This paper studies plug-and-play (PnP) Langevin sampling strategies 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 offers 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. Herein, we leverage and adapt recent developments in this framework to tackle challenging imaging problems involving weakly informative Poisson data. 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 explore 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 evaluated and contrasted through extensive numerical experiments and comparisons with state-of-the-art methods. The source code accompanying this paper is available at https://github.com/freyyia/pnp-langevin-poisson.

翻译：本文研究用于低光子泊松成像问题中贝叶斯推断的即插即用（PnP）朗之万采样策略，该类问题在天文学、医学和生物学领域具有重要应用价值且极具挑战性。PnP朗之万采样为贝叶斯图像复原提供了强大框架，既能实现精确的点估计，又能执行包括不确定性量化与可视化分析在内的高级推断任务，以及用于自动模型参数调优的经验贝叶斯推断。本文利用并调整该框架的最新进展，以应对涉及弱信息泊松数据的挑战性成像问题。现有PnP朗之万算法因解的高不确定性和不良正则性（如梯度爆炸和非负约束）而不适用于低光子泊松成像。为解决这些挑战，我们探索了两种将朗之万PnP采样扩展至泊松成像模型的策略：（i）融合边界反射与泊松似然近似的加速PnP朗之万方法；（ii）利用黎曼几何处理约束及似然不良正则性且无需近似的镜像采样算法。通过大量数值实验以及与前沿方法的对比，评估并比较了这些方法的有效性。本文随附的源代码可在 https://github.com/freyyia/pnp-langevin-poisson 获取。