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.

翻译：本文针对低光子泊松成像问题中的贝叶斯推断，提出了一种新颖的即插即用（PnP）朗之万采样方法。这类问题在天文学、医学和生物学领域具有重要应用价值，但求解极具挑战性。PnP朗之万采样算法为贝叶斯图像复原提供了强大框架，不仅能实现精确的点估计，还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。然而，由于解的高不确定性和较差的正则性特性（如梯度爆炸和非负约束），现有PnP朗之万算法难以适用于低光子泊松成像。为应对这些挑战，我们提出了两种将朗之万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条件和似然函数不良正则性的镜像采样算法（无需近似）。通过大量数值实验以及与前沿方法的对比，验证了所提方法的有效性。