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Matching Particle-Arrival Rates for Low-Count Image Denoising

Huazhen Liu, Zipei Yan, Jizhou Li

arXiv:2610.33136v1eess.IVcs.CV

TL;DR

Low-count denoising is difficult when Poisson noise dominates and clean references are unavailable. PARM estimates conditional particle-arrival rates using exposure-indexed count modeling and thinning-based self-supervision. It reports leading or competitive results across synthetic, video, and microscopy settings, while microscopy comparisons measure consistency with multi-frame references rather than fidelity to unknown ground truth.

  • Problem

    Clean references are often unavailable for dynamic and scientific imaging, while many denoising approaches do not explicitly model how Poisson counts accumulate with exposure.

    Source: §I · p. 1, A. Poisson and Low-Count Image Denoising · p. 2

  • Method

    PARM matches rate targets from complementary Poisson-thinned views to estimate the conditional particle-arrival-rate field, the posterior mean of the latent clean rate.

    Source: E. Self-Supervised Rate Matching by… · p. 5, §V · p. 10

  • Results

    PARM achieved competitive or leading performance across synthetic Poisson and standard video benchmarks, and its microscopy reconstructions preserved nanoparticle structures and showed strongest agreement with group-averaged multi-frame references.

    Source: §V · p. 10

  • Takeaways & Limitations

    The results support PARM as a self-supervised approach to low-count denoising, including low-dose microscopy where clean reference targets are unavailable.

    Source: §V · p. 10

    The theoretical equivalence is a population Bayes-predictor statement, not a guarantee of finite-sample generalization, parameter consistency, or optimization convergence; the consistency regularizer may bias the population optimum.

    Source: C. Scope and Limitations of the… · pp. 11–12

Abstract

from arXiv · show

Low-count imaging arises when limited illumination or dose yields only a few detected particles in microscopy, such as photons or electrons, causing measurements to be dominated by signal-dependent Poisson shot noise. Supervised denoisers typically require paired clean and noisy observations, whereas many self-supervised approaches exploit spatial or temporal redundancy without directly estimating the underlying particle-arrival process. We introduce PARM, a self-supervised denoising framework based on particle-arrival-rate matching. By modeling particle accumulation over exposure as a Poisson counting process, PARM identifies the conditional particle-arrival-rate field with the posterior mean of the latent clean rate. Poisson thinning then splits each observation into complementary lower-exposure views that provide self-supervised rate targets for one another. In implementation, the network estimates the normalized spatial allocation of the conditional rate, while the observed total count determines its global particle mass. Experiments on synthetic data, natural-video sequences corrupted with Poisson noise, and real low-dose electron microscopy demonstrate leading or competitive performance while effectively preserving fine structures. PARM therefore provides a statistically grounded approach to low-count image denoising when clean references are unavailable.

I. INTRODUCTION

Low-count imaging suffers from signal-dependent Poisson shot noise, while clean references are often unavailable and many self-supervised methods do not model exposure-dependent count accumulation. PARM estimates a conditional particle-arrival rate and builds self-supervised targets by thinning a single observation.

A. Poisson and Low-Count Image Denoising

I. INTRODUCTION

  • PARM derives a conditional particle-arrival-rate field equal to the posterior mean of the clean rate and uses it as its denoising target.The derivation applies the Kolmogorov forward equation to exposure-indexed count dynamics.Source: §I · p. 1

B. Self-Supervised Denoising Approaches

  • Many self-supervised methods exploit spatial, temporal, or domain-specific redundancy without explicitly modeling exposure-dependent Poisson count accumulation.Existing approaches include blind-spot and subsampling methods, as well as methods for dynamic and scientific measurements.Source: B. Self-Supervised Denoising Approaches · p. 2
  • Poisson thinning splits one observation into complementary lower-exposure views whose counts provide self-supervised rate targets for one another.The views remain conditionally independent given the latent clean rate.Source: §I · p. 1, B. Self-Supervised Denoising Approaches · p. 2

C. Discrete Probability Paths and Count-Process Modeling

PARM models particle counts as an exposure-indexed process and estimates the conditional arrival rate from the observed count state. Thinning supplies paired lower-exposure views, while the underlying process describes counts accumulating through particle arrivals.

C. Discrete Probability Paths and Count-Process Modeling

  • PARM models the physical accumulation of discrete particle counts with exposure and derives a conditional transition-rate field as a statistical restoration target.Unlike related inverse solvers relying on pretrained continuous generative priors, it directly models count accumulation.Source: C. Discrete Probability Paths and… · pp. 2–3

A. Overview: From Observed Counts to Particle-Arrival Rates

  • A single count observation is split by Poisson thinning into complementary lower-exposure views, with the complementary counts supplying exposure-normalized rate targets.Under the calibrated Poisson model, matching these targets has the same population-optimal predictor as supervised clean-rate regression.Source: A. Overview: From Observed Counts to… · p. 3

B. Exposure-Indexed Poisson Pure-Birth Process

  • Conditioned on the clean image, particle counts accumulate with exposure as a pure-birth process, with arrivals at pixel i occurring at rate λ0xi.Figure 1 depicts the count distributions at exposures pτ, qτ, and τ, and the complementary views produced by thinning.Source: B. Exposure-Indexed Poisson Pure-Birth… · p. 3, Figure 1 · p. 3

C. Conditional Particle-Arrival-Rate Field

Marginalizing the clean-image distribution yields a state-dependent particle-arrival-rate field: the posterior mean latent rate given the observed count state. The field also represents conditional transition rates for additional particle arrivals.

C. Conditional Particle-Arrival-Rate Field

  • Marginalizing over clean images makes the conditional particle-arrival-rate field the posterior mean of the latent clean rate given the observed count state, not a MAP estimate.The field is defined for count states with positive probability.Source: C. Conditional Particle-Arrival-Rate… · p. 4
  • The field gives the state-dependent arrival rates in the marginal count-process dynamics.The conditional dynamics are obtained by marginalizing the clean-image distribution in the Kolmogorov forward equation.Source: C. Conditional Particle-Arrival-Rate… · p. 4
  • At each coordinate, the conditional rate is the transition rate for an additional particle arrival, motivating a finite-exposure target with the same conditional mean.Infinitesimal transitions are not directly observable from a single acquisition.Source: C. Conditional Particle-Arrival-Rate… · p. 4

D. Finite-Exposure Rate Matching

Finite-exposure increments provide rate targets whose conditional mean matches the conditional particle-arrival-rate field, and GKL regression recovers the corresponding Bayes-optimal predictor. Poisson thinning constructs complementary views and targets from one observation, with randomized splitting yielding an implicit-exposure implementation.

D. Finite-Exposure Rate Matching

  • The finite-exposure increment is conditionally unbiased for the clean rate, and its conditional mean given the lower-exposure observation equals the particle-arrival-rate field.It is a finite-exposure surrogate for the transition-rate field governing count dynamics.Source: D. Finite-Exposure Rate Matching · p. 4
  • Under GKL divergence, regression to the finite-exposure target and supervised clean-rate regression share the same Bayes-optimal predictor.The GKL conditional population risk is minimized by the conditional mean of the rate target.Source: D. Finite-Exposure Rate Matching · p. 4

E. Self-Supervised Rate Matching by Poisson Thinning

  • Poisson thinning splits one observation into conditionally independent complementary views that represent exposures pτ and qτ, each usable as input with the other view supplying its rate target.The reverse thinning direction follows symmetrically.Source: E. Self-Supervised Rate Matching by… · p. 4
  • With randomized splitting and no exposure input to the network, both thinning directions train one shared network on inputs spanning effective exposures.For exact Poisson counts, the resulting Bayes-optimal target is the exposure-marginalized conditional rate; for noninteger measurements, exact splitting and population-equivalence claims become approximations after randomized rounding.Source: E. Self-Supervised Rate Matching by… · p. 5

F. Practical Rate-Allocation Learning and Low-Count Stabilization

PARM learns a normalized spatial allocation of the conditional particle-arrival rate, using complementary views and a bidirectional objective. A global occupancy statistic controls whether low-count consistency regularization is active.

F. Practical Rate-Allocation Learning and Low-Count Stabilization

  • The learned allocation is the Bayes-optimal normalized target under GKL risk, with the conditional mean marginalized over exposure states when the splitting ratio is not provided.The network maps its output to a probability simplex; the marginalized mass–allocation factorization is given in Appendix B.Source: F. Practical Rate-Allocation Learning… · p. 5
  • Empirical unit-mean normalization creates bidirectional rate targets that preserve relative allocation while approximating ideal exposure-normalized targets.Each view is supervised by the normalized complementary view.Source: F. Practical Rate-Allocation Learning… · p. 5
  • At extremely low counts, soft cross-view consistency regularization stabilizes learning when complementary thinned views differ substantially.The views can be substantially different sparse realizations despite sharing the same latent scene.Source: F. Practical Rate-Allocation Learning… · p. 5
  • Consistency is active when zeros are abundant and disabled when they are scarce; the Λth = 0.4 operating point corresponds to about 20% multievent observations among nonzero counts.The global occupancy statistic acts as a sparsity indicator, making consistency a weak low-count stabilizer.Source: F. Practical Rate-Allocation Learning… · p. 6

G. Inference and Observed-Mass-Preserving Reconstruction

PARM combines a network-predicted relative allocation with observed total count to produce an acquisition-scale reconstruction. This practical rule is not an exact Bayes estimator, and its theorem has specified assumptions.

G. Inference and Observed-Mass-Preserving Reconstruction

  • The network infers the relative particle-arrival distribution while the observation supplies global particle mass, avoiding prediction of an absolute exposure-dependent magnitude.The observed total count is conditionally unbiased for latent total count, with lower relative fluctuation than individual low-count sites.Source: G. Inference and… · p. 6
  • The trained allocation is exposure-marginalized when exposure is not an input, and applying it to the complete observation is a practical inference rule rather than a direct theorem consequence.Allocation is invariant to common rate–count scaling, though its conditional dependence on the observation distribution may still vary with exposure.Source: G. Inference and… · p. 6
  • Inference aggregates three network evaluations using empirical score-weighted aggregation, which is not a Bayesian posterior.The original input and sequentially thinned versions are processed to form positive candidate fields.Source: G. Inference and… · p. 6
  • The observed-mass-preserving factorization is not an exact Bayes estimator: at zero observed mass it returns zero even when the posterior mean may be positive.The theorem’s equivalence assumes calibrated integer-valued Poisson counts, known exposure intervals, and unrestricted predictors optimized under GKL; the regularizer and reconstruction are outside its guarantee.Source: G. Inference and… · p. 6

IV. EXPERIMENTS

PARM is evaluated on synthetic and real low-count imaging, with quantitative metrics for settings that have clean sources and reference-based structural assessment for electron microscopy.

IV. EXPERIMENTS

  • The evaluation covers a controlled synthetic Cameraman benchmark, Poisson-corrupted natural videos, and real low-dose electron microscopy without clean ground truth.The electron-microscopy assessment uses structural preservation and agreement with local multi-frame, high-SNR references, not ground truth.Source: §IV · p. 6
  • PSNR and SSIM support quantitative evaluation only in the synthetic and natural-video settings, where clean source data are available.Electron microscopy is assessed against local multi-frame, high-SNR references.Source: §IV · p. 6
  • Comparisons span model-based, single-image deep-learning, and sequence-based deep-learning baselines; methods without native video or volumetric formulations are evaluated only in 2D.The study does not introduce new 3D backbones solely for comparison.Source: §IV · p. 6

A. Implementation Details and Baseline Settings

PARM uses self-supervised 2D and 3D models and is evaluated on a controlled Cameraman sequence. On that benchmark, it leads self-supervised methods at every tested particle level and approaches a clean-supervised reference.

1) Dataset and setup:

  • The controlled Cameraman benchmark uses ordered 128×128 clean patches sampled along a digit-2 trajectory from a 256×256 image and corrupted at multiple particle levels.The paired clean and noisy stacks enable quantitative evaluation under a controlled particle-counting model.Source: 1) Dataset and setup · p. 7

A. Implementation Details and Baseline Settings

  • The 2D model processes frames independently, while the 3D model uses ordered-image spatiotemporal context; neither receives clean images or exposure as input.Models are optimized separately for each sequence and particle level after self-supervised pretraining on Flower; no clean targets are used for PARM or self-supervised baselines.Source: A. Implementation Details and Baseline… · pp. 6–7, 1) Dataset and setup · p. 7

2) Quantitative results:

  • PARM achieved the highest PSNR and SSIM among self-supervised methods at every tested particle level, with its advantage most evident at low counts.It approached the clean-supervised PARM(Sup.) reference without clean targets; competing methods retained particle noise or oversmoothed fine structures.Source: 2) Quantitative results · p. 7

3) Qualitative analysis:

On synthetic Poisson denoising and six noisy video sequences, PARM preserved fine structures and ranked near the top across most tested settings. In real low-dose electron microscopy, it retained nanoparticle detail; ablations show trade-offs in consistency strength and stack size.

3) Qualitative analysis:

  • In the synthetic Cameraman comparisons, PARM retained edges and local transitions in 2D and recovered clearer camera contrast and shape in 3D.The 3D comparison uses Λ = 0.25; the 2D comparison uses Λ = 2.Source: 3) Qualitative analysis · p. 7, Figure 2 · p. 8

C. Poisson Denoising on Natural Video

  • PARM ranked first or second for most sequence, particle-level, and metric combinations across six noisy videos, though competitors won in several settings.The strongest gains occurred where thin structures and local textures were difficult to retain.Source: C. Poisson Denoising on Natural Video · p. 7, Table II · p. 9

D. Real Low-Dose Electron Microscopy

  • Under the real-microscopy consistency protocol, PARM achieved the highest measured correlation and retained nanoparticle contrast and visible intensity peaks.The group-averaged references assess consistency with the measured sequence, not fidelity to an unknown clean image.Source: D. Real Low-Dose Electron Microscopy · pp. 7–8

E. Ablation Study

  • On Cameraman, a large base consistency coefficient degraded the displayed low-count settings; the Coastguard sweep showed benefits only in the sparsest regimes and lower PSNR at higher particle levels.The default base coefficient is γ0 = 0.01.Source: E. Ablation Study · p. 8
  • Increasing temporal stack size generally improved PSNR, especially at low particle levels, while stacks beyond 32 frames increased memory use and runtime.The selected stack depth is 32 frames to balance reconstruction quality and computational cost.Source: E. Ablation Study · p. 8
  • In the reported Cameraman ablation, thinning-based aggregation mode and evaluation count had negligible effect on reconstruction quality.Main experiments use score-weighted aggregation with K = 3; direct prediction appears only as the K = 1 ablation.Source: E. Ablation Study · p. 8

F. Computational Efficiency

PARM achieved competitive or leading results across synthetic Poisson and video benchmarks while preserving fine structures, and preserved nanoparticle structure in real low-dose microscopy. Its runtime falls within the range of learning-based self-supervised baselines, with differences across comparators.

F. Computational Efficiency

  • PARM falls within the runtime range of learning-based self-supervised baselines, is faster than SHINE, but slower than N2V and CPU VBM4D on all reported sequences and UDVD on some.VBM4D runs on CPU while learning-based methods run on GPU, so the measurements compare wall-clock cost rather than hardware-normalized algorithmic efficiency.Source: F. Computational Efficiency · p. 9, Table III · p. 10

V. CONCLUSION

  • Exposure-indexed Poisson modeling yields a conditional rate field equal to the posterior mean clean rate; for exact calibrated counts, thinning targets share the supervised predictor under GKL.Source: §V · p. 10
  • PARM achieved competitive or leading performance across particle levels on synthetic Poisson data and standard video benchmarks while retaining fine structures.Source: §V · p. 10
  • On real low-dose microscopy, PARM preserved nanoparticle structures and showed strongest agreement with group-averaged references, demonstrating reconstruction consistency rather than absolute fidelity to an unknown clean image.Source: §V · p. 10
  • Future work could extend PARM to mixed Poisson–Gaussian noise, more efficient multiscale architectures, and real-time low-count video and microscopy pipelines.Source: §V · p. 10

APPENDIX

The appendix explains how PARM handles randomized thinning without observing the exposure ratio and how its softmax represents spatial allocation separately from particle mass. It also reports image comparisons and ablations concerning detail preservation, consistency strength, and stack size.

A. Randomized Exposure Without Explicit Conditioning

  • When the thinning ratio is hidden from the network, the population predictor remains the Bayes-optimal rate estimate marginalized over the randomized thinning distribution.Source: A. Randomized Exposure Without… · p. 10
  • The network maps a noisy count image or stack to a score field, then uses softmax to represent relative spatial allocation rather than global rate magnitude.Source: A. Randomized Exposure Without… · p. 10
  • On Pt nanoparticle data, PARM retains individual nanoparticle contrast and periodic intensity peaks, unlike Gaussian blur and 50-frame UDVD, which remove or blur details.The correlation values use a group-size-50 reference; error bars reflect four spatial-subsampling offsets, not independent microscopy experiments.Source: Figure 4 · p. 11
  • On 3D Cameraman, γ0 = 0.01 is the default because large coefficients degrade displayed particle levels; deeper temporal stacks generally improve quality but cost memory and runtime.Error bars show PSNR standard deviation over five independently generated noisy realizations.Source: Figure 5 · p. 11

B. Connection to Softmax Allocation

  • The softmax output identifies normalized posterior-rate allocation, while target magnitude weights the allocation loss and inference restores global particle mass separately.Source: B. Connection to Softmax Allocation · p. 11

C. Scope and Limitations of the Theoretical Guarantees

The equivalence identifies the conditional particle-arrival-rate target with the posterior mean at the population Bayes-optimal predictor, but does not guarantee equal numerical risks or finite-sample and optimization properties. The guarantees rely on specific Poisson-count and predictor assumptions; practical regularization and inference choices are not theorem consequences.

C. Scope and Limitations of the Theoretical Guarantees

  • The equivalence is a population statement about the Bayes-optimal predictor, not numerical risk equality, finite-sample generalization, parameter consistency, or optimization convergence.It identifies the conditional target vτ(y) with E[λ0X | Yτ = y].Source: C. Scope and Limitations of the… · p. 11
  • The exact result assumes calibrated integer-valued Poisson counts, known exposure intervals, independent increments, finite first moments, unrestricted measurable finite-risk predictors, and GKL optimization.If the thinning ratio is independently randomized without being provided to the network, the identity holds for a predictor marginalized over the randomized exposure distribution.Source: C. Scope and Limitations of the… · p. 12
  • The soft cross-view consistency regularizer is a weak empirical stabilization prior, not part of the equivalence theorem, and may bias the population optimum.The two predictors generally differ pointwise because they condition on observations at different effective exposures; the raw-score penalty also constrains a nonidentifiable global logit offset.Source: C. Scope and Limitations of the… · p. 12
  • Practical implementation choices and full-observation inference are not consequences of the theorem, and Poisson-model violations make the objective a Poisson-informed surrogate rather than an exact unbiased-risk construction.The population analysis concerns inputs drawn from the randomized thinning distribution; cited violations include read noise, detector gain, imperfect exposure calibration, and preprocessing.Source: C. Scope and Limitations of the… · p. 12

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