Muon Tomo

Muon Tomography

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Standard reconstruction benchmark — forward model perfectly known, no calibration needed. Score = 0.5 × clip((PSNR−15)/30, 0, 1) + 0.5 × SSIM

# Method Score PSNR (dB) SSIM Source
🥇 PETFormer 0.816 34.92 0.967 ✓ Certified Li et al., ECCV 2024
🥈 TransEM 0.781 33.7 0.938 ✓ Certified Xie et al., 2023
🥉 PET-ViT 0.766 32.53 0.948 ✓ Certified Smith et al., ICCV 2024
4 DeepPET 0.749 32.4 0.918 ✓ Certified Haggstrom et al., MIA 2019
5 U-Net-PET 0.704 29.81 0.914 ✓ Certified Ronneberger et al. variant, MICCAI 2020
6 MAPEM-RDP 0.632 28.5 0.815 ✓ Certified Nuyts et al., 2002
7 ML-EM 0.625 26.85 0.854 ✓ Certified Shepp & Vardi, IEEE TPAMI 1982
8 FBP-PET 0.592 25.75 0.825 ✓ Certified Analytical baseline
9 OS-EM 0.533 23.97 0.767 ✓ Certified Hudson & Larkin, IEEE TMI 1994
10 OSEM 0.508 24.8 0.690 ✓ Certified Hudson & Larkin, IEEE TMI 1994

Dataset: PWM Benchmark (10 algorithms)

Blind Reconstruction Challenge — forward model has unknown mismatch, must calibrate from data. Score = 0.4 × PSNR_norm + 0.4 × SSIM + 0.2 × (1 − ‖y − Ĥx̂‖/‖y‖)

# Method Overall Score Public
PSNR / SSIM
 Dev
PSNR / SSIM
 Hidden
PSNR / SSIM
 Trust Source
🥇 TransEM + gradient 0.722
0.767
31.72 dB / 0.940
0.729
29.62 dB / 0.911
0.671
26.18 dB / 0.837
✓ Certified Xie et al., 2023
🥈 PETFormer + gradient 0.720
0.805
33.42 dB / 0.956
0.716
28.42 dB / 0.889
0.640
25.16 dB / 0.807
✓ Certified Li et al., ECCV 2024
🥉 PET-ViT + gradient 0.677
0.750
30.59 dB / 0.925
0.676
27.05 dB / 0.859
0.605
23.58 dB / 0.753
✓ Certified Smith et al., ICCV 2024
4 DeepPET + gradient 0.663
0.768
30.82 dB / 0.929
0.643
25.46 dB / 0.816
0.579
22.54 dB / 0.713
✓ Certified Haggstrom et al., MIA 2019
5 MAPEM-RDP + gradient 0.637
0.706
27.48 dB / 0.869
0.619
24.5 dB / 0.786
0.587
23.34 dB / 0.744
✓ Certified Nuyts et al., IEEE TMI 2002
6 FBP-PET + gradient 0.608
0.619
23.97 dB / 0.767
0.616
24.18 dB / 0.775
0.589
22.66 dB / 0.717
✓ Certified Analytical baseline
7 ML-EM + gradient 0.601
0.631
24.16 dB / 0.774
0.595
22.86 dB / 0.726
0.577
22.71 dB / 0.720
✓ Certified Shepp & Vardi, IEEE TPAMI 1982
8 OSEM + gradient 0.589
0.619
23.41 dB / 0.747
0.577
22.62 dB / 0.716
0.570
22.39 dB / 0.706
✓ Certified Hudson & Larkin, IEEE TMI 1994
9 U-Net-PET + gradient 0.579
0.700
27.47 dB / 0.869
0.567
21.74 dB / 0.679
0.470
18.68 dB / 0.534
✓ Certified Ronneberger et al. variant, MICCAI 2020
10 OS-EM + gradient 0.534
0.559
21.46 dB / 0.666
0.540
21.11 dB / 0.651
0.504
20.08 dB / 0.603
✓ Certified Hudson & Larkin, IEEE TMI 1994

Complete score requires all 3 tiers (Public + Dev + Hidden).

Join the competition →
Scoring: 0.4 × PSNR_norm + 0.4 × SSIM + 0.2 × (1 − ‖y − Ĥx̂‖/‖y‖) PSNR 40% · SSIM 40% · Consistency 20%
Public 3 scenes

Full-access development tier with all data visible.

What you get & how to use

What you get: Measurements (y), ideal forward operator (H), spec ranges, ground truth (x_true), and true mismatch spec.

How to use: Load HDF5 → compare reconstruction vs x_true → check consistency → iterate.

What to submit: Reconstructed signals (x_hat) and corrected spec as HDF5.

Public Leaderboard
# Method Score PSNR SSIM
1 PETFormer + gradient 0.805 33.42 0.956
2 DeepPET + gradient 0.768 30.82 0.929
3 TransEM + gradient 0.767 31.72 0.94
4 PET-ViT + gradient 0.750 30.59 0.925
5 MAPEM-RDP + gradient 0.706 27.48 0.869
6 U-Net-PET + gradient 0.700 27.47 0.869
7 ML-EM + gradient 0.631 24.16 0.774
8 FBP-PET + gradient 0.619 23.97 0.767
9 OSEM + gradient 0.619 23.41 0.747
10 OS-EM + gradient 0.559 21.46 0.666
Spec Ranges (3 parameters)
Parameter Min Max Unit
angular_resolution -2.0 4.0 mrad
momentum_estimate -10.0 20.0 %
detector_efficiency -3.0 6.0 %
View Details →
Dev 3 scenes

Blind evaluation tier — no ground truth available.

What you get & how to use

What you get: Measurements (y), ideal forward operator (H), and spec ranges only.

How to use: Apply your pipeline from the Public tier. Use consistency as self-check.

What to submit: Reconstructed signals and corrected spec. Scored server-side.

Dev Leaderboard
# Method Score PSNR SSIM
1 TransEM + gradient 0.729 29.62 0.911
2 PETFormer + gradient 0.716 28.42 0.889
3 PET-ViT + gradient 0.676 27.05 0.859
4 DeepPET + gradient 0.643 25.46 0.816
5 MAPEM-RDP + gradient 0.619 24.5 0.786
6 FBP-PET + gradient 0.616 24.18 0.775
7 ML-EM + gradient 0.595 22.86 0.726
8 OSEM + gradient 0.577 22.62 0.716
9 U-Net-PET + gradient 0.567 21.74 0.679
10 OS-EM + gradient 0.540 21.11 0.651
Spec Ranges (3 parameters)
Parameter Min Max Unit
angular_resolution -2.4 3.6 mrad
momentum_estimate -12.0 18.0 %
detector_efficiency -3.6 5.4 %
Submit Reconstruction View Details →
Hidden 3 scenes

Fully blind server-side evaluation — no data download.

What you get & how to use

What you get: No data downloadable. Algorithm runs server-side on hidden measurements.

How to use: Package algorithm as Docker container / Python script. Submit via link.

What to submit: Containerized algorithm accepting y + H, outputting x_hat + corrected spec.

Hidden Leaderboard
# Method Score PSNR SSIM
1 TransEM + gradient 0.671 26.18 0.837
2 PETFormer + gradient 0.640 25.16 0.807
3 PET-ViT + gradient 0.605 23.58 0.753
4 FBP-PET + gradient 0.589 22.66 0.717
5 MAPEM-RDP + gradient 0.587 23.34 0.744
6 DeepPET + gradient 0.579 22.54 0.713
7 ML-EM + gradient 0.577 22.71 0.72
8 OSEM + gradient 0.570 22.39 0.706
9 OS-EM + gradient 0.504 20.08 0.603
10 U-Net-PET + gradient 0.470 18.68 0.534
Spec Ranges (3 parameters)
Parameter Min Max Unit
angular_resolution -1.4 4.6 mrad
momentum_estimate -7.0 23.0 %
detector_efficiency -2.1 6.9 %
Submit Algorithm View Details →

Blind Reconstruction Challenge

Challenge

Given measurements with unknown mismatch and spec ranges (not exact params), reconstruct the original signal. A method must be evaluated on all three tiers for a complete score. Scored on a composite metric: 0.4 × PSNR_norm + 0.4 × SSIM + 0.2 × (1 − ‖y − Ĥx̂‖/‖y‖).

Input

Measurements y, ideal forward model H, spec ranges

Output

Reconstructed signal x̂

About the Imaging Modality

Muon tomography uses naturally occurring cosmic-ray muons (mean energy ~4 GeV, flux ~1/cm2/min at sea level) to image the interior of large, dense objects by measuring the scattering angle of each muon as it traverses the object. High-Z materials (uranium, plutonium, lead) cause large-angle scattering that is readily distinguished from low-Z materials. Position-sensitive detectors (drift tubes, RPCs) above and below the object track each muon's trajectory. The scattering density is proportional to Z^2/A. Reconstruction uses the point-of-closest-approach (POCA) algorithm or maximum-likelihood/expectation-maximization (ML-EM). Long exposure times (minutes to hours) are needed due to the low natural muon flux. Applications include nuclear material detection and volcano interior imaging (muography).

Principle

Muon tomography uses naturally occurring cosmic-ray muons to image the internal density structure of large objects (buildings, volcanoes, cargo containers). Muons undergo multiple Coulomb scattering, with the scattering angle proportional to the areal density and atomic number of the traversed material. By measuring the incoming and outgoing muon trajectories, the density distribution inside the object can be tomographically reconstructed.

How to Build the System

Place tracking detectors (drift tubes, scintillator strips, resistive plate chambers, or GEM detectors) above and below (or around) the object to be imaged. Each detector station measures the position and angle of each cosmic-ray muon before and after it traverses the object. Typical cosmic-ray muon flux is ~10,000 muons/m²/min at sea level. Exposure times range from minutes (for dense nuclear materials) to months (for geological structures like volcanoes).

Common Reconstruction Algorithms

  • Point of Closest Approach (POCA) voxel reconstruction
  • Maximum Likelihood / Expectation Maximization (ML/EM) scattering tomography
  • Angle Statistics Reconstruction (ASR) for material discrimination
  • Binned scattering density reconstruction
  • Deep-learning muon tomography for faster convergence with fewer muons

Common Mistakes

  • Insufficient muon statistics for the desired spatial resolution (need long exposure)
  • Detector alignment errors causing incorrect scattering angle measurements
  • Not accounting for muon momentum spectrum (affects scattering angle distribution)
  • Background tracks (electrons, low-momentum muons) contaminating the data
  • POCA algorithm limitations in complex, non-point-like geometries

How to Avoid Mistakes

  • Calculate required exposure time based on object size, density, and desired resolution
  • Align detectors carefully using straight-through cosmic ray tracks as calibration
  • Use momentum measurement (from curvature in a magnetic field) or momentum-dependent MCS model
  • Apply track quality cuts (chi-squared, minimum number of detector hits) to reject background
  • Use iterative reconstruction (ML/EM) rather than POCA for quantitative density imaging

Forward-Model Mismatch Cases

  • The widefield fallback applies Gaussian blur, but muon tomography measures the scattering angle of cosmic-ray muons passing through the object — the scattering angle (Highland formula) encodes radiation length and density, not image blur
  • Muon tomography uses natural cosmic-ray flux (~10,000 muons/m^2/min) with tracking detectors above and below the object — the widefield optical model has no connection to high-energy particle tracking or multiple Coulomb scattering physics

How to Correct the Mismatch

  • Use the muon tomography operator that models multiple Coulomb scattering: incoming and outgoing muon tracks are measured, and the scattering angle distribution at each voxel encodes the local radiation length (related to material Z and density)
  • Reconstruct using POCA (Point of Closest Approach) for quick imaging, or ML/EM iterative methods for quantitative density/Z mapping, using the correct scattering probability forward model

Experimental Setup — Signal Chain

Experimental setup diagram for Muon Tomography

Experimental Setup

Instrument: Los Alamos muon radiography / Decision Sciences MMS
Mean Energy Gev: 4.0
Flux Per Cm2 Per Min: 1.0
Detector: drift tube / RPC panels (2 tracking planes above + 2 below)
Position Resolution Mm: 1.0
Angular Resolution Mrad: 3.0
Exposure Min: 60
Technique: multiple scattering tomography (MST)
Application: nuclear material detection / volcano imaging

Key References

  • Borozdin et al., 'Radiographic imaging with cosmic-ray muons', Nature 422, 277 (2003)
  • Tanaka et al., 'Imaging the conduit size of the dome with cosmic-ray muons: The structure beneath Showa-Shinzan Lava Dome', Geophysical Research Letters 34, L22311 (2007)

Canonical Datasets

  • Los Alamos muon tomography simulation benchmarks
  • IAEA muon imaging reference data

Spec DAG — Forward Model Pipeline

R(θ_cosmic) → Π(muon) → D(g, η₁)

R Cosmic Muon Incidence (θ_cosmic)
Π Muon Scattering Projection (muon)
D Drift Tube Tracker (g, η₁)

Mismatch Parameters

Symbol Parameter Description Nominal Perturbed
Δθ angular_resolution Angular resolution error (mrad) 0 2.0
Δp momentum_estimate Muon momentum error (%) 0 10.0
Δε detector_efficiency Detector efficiency error (%) 0 3.0

Credits System

40%
Platform Profit Pool
Revenue allocated to benchmark rewards
30%
Winner Share
Top algorithm receives from pool
$100
Min Withdrawal
Minimum payout threshold
Spec Primitives Reference (11 primitives)
P Propagation

Free-space or medium propagation kernel (Fresnel, Rayleigh-Sommerfeld).

M Mask / Modulation

Spatial or spatio-temporal amplitude modulation (coded aperture, SLM pattern).

Π Projection

Geometric projection operator (Radon transform, fan-beam, cone-beam).

F Fourier Sampling

Sampling in the Fourier / k-space domain (MRI, ptychography).

C Convolution

Shift-invariant convolution with a point-spread function (PSF).

Σ Summation / Integration

Summation along a physical dimension (spectral, temporal, angular).

D Detector

Sensor readout with gain g and noise model η (Gaussian, Poisson, mixed).

S Structured Illumination

Patterned illumination (block, Hadamard, random) applied to the scene.

W Wavelength Dispersion

Spectral dispersion element (prism, grating) with shift α and aperture a.

R Rotation / Motion

Sample or gantry rotation (CT, electron tomography).

Λ Wavelength Selection

Spectral filter or monochromator selecting a wavelength band.