Efficient Bayesian computation for low-photon imaging problems

TL;DR

This paper introduces a new efficient MCMC method using reflected and regularised Langevin SDEs for Bayesian inference in low-photon imaging, effectively handling non-Gaussian noise and regularity challenges.

Contribution

It proposes a novel reflected proximal Langevin MCMC algorithm tailored for low-photon imaging, addressing regularity issues and enabling robust Bayesian computation.

Findings

Effective in image deblurring, denoising, and inpainting tasks

Handles non-Gaussian noise like binomial, geometric, Poisson

Demonstrates improved stability and convergence

Abstract

This paper studies a new and highly efficient Markov chain Monte Carlo (MCMC) methodology to perform Bayesian inference in low-photon imaging problems, with particular attention to situations involving observation noise processes that deviate significantly from Gaussian noise, such as binomial, geometric and low-intensity Poisson noise. These problems are challenging for many reasons. From an inferential viewpoint, low-photon numbers lead to severe identifiability issues, poor stability and high uncertainty about the solution. Moreover, low-photon models often exhibit poor regularity properties that make efficient Bayesian computation difficult; e.g., hard non-negativity constraints, non-smooth priors, and log-likelihood terms with exploding gradients. More precisely, the lack of suitable regularity properties hinders the use of state-of-the-art Monte Carlo methods based on numerical approximations of the Langevin stochastic differential equation (SDE), as both the SDE and its numerical approximations behave poorly. We address this difficulty by proposing an MCMC methodology based on a reflected and regularised Langevin SDE, which is shown to be well-posed and exponentially ergodic under mild and easily verifiable conditions. This then allows us to derive four reflected proximal Langevin MCMC algorithms to perform Bayesian computation in low-photon imaging problems. The proposed approach is demonstrated with a range of experiments related to image deblurring, denoising, and inpainting under binomial, geometric and Poisson noise.