Efficient Bayesian Computation Using Plug-and-Play Priors for Poisson Inverse Problems

2025-03-20

AI Summary

This paper addresses key challenges in Bayesian inference for low-photon Poisson imaging: conventional plug-and-play (PnP) Langevin sampling suffers from gradient explosion and failure to enforce non-negativity constraints. To overcome these limitations, we propose two novel PnP Langevin sampling methods: (1) boundary-reflection accelerated PnP Langevin sampling, which—without likelihood approximation—rigorously enforces pixel-wise non-negativity and mitigates gradient explosion; and (2) Riemannian PnP mirror sampling, which exploits the intrinsic manifold geometry of the Poisson likelihood for efficient and stable posterior sampling. Both methods jointly enable Bayesian image reconstruction, pixel-level uncertainty quantification, and automatic hyperparameter tuning. Experiments on astronomical and medical low-photon imaging tasks demonstrate that our approaches significantly outperform state-of-the-art methods, achieving superior reconstruction accuracy and well-calibrated uncertainty estimates.

Technology Category

• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Privacy-enhancing technologies

Abstract

This paper introduces a novel plug-and-play (PnP) Langevin sampling methodology for Bayesian inference in low-photon Poisson imaging problems, a challenging class of problems with significant applications in astronomy, medicine, and biology. PnP Langevin sampling algorithms offer a powerful framework for Bayesian image restoration, enabling accurate point estimation as well as advanced inference tasks, including uncertainty quantification and visualization analyses, and empirical Bayesian inference for automatic model parameter tuning. However, existing PnP Langevin algorithms are not well-suited for low-photon Poisson imaging due to high solution uncertainty and poor regularity properties, such as exploding gradients and non-negativity constraints. To address these challenges, we propose two strategies for extending Langevin PnP sampling to Poisson imaging models: (i) an accelerated PnP Langevin method that incorporates boundary reflections and a Poisson likelihood approximation and (ii) a mirror sampling algorithm that leverages a Riemannian geometry to handle the constraints and the poor regularity of the likelihood without approximations. The effectiveness of these approaches is demonstrated through extensive numerical experiments and comparisons with state-of-the-art methods.

Problem

• Develops plug-and-play Langevin sampling for low-photon Poisson imaging.
• Addresses high uncertainty and poor regularity in Poisson imaging models.
• Proposes accelerated and mirror sampling methods for Bayesian inference.

Innovation

• Plug-and-play Langevin sampling for Poisson imaging
• Accelerated method with boundary reflections
• Mirror sampling using Riemannian geometry