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Deep-Unfolded Penalized MLEM for Rapid Poisson Image Reconstruction
Zohar Davidov, Alon Osovizky, Max Ghelman, Nir Shlezinger
TL;DR
Poisson MLEM can require many iterations for accurate reconstruction, while model mismatch can impair its behavior. This paper unfolds penalized MLEM into a finite-depth trainable architecture that retains analytical projections and adapts reconstruction parameters and sensitivity normalization. Experiments report improved reconstruction quality with substantially fewer update stages than iterative MLEM and task-adapted direct models.
Problem
Poisson MLEM may require many iterations for accurate recovery and can amplify measurement noise under model mismatch.
Source: §1 · p. 1Method
The method unfolds a prescribed small number of penalized-MLEM iterations into trainable layers, learning layer-dependent parameters and a data-dependent sensitivity correction while retaining analytical projections.
Source: §1 · p. 1, §3.1 · p. 2, §3.3 · p. 3Results
Takeaways & Limitations
The architecture combines model-based reconstruction with data-driven adaptation for recovery under limited iteration budgets and model mismatch.
Source: §5 · p. 4Training assumes access to a dataset of measurement and ground-truth image pairs.
Source: §2 · p. 1
Abstract
from arXiv · showhide
Maximum likelihood expectation maximization (MLEM) is a common approach for Poisson image reconstruction, but accurate recovery requires many iterations and is sensitive to mismatch in the assumed acquisition model. We propose a deep-unfolded penalized MLEM framework that maps a prescribed small number of iterations into trainable layers while retaining the analytical forward/backward projections and the multiplicative structure of MLEM. The unfolded architecture learns layer-dependent regularization and Poisson-model parameters, together with a data-dependent correction of the sensitivity normalization to mitigate model mismatch. We numerically show that our method improves reconstruction resolution at a fixed iteration budget and attains reconstruction quality comparable to long MLEM runs and direct data-driven models.
1. INTRODUCTION
Poisson MLEM can require many iterations and remains vulnerable to acquisition-model mismatch. The paper unfolds penalized MLEM into a trainable finite-depth architecture that adapts regularization and sensitivity normalization while retaining the physical projections.
1. INTRODUCTION
- Poisson MLEM may need many iterations for accurate reconstruction, and continued iterations can amplify measurement noise under model mismatch.Penalized-likelihood variants add prior structure but require regularization parameters that are typically fixed across iterations.Source: §1 · p. 1
- Prior acceleration methods modify measurement or variable processing, regularized-likelihood methods use spatial priors, and aggregation exploits acquisition structure, yet updates remain analytically prescribed.The conventional iterative behavior relies on a known forward operator and numerous iterations.Source: §1 · p. 1
- Deep unfolding maps a prescribed iteration count into trainable layers, while prior Poisson deconvolution work cited here uses a weighted least-squares surrogate instead of Poisson data fidelity.Plug-and-play methods are a complementary model-based approach that incorporate learned priors through denoisers.Source: §1 · p. 1
- The proposed unfolded method improves reconstruction quality over direct data-driven models using a fixed number of layers, versus substantially longer classical and penalized-MLEM runs.It is evaluated on Richardson-Lucy deconvolution and Compton camera imaging.Source: §1 · p. 1
2. POISSON IMAGING AND MLEM
The section formulates photon-counting reconstruction under a Poisson model and describes MLEM and its penalized variant. Their iteration costs and dependence on the assumed acquisition model motivate limited-latency and model-mismatch challenges.
2. POISSON IMAGING AND MLEM
- Photon counts are modeled as Poisson with mean Ax+b, with a nonnegative forward operator A and known background contribution b.The reconstruction goal is to recover x from measurements y using the physical forward operator A.Source: §2 · p. 1
- The formulation targets limited reconstruction latency and an available A that may only approximate the acquisition model, assuming access to paired measurements and ground-truth images.These are identified as challenges C1 and C2.Source: §2 · p. 1
- Each MLEM iteration requires one forward projection through A and one backward projection through A^T, making many iterations costly.The classical update uses sensitivity s = A^T1 and initializes x(0) > 0.Source: §2 · p. 2
- A spatial regularizer can suppress noise amplification, but conventional penalized MLEM fixes γ across iterations and relies on A and its derived sensitivity, leaving it susceptible to model mismatch.γ controls the trade-off between data fidelity and spatial smoothness.Source: §2 · p. 2
3. DEEP-UNFOLDED MLEM
The section unfolds penalized MLEM into a fixed-depth architecture that retains analytical projections while learning layer-specific update components. It also describes progressive training followed by end-to-end fine-tuning and analyzes inference costs.
3.1. Architecture
- Each layer unfolds a penalized MLEM step, retaining A and A^T while learning layer-specific Poisson scales, regularization, and a sensitivity correction.The learned correction is intended to mitigate discrepancies between nominal and actual acquisition models.Source: §3.1 · p. 2
3.2. Training
- Training first adds layers sequentially with preceding layers fixed, then jointly fine-tunes all layers using the final reconstruction.The progressive-stage hybrid loss combines image-shape, source-localization, and Poisson data-consistency terms.Source: §3.2 · pp. 2–3
3.3. Discussion
- The prescribed depth K bounds the number of forward and backward projections, while learned parameters adapt the reconstruction trajectory and sensitivity normalization to model discrepancies.The imaging model remains explicitly embedded in every layer rather than being replaced by learning.Source: §3.3 · p. 3
- When K ≪ T, unfolded inference can reduce overall cost: each layer applies A and A^T at O(MN), while the CNN cost scales as O(N).The comparison concerns inference complexity and runtime; the stated cost reduction applies when K is much smaller than T.Source: §3.3 · p. 3
4. EXPERIMENTAL STUDY
The study evaluates a ten-layer unfolded MLEM method in synthetic photon-limited deconvolution and Compton-camera imaging, comparing reconstruction quality with iterative and direct-learning methods.
Experimental setup
- Both settings use ten unfolded layers: synthetic RL deconvolution and CC imaging are assessed against iterative MLEM variants and task-adapted direct models.The RL test set has 500 images; the CC test set has 500 sets of 1000 events.Source: §4 · pp. 3–4
CC experiment
- The ten-layer reconstruction resolves both target intersections more clearly than ComptonNet, whose filled-in rectangle yields only a shallow central dip.Panels (a)-(g) show reconstructions, while (h)-(m) show central y = 0 profiles.Source: Figure 3 · p. 4, §4 · p. 4
- Approximately 0.15 NMSE after ten unfolded layers compares with roughly 0.35-0.4 for conventional methods after ten iterations and approximately 0.23 for penalized MLEM after 100.Classical MLEM begins to deteriorate after approximately 15 iterations, while ComptonNet achieves approximately 0.70 NMSE.Source: Figure 4 · p. 4, §4 · p. 4
RL experiment
- An NMSE of 0.11 after ten unfolded layers compares with 0.18 for penalized MLEM after 300 iterations and 0.27 for DenseDeconNet; classical MLEM is non-converging.The experiment uses synthetic photon-limited deconvolution with a known additive background.Source: §4 · p. 4
5. CONCLUSIONS
The conclusion presents deep-unfolded penalized MLEM as a finite-depth Poisson reconstruction method that preserves MLEM structure while learning parameters and a sensitivity correction. Across two imaging experiments, it reports improved quality with fewer update stages than iterative MLEM and task-adapted direct models.
Conclusion
- The framework preserves analytical forward and backward projections and multiplicative updates while learning iteration-dependent parameters and a sensitivity correction.It is intended for finite-depth Poisson image reconstruction.Source: §5 · p. 4
- The architecture combines model-based reconstruction with data-driven adaptation for limited iteration budgets and acquisition-model mismatch.Source: §5 · p. 4
- Experiments on photon-limited deconvolution and CC imaging report improved reconstruction quality with substantially fewer update stages than iterative MLEM and task-adapted direct models.Source: §5 · p. 4