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Machine learning methods for subpixel trajectory reconstruction in discretized position detectors

This study demonstrates that transformer-based machine learning architectures significantly outperform traditional centroid methods in reconstructing subpixel particle trajectories and angular resolution within discretized scintillator detectors, achieving a 2.22-fold improvement in angular accuracy and a 6.33-fold improvement in position precision.

Original authors: Matthew Mark Romano, Zhengzhi Liu, JungHyun Bae

Published 2026-07-16

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CC BY 4.0

Original authors: Matthew Mark Romano, Zhengzhi Liu, JungHyun Bae

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to find the exact path of a ghostly, invisible bullet zipping through a dark room. This is the daily challenge of scientists who study muons, tiny particles that rain down on Earth from space like cosmic snow. Because these particles can punch through mountains and lead shielding, they are like nature's own X-ray machines. By tracking how muons bend as they pass through different materials, scientists can "see" inside volcanoes, ancient pyramids, or even check for hidden nuclear secrets without ever drilling a hole.

To do this, they use detectors made of a grid of square tiles, kind of like a giant, high-tech floor made of glowing tiles. When a muon hits a tile, that tile lights up. The problem is, the tiles are big—about the size of a large pizza box. If a muon zips right through the middle of a tile, the detector knows exactly where it is. But if it skims the edge or hits at a weird angle, the light might spill over into neighboring tiles. The old way of figuring out where the muon actually went was to just guess the "center of gravity" of the light, like balancing a seesaw. But this guess often misses the mark, especially near the edges of the tiles, leaving the scientists with a blurry picture. The big question was: Could we teach a computer to look at the messy pattern of light and figure out the muon's true path with much higher precision, even if the muon didn't hit the center of a tile?

This paper sets out to answer that question by treating the detector's light pattern like a puzzle that a smart computer can solve. The researchers simulated a stream of cosmic muons hitting an 8-by-8 grid of glowing tiles (each tile being 6.25 cm wide) and tested four different ways to guess the muon's path. The first method was the old-school "balance the seesaw" approach (called the centroid method). The other three were different types of artificial intelligence: a standard neural network (MLP), a picture-recognizing network (CNN), and a fancy new type of network called a Transformer, which is the same kind of brainpower that helps computers understand language.

The results were a game-changer for precision. The old "seesaw" method was often off by about 1.43 cm, which is a huge miss when you're trying to see fine details. In contrast, the AI methods learned to spot the subtle clues in the light pattern that humans and simple math missed. The Transformer network was the star of the show, narrowing the error down to just 0.18 cm. To put that in perspective, the tiles are 6.25 cm wide, so the Transformer could pinpoint the muon's location to within about 3% of a single tile's width. That's like being able to tell if a fly landed on the very tip of a ruler's edge, even though you can only see the whole ruler. The CNN was also excellent, coming in at 0.23 cm error.

The study also looked at how well these methods could guess the angle of the muon's flight. Here, the Transformer and CNN reduced the error by 5 to 8 times compared to the old method. The researchers found that while the AI was incredibly good at the "typical" cases, it still struggled a tiny bit with very rare, tricky events where secondary particles created confusing light patterns, leading to a few large mistakes. However, for the vast majority of muons, the AI proved that you don't need to build smaller, more expensive tiles to get a sharper picture; you just need a smarter way to read the ones you already have.

It is important to note that these results come from computer simulations, not a physical experiment in a lab. The researchers built a virtual world with virtual muons and virtual detectors to test their ideas. While the numbers are promising, the paper suggests that real-world tests would need to account for things like electronic noise and imperfect sensors, which weren't part of this simulation. Nevertheless, the study strongly suggests that machine learning can unlock "sub-pixel" secrets hidden in coarse detectors, potentially making muon tomography sharper and more powerful for exploring the world's hidden structures.

Technical Summary: Machine Learning Methods for Sub-Pixel Trajectory Reconstruction in Discretized Position Detectors

Problem Statement
Muon tomography relies on reconstructing particle trajectories from hit positions measured in multiple detector planes. In pixelated scintillator detectors, the spatial resolution is fundamentally limited by the discrete geometry of the sensor cells. When a muon traverses a cell at an angle or produces secondary particles, the resulting scintillation light is often distributed across multiple cells. Conventional reconstruction methods, specifically the energy-weighted centroid, assume a symmetric signal distribution around the crossing point. However, this assumption is frequently violated near cell boundaries or for inclined tracks, leading to systematic position errors that degrade the angular resolution of the reconstructed trajectory. The challenge is to recover sub-pixel position information directly from coarse, multi-cell signal patterns without altering the detector hardware.

Methodology
The authors evaluated four reconstruction approaches using a dataset of 100,000 cosmic-ray muon events simulated in Geant4. The simulation modeled an 8×8 segmented polyvinyltoluene (PVT) scintillator array (6.25 cm cell width) and tracked optical photon transport to generate 64-cell signal patterns. The dataset was split into training (64%), validation (16%), and test (20%) sets.

The four methods compared were:

Energy-Weighted Centroid: A conventional baseline calculating the weighted average of cell coordinates based on recorded energy.
Multilayer Perceptron (MLP): A fully connected network mapping a flattened 64-element signal vector to $(x, y)$ coordinates.
Convolutional Neural Network (CNN): A network preserving the 2D spatial structure of the signal map using two convolutional layers followed by fully connected regression layers.
Hybrid Transformer: A novel architecture that decomposes the task into two stages:
- Coarse Cell Classification: Identifying the specific cell containing the muon crossing.
- In-Cell Offset Regression: Estimating the precise position within that cell.
  The transformer utilizes a convolutional stem to generate tokens, a four-layer encoder with self-attention to capture non-local correlations, and a composite loss function ( $\mathcal{L} = \mathcal{L}_{cell} + 2\mathcal{L}_{offset} + 0.5\mathcal{L}_{abs}$ ) to jointly optimize classification and regression.

Key Results
The methods were evaluated on a held-out test set of 19,053 events. The performance metrics highlight significant improvements of learned models over the centroid baseline:

Position Reconstruction:
- The Transformer achieved the lowest median absolute position error at 0.18 cm (approx. 2.9% of the 6.25 cm cell width).
- The CNN achieved a median error of 0.23 cm.
- The MLP achieved a median error of 0.30 cm.
- The Centroid method resulted in a median error of 1.43 cm.
- Relative to the centroid, the Transformer and CNN reduced median position errors by factors of 7.9× and 6.2×, respectively.
- While the CNN had the lowest Root Mean Square Error (RMSE) at 0.64 cm compared to the Transformer's 0.72 cm, the 95% bootstrap confidence intervals for the RMSE of the learned methods overlapped, indicating no statistically significant ordering among them.
Angular Reconstruction:
- The Transformer achieved a median angular error of 0.21°, and the CNN achieved 0.29°.
- The Centroid method yielded a median angular error of 1.50°.
- This represents a 7.1× and 5.1× reduction in median angular error for the Transformer and CNN, respectively, compared to the centroid.
Error Distribution Analysis:
- All learned methods demonstrated heavy-tailed error distributions. While typical events were reconstructed with high precision, a small subset of events (approx. 0.92% on the second detector plane) exhibited large errors.
- These large errors occurred when the brightest signal cell was displaced by more than one cell width from the true crossing point, often due to secondary particle activity creating competing peaks. This ambiguity in the input signal map limited the performance of all methods, including the learned models, particularly on the second detector plane where RMSE values were significantly higher than median errors.

Significance and Claims
The paper presents a proof-of-concept demonstrating that learned reconstruction methods can recover sub-pixel position information from coarse, segmented scintillator arrays without modifying the detector geometry. The authors claim that the hybrid transformer architecture, by separating cell classification from offset regression, effectively captures spatial correlations that fixed functional forms (like the centroid) miss.

The study emphasizes that while the improvements are most pronounced in the median and lower percentiles of the error distribution, the presence of a heavy error tail—driven by ambiguous signal patterns from secondary particles—limits the reduction in RMSE. The authors explicitly state that this work is limited to simulated data with a simplified source distribution and does not model photodetector electronics or noise. Consequently, they conclude that while the method shows promise for muon tomography and cosmic-ray measurements, experimental validation is required to assess performance under realistic conditions, including calibration variations and electronic noise. The work does not claim to solve the problem of trajectory reconstruction in all conditions but establishes that neural networks can significantly outperform traditional centroid methods in idealized, simulated scenarios.

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