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

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.