribution of each sub-module and the total time distribution in the baseline experiment are shown in Fig. 8 [FIGURE:8]. The average time to obtain the final result using the point kernel integration method and Geant4 simulation was 172.22 s and 174.28 s, respectively. The time consumption of each sub-module satisfies the order MLE-1 > MLE-3 > MLE-4 > EM > MLE-2. The MLE-1 sub-module contributed the most time, accounting for approximately 65% of the total time. The EM sub-module can provide an initial estimate in ∼2s, enabling users to identify the area where radiation sources may exist quickly. The computation time of MLE-4 is between one-quarter and one-third of that of MLE-3. Since the number of calls for MLE-3 and MLE-4 is the same, using geometric constraints has enhanced convergence efficiency.

Parameter Influence Analysis: To comprehensively evaluate the performance and applicability of the algorithm, experiments were conducted to assess its performance under different pipeline shielding parameters (material and thickness), pipeline shapes, source parameters (number and intensity range), detector parameters (number and distribution), and algorithm hyperparameters (source fusion distance $d_{min}$ and activity threshold ratio).

Pipeline Parameters: With six radiation sources randomly placed in the baseline pipeline model, a minimum-to-maximum source activity ratio of 1:2, and 164 detectors arranged in a GRID distribution, the success rate $p$, average position error $\Delta r$, and average relative activity error $\delta\lambda$ of the algorithm were tested under different pipeline shielding materials and thicknesses. The results are shown in Fig. 9 [FIGURE:9]. As the linear attenuation coefficient of the shielding material or the wall thickness increases, the algorithm's performance slightly decreases. However, the success rate remains above 92%, and the average position error and the average relative activity error are also maintained below 8.58 mm and 4.96%, respectively. The result indicates that, with accurate modeling and calculation of the attenuation effects, the Geo-EM-ML algorithm has a certain degree of robustness to changes in the shielding material and wall thickness.

To further validate the adaptability of the algorithm to different pipeline models, we generated four additional complex pipeline models using a random walk algorithm, as shown in Fig. 10 [FIGURE:10]. On these models, we repeated the same experimental setup as the baseline model. The performance of the algorithm under different pipeline models is shown in Fig. 11 [FIGURE:11]. The results indicate that even as the pipeline structure becomes more complex, the algorithm's performance metrics do not show significant changes, demonstrating the robustness of the Geo-EM-ML algorithm to pipeline shape.

Source Parameters: With the baseline pipeline model (stainless steel, 5 mm wall thickness) and 164 detectors in a GRID distribution, the success rate $p$, average position error $\Delta r$, and average relative activity error $\delta\lambda$ of the algorithm were tested under different numbers of sources and ranges of activity ratios. The results are shown in Fig. 12 [FIGURE:12]. As the number of sources and the range of relative intensities increase, the algorithm's performance degrades: the success rate decreases, while both the average position error and the average relative activity error increase. This degradation is primarily due to the expanded parameter space for the inverse problem, which increases the problem's ill-posedness. Moreover, when the activity range widens, detector responses from weak sources are more likely to be masked by those from strong sources, further reducing the algorithm's success rate.

To better understand these phenomena, we analyze the problem from an information-theoretic perspective. The essence of inverse problem solving is to infer unknown parameters from limited observational data. When the dimensionality of the parameter space increases, the complexity of the problem grows exponentially. Assuming the number of detectors is $N$ and the number of radiation sources is $M$, the amount of information provided by the observational data is $\mathcal{O}(N)$, while the number of parameters to be determined is $\mathcal{O}(4M)$ (including three-dimensional positions and activities). When $M$ increases, the volume of the parameter space grows as $(V_{max} - V_{min})^{3M} \times (\lambda_{max} - \lambda_{min})^M$, where $V_{max}$ and $V_{min}$ are the upper and lower bounds of the position search space, and $\lambda_{max}$ and $\lambda_{min}$ are the upper and lower bounds of the activity range. According to Shannon information theory, in the presence of noise, the amount of information that can be extracted from given observational data is limited. When the complexity of the parameter space exceeds the carrying capacity of the observable information, the problem becomes underdetermined, and multiple parameter combinations may produce similar observational results, leading to non-uniqueness of solutions. Furthermore, when the activity ratio range expands, the system's dynamic range increases, the probability of weak signals being masked by strong signals rises, and the effective signal-to-noise ratio decreases, further reducing the amount of usable information.

Detector Parameters: With stainless steel as the shielding material of the pipeline model, a wall thickness of 5 mm, six radiation sources randomly placed, and a minimum-to-maximum ratio of 1:2 for the source activity parameters, the success rate $p$, average position error $\Delta r$, and average relative activity error $\delta\lambda$ of the algorithm are tested under different numbers of detectors and distribution methods, as shown in Fig. 13 [FIGURE:13]. As the number of detectors increases, the constraint equations for the localization task increase, reducing the ill-posedness of the inverse problem and improving the algorithm's performance. Whether the detectors are arranged in a GRID or RANDOM pattern, the success rate of localizing six radiation sources exceeds 92% when the number of detectors exceeds 135. The result indicates that the algorithm does not rely on a structured detector arrangement.

Hyperparameters: With the pipeline model, source, and detector parameters consistent with the baseline experiment, we tested the influence of algorithm hyperparameters (source fusion distance $d_{min}$ and activity threshold ratio) on performance, as shown in Fig. 14 [FIGURE:14]. In this experiment, we varied the set values of these two parameters by -50%, -25%, -10%, +10%, +50%, and +100% from their true values.

From the left column of Fig. 14, it is evident that underestimating the minimum inter-source distance $d_{min}$ (i.e., the set value is less than the true value) has minimal impact on the algorithm's success rate, position error, and activity error. However, overestimating this distance, especially by 100% (i.e., setting it to twice the true value), leads to a sharp decline in the success rate. This is because an excessively large $d_{min}$ setting can cause distinct, closely-spaced sources to be incorrectly merged into a single source during the fusion step, resulting in localization failure.

The right column of Fig. 14 shows that overestimating the activity threshold ratio (i.e., allowing a wider dynamic range than reality) has little effect on performance. Conversely, underestimating this ratio, particularly by 50%, causes the success rate to plummet to zero. This occurs because a too-low activity threshold ratio leads to the erroneous filtering of true, low-activity sources as pseudo-sources, again resulting in localization failure.

Overall, these experimental results offer clear practical guidance. Underestimating $d_{min}$ or overestimating the activity ratio is equivalent to making a more "stringent" or "conservative" estimation of the problem environment. While this conservative approach might slightly increase computation time by expanding the search space for the optimization algorithm, it effectively prevents the correct solution from being filtered out. Conversely, an overly "optimistic" estimation (overestimating $d_{min}$ or underestimating the activity ratio), though potentially faster, is highly prone to erroneous pruning, preventing the algorithm from finding the correct solution. Therefore, in practical applications, it is recommended that operators make conservative estimates of environmental parameters based on their experience, i.e., moderately underestimating inter-source distances and overestimating the dynamic range of activities.

In most cases, when using the same pipeline model, radiation source parameters, number of detectors, distribution method, and algorithm hyperparameters, the algorithm performs better with detector response generated by the point kernel integration compared to that generated by Geant4. This result is consistent with the analysis in Sec. III A.

C. Ablation Study

In this section, ablation experiments are conducted on each module of the Geo-EM-ML algorithm to demonstrate the necessity of using these modules. All ablation experiments use the same experimental settings, which are the same as the settings of the baseline experiment in Sec. III B 1. The algorithm settings and results of the ablation study are shown in table 1 [TABLE:1], with the first row showing the results of the baseline experiment.

Table 1. Ablation Study of Geo-EM-ML Algorithm

Compared to the baseline experiment, the results of the second group show a significant decrease in both success rate and positioning accuracy. The results of the third group show a slight decrease in both success rate and positioning accuracy compared to the baseline experiment. The algorithms used in the second and third groups replace the maximum likelihood estimation algorithm with the least squares method in the pre-optimization and joint optimization stages, respectively. The results indicate that using the maximum likelihood estimation algorithm can achieve better positioning results.

The results of the fourth and fifth groups of experiments show that using the maximum likelihood estimation algorithm, either with or without geometric constraints alone, will significantly increase positioning error when using the detector response data generated by point kernel integration. However, for the detector response data generated by Geant4, the change in the results of these two groups of experiments compared to the baseline experiment is insignificant. In the sixth and seventh groups of experiments, the radiation source positions sampled by the SPACE method in the initial estimation and pre-optimization stages, and in the seventh group of experiments, the geometric constraints are not used in the joint optimization stage. The results show that the radiation source positions sampled by the SPACE method significantly reduce the success rate and positioning accuracy of the algorithm. Furthermore, omitting geometric constraints in the joint optimization stage further significantly reduces the performance of the algorithm. The PIPE sampling method utilizes geometric constraints and samples fewer positions (SPACE samples 1540 positions, while PIPE samples 916 positions). By comparing experiments 4, 5, 6, 7, and the baseline experiment, it can be seen that using geometric constraints significantly improves the performance of the algorithm.

The results of the eighth group of experiments show that neglecting the shielding effects in a shielded scenario significantly decreases both positioning accuracy and success rate. Compared to previous studies, the Geo-EM-ML algorithm accurately models the attenuation based on ray tracing, effectively improving positioning accuracy and success rate.

Discussion

Despite the promising performance of the Geo-EM-ML algorithm in multi-source localization problems, several limitations remain that should be considered in practical applications.

First, in terms of detector deployment, the algorithm's positioning accuracy is sensitive to the number of detectors. Experimental results indicate that at least 135 detectors are required for stable localization performance, which may present challenges in terms of cost and deployment convenience in practical scenarios. To address this issue, sequential measurements can be performed using a single detector mounted on a mobile robot or robotic arm, or carried manually as a handheld device, effectively creating a virtual multi-detector array.

Second, regarding source intensity variations, the algorithm's performance degrades significantly when the activity ratio between sources becomes too large. Experimental results show that when the minimum-to-maximum activity ratio exceeds 1:20, weak source signals can be masked by strong sources, leading to reduced localization success rates. In practical multi-source scenarios with substantial activity differences, a staged localization strategy is advisable: first, localize the strong sources and subtract their contributions from the measurement data before proceeding to localize the weaker sources.

Third, in terms of scalability, the current algorithm is primarily designed for single-nuclide scenarios. In multi-nuclide environments, spectral analysis can identify the full-energy peaks of each nuclide, allowing the Geo-EM-ML algorithm to be applied separately to each. Building upon the practical observation that environmental hotspots typically maintain consistent nuclide composition ratios, the results from multiple single-nuclide localization problems can be cross-referenced and fused, further improving overall localization accuracy through cross-validation of position information and consistency constraints from activity ratios. For example, when a nuclide has a weak signal with high localization uncertainty, the strong signal localization results from other nuclides can be used as prior information to optimize the weak signal nuclide localization through Bayesian inference and similar methods.

Despite these limitations, the Geo-EM-ML algorithm still provides an effective solution for multi-source localization problems in nuclear facilities. Through reasonable implementation strategies and subsequent improvements, these limitations can be mitigated or resolved, enabling the algorithm to provide greater value in practical applications.

IV. Conclusions

This study addresses the problem of multi-source localization in complex pipeline systems and proposes a novel Geometrically Constrained Expectation-Maximization Maximum Likelihood (Geo-EM-ML) algorithm. The algorithm integrates the Expectation-Maximization algorithm and Maximum Likelihood Estimation, incorporating constraints based on the geometric structure of the pipeline system. The algorithm effectively addresses the challenges of limited solution spaces, shielding effects, and the ill-posed nature of inverse problems encountered by traditional methods in multi-source localization.

This study proposed a multi-source localization algorithm effectively integrating geometric constraints, breaking through the limitations of traditional two-dimensional or discrete grid spaces, and achieving efficient localization in three-dimensional continuous space. The algorithm can handle more radiation hotspots, expanding its applicability in practical applications. It uses ray tracing to accurately model radiation shielding effects, ensuring the accuracy and stability of the algorithm in complex pipeline systems. Additionally, the algorithm designs an effective source fusion and filtering strategy, which reduces the computational complexity of the algorithm while accurately estimating the number of radiation sources, providing faster and more accurate localization results for practical applications.

The experimental results of this study show that the Geo-EM-ML algorithm exhibits high success rates and precise localization capabilities under different pipe thicknesses, materials, source activities, and detector configurations. In addition, ablation studies further validate the necessity of each module, demonstrating the critical role of geometric constraints and accurate attenuation calculations in improving algorithm performance.

The Geo-EM-ML algorithm has crucial practical value in nuclear facility maintenance, enabling rapid and accurate localization of radiation sources, significantly reducing the risk of radiation exposure to workers, and optimizing maintenance path planning. Its efficiency and robustness make it highly scalable and applicable in large-scale sensor network environments.

Future work will focus on further improving the Geo-EM-ML algorithm from four key aspects. First, geometric constraint methods will be extended from pipeline-specific scenarios to general three-dimensional environments. A unified framework for geometric information representation and probabilistic modeling will be established to improve adaptability across various nuclear facility types and structural configurations. Second, intelligent optimization algorithms—such as genetic algorithms, particle swarm optimization, and simulated annealing—will be integrated into the existing Geo-EM-ML framework to enhance global search capability and robustness in complex, multimodal optimization problems. Third, heterogeneous detector technologies will be integrated by establishing a unified multimodal detector response model, accommodating the simultaneous deployment of counting detectors, gamma cameras, and other types commonly used in nuclear facilities. By fully leveraging the strengths of different detectors, the overall localization accuracy of the system can be improved. Finally, autonomous detection and multi-source 3D localization techniques guided by minimum detectable activity (MDA) will be developed. This involves constructing MDA evaluation models that account for detector sensitivity, background radiation, and shielding effects, and incorporating reinforcement learning methods to enable autonomous path planning—ensuring detection efficiency while minimizing radiation exposure to personnel.

References

[1] O. Kadri, A. Alfuraih, Monte Carlo assessment of coded aperture tool for breast imaging: a Mura-mask case study. Nucl. Sci. Tech. 30, 164 (2019). doi: 10.1007/s41365-019-0682-3

[2] T. Zhang, L. Wang, J. Ning, et al., Simulation of an imaging system for internal contamination of lungs using MPA-MURA coded-aperture collimator. Nucl. Sci. Tech. 32, 17 (2021). doi: 10.1007/s41365-021-00849-3

[3] R.-Y. Wu, C.-R. Geng, F. Tian, et al., GPU-accelerated three-dimensional reconstruction method of the Compton camera and its application in radionuclide imaging. Nucl. Sci. Tech. 34, 52 (2023). doi: 10.1007/s41365-023-01199-y

[4] W. Lu, H.-W. Zhang, M.-Z. Liu, et al., Generalization ability of a CNN -ray localization model for radiation imaging. Nucl. Sci. Tech. 34, 185 (2023). doi: 10.1007/s41365-023-01323-y

[5] D. Zhao, X.-W. Liang, P.-K. Cai, et al., Design and performance evaluation of a large field-of-view dual-particle time-encoded imager based on a depth-of-interaction detector. Nucl. Sci. Tech. 35, 67 (2024). doi: 10.1007/s41365-024-01416-2

[6] E.-W. Bai, A. Heifetz, P. Raptis, et al., Maximum likelihood localization of radioactive sources against a highly fluctuating background. IEEE T. Nucl Sci. 62, 3274–3282 (2015). doi: 10.1109/TNS.2015.2497327

[7] Z. Liu, S. Abbaszadeh, Double Q-Learning for radiation source detection. Sensors 19, 960 (2019). doi: 10.3390/s19040960

[8] C. Q. Wu, M. L. Berry, K. M. Grieme, et al., Network detection of radiation sources using localization-based approaches. IEEE T. Ind. Inform. 15, 2308–2320 (2019). doi: 10.1109/TII.2019.2891253

[9] J. Zhao, Z. Zhang, C. J. Sullivan, Identifying anomalous nuclear radioactive sources using Poisson kriging and mobile sensor networks. PLOS ONE 14, e0216131 (2019). doi: 10.1371/journal.pone.0216131

[10] P. Gong, X.-B. Tang, X. Huang, et al., Locating lost radioactive sources using a UAV radiation monitoring system. Appl. Radiat. Isotopes 150, 1–13 (2019). doi: 10.1016/j.apradiso.2019.04.037

[11] J. Huo, M. Liu, K. A. Neusypin, et al., Autonomous search of radioactive sources through mobile robots. Sensors 20, 3461 (2020). doi: 10.3390/s20123461

[12] Y. Ji, Y. Zhao, B. Chen, et al., Source searching in unknown obstructed environments through source estimation, target determination, and path planning. Build. Environ. 221, 109266 (2022). doi: 10.1016/j.buildenv.2022.109266

[13] M. Ling, J. Huo, G. V. Moiseev, et al., Multi-robot collaborative radioactive source search based on particle fusion and adaptive step size. Ann. Nucl. Energy 173, 109104 (2022). doi: 10.1016/j.anucene.2022.109104

[14] H. Hu, J. Wang, A. Chen, et al., An autonomous radiation source detection policy based on deep reinforcement learning with generalized ability in unknown environments. Nucl. Eng. Technol. 55, 285–294 (2023). doi: 10.1016/j.net.2022.09.010

[15] J. Huo, X. Hu, J. Wang, et al., ACA: Automatic search strategy for radioactive source. Nucl. Eng. Technol. 55, 3030–3038 (2023). doi: 10.1016/j.net.2023.05.017

[16] H.-L. Liu, H.-B. Ji, J.-M. Zhang, et al., Novel algorithm for detection and identification of radioactive materials in an urban environment. Nucl. Sci. Tech. 34, 154 (2023). doi: 10.1007/s41365-023-01304-1

[17] M. Luo, J. Huo, M. Liu, et al., Distributed collaboration: Cognitive difference and collaborative decision for multi-robot radioactive source search. Ann. Nucl. Energy 196, 110210 (2024). doi: 10.1016/j.anucene.2023.110210

[18] Z. Hong, F. Cong, B. Yang, et al., Localization of a radioactive source with the predictive range-whale optimization algorithm method. Nucl. Eng. Technol. , 103562 (2025). doi: 10.1016/j.net.2025.103562

[19] L.-C. Chen, N. Van Thai, H.-F. Shyu, et al., In situ clouds-powered 3-D radiation detection and localization using novel color-depth-radiation (CDR) mapping. Adv. Robotics 28, 841–857 (2014). doi: 10.1080/01691864.2014.894942

[20] B. Deb, J. A. F. Ross, A. Ivan, et al., Radioactive source estimation using a system of directional and non-directional detectors. IEEE T. Nucl Sci. 58, 3281–3290 (2011). doi: 10.1109/TNS.2011.2165558

[21] B. Deb, Iterative estimation of location and trajectory of radioactive sources with a networked system of detectors. IEEE T. Nucl. Sci. 60, 1315–1326 (2013). doi: 10.1109/TNS.2013.2247060

[22] D. Hellfeld, T. H. Y. Joshi, M. S. Bandstra, et al., Gamma-ray point-source localization and sparse image reconstruction using Poisson likelihood. IEEE T. Nucl. Sci. 66, 2088–2099 (2019). doi: 10.1109/TNS.2019.2930294

[23] J. R. Vavrek, D. Hellfeld, M. S. Bandstra, et al., Reconstructing the position and intensity of multiple gamma-ray point sources with a sparse parametric algorithm. IEEE T. Nucl. Sci. 67, 2421–2430 (2020). doi: 10.1109/TNS.2020.3024735

[24] M. S. Bandstra, D. Hellfeld, J. R. Vavrek, et al., Improved gamma-ray point source quantification in three dimensions by modeling attenuation in the scene. IEEE T. Nucl Sci. 68, 2637–2646 (2021). doi: 10.1109/TNS.2021.3113588

[25] A. Abdelhakim, Heuristic techniques for maximum likelihood localization of radioactive sources via a sensor network. Nucl. Sci. Tech. 34, 127 (2023). doi: 10.1007/s41365-023-01267-3

[26] M. Morelande, et al., Detection and parameter estimation of multiple radioactive sources. in 2007 10th International Conference on Information Fusion, pp. 1–7. July 2007 doi: 10.1109/ICIF.2007.4408094

[27] B. Ristic, et al., A controlled search for radioactive point sources. in 2008 11th International Conference on Information Fusion, pp. 1–5. June 2008

[28] M. R. Morelande, B. Ristic, Radiological source detection and localisation using Bayesian techniques. IEEE T. Signal Proces. 57, 4220–4231 (2009). doi: 10.1109/TSP.2009.2026618

[29] B. Ristic, M. Morelande, A. Gunatilaka, Information driven search for point sources of gamma radiation. Signal Process. 90, 1225–1239 (2010). doi: 10.1016/j.sigpro.2009.10.006

[30] J.-C. Chin, et al., Efficient and robust localization of multiple radiation sources in complex environments. in 2011 31st International Conference on Distributed Computing Systems, pp. 780–789. June 2011 doi: 10.1109/ICDCS.2011.94

[31] E. Yee, A. Gunatilaka, B. Ristic, Comparison of two approaches for detection and estimation of radioactive sources. Int. Sch. Res. Notices. 2011, 189735 (2011). doi: 10.5402/2011/189735

[32] R. B. Anderson, M. Pryor, A. Abeyta, et al., Mobile robotic radiation surveying with recursive Bayesian estimation and attenuation modeling. IEEE. T. Autom. Sci. Eng. 19, 410–424 (2022). doi: 10.1109/TASE.2020.3036808

[33] T. Lazna, L. Zalud, Localizing multiple radiation sources actively with a particle filter. Nucl. Eng. Technol. 57, 103171 (2025). doi: 10.1016/j.net.2024.08.040

[34] E. Wacholder, E. Elias, Y. Merlis, Artificial neural networks optimization method for radioactive source localization. Nucl. Technol. 110, 228–237 (1995). doi: 10.13182/NT95-A35120

[35] F. Mendes, M. Barros, A. Vale, et al., Radioactive hot-spot localisation and identification using deep learning. J. Radiol. Prot. 42, 011516 (2022). doi: 10.1088/1361-6498/ac1a5c

[36] X. Hu, J. Huo, J. Wang, et al., Research on a localization method of multiple unknown gamma radioactive sources. Ann. Nucl. Energy 177, 109302 (2022). doi: 10.1016/j.anucene.2022.109302

[37] A. Abdelhakim, Machine learning for localization of radioactive sources via a distributed sensor network. Soft Comput. 27, 10493–10508 (2023). doi: 10.1007/s00500-023-08447-8

[38] Y.-S. Hao, Z. Wu, Y.-H. Pu, et al., Research on inversion method for complex source-term distributions based on deep neural networks. Nucl. Sci. Tech. 34, 195 (2023). doi: 10.1007/s41365-023-01327-8

[39] R. Okabe, S. Xue, J. R. Vavrek, et al., Tetris-inspired detector with neural network for radiation mapping. Nat. Commun. 15, 3061 (2024). doi: 10.1038/s41467-024-47338-w

[40] F. Hautot, P. Dubart, C.-O. Bacri, et al., Visual simultaneous localization and mapping (VSLAM) methods applied to indoor 3D topographical and radiological mapping in real-time. EPJ Nuclear Sci. Technol. 3, 15 (2017). doi: 10.1051/epjn/2017010

[41] W. Khan, C.-H. He, Q.-M. Zhang, et al., Design of CsI(TI) detector system to search for lost radioactive source. Nucl. Sci. Tech. 30, 132 (2019). doi: 10.1007/s41365-019-0658-3

[42] Q.-Y. Zhou, J. Park, V. Koltun, Open3D: a modern library for 3D data processing (2018), http://arxiv.org/abs/1801.09847. Accessed 26 Nov 2024