Bayesian Image Restoration under Poisson Noise and Log-concave Prior

In recent years, much research has been devoted to the restoration of Poissonian images using optimization-based methods. On the other hand, the derivation of efficient and general fully Bayesian approaches is still an active area of research and especially if standard regularization functions are used, e.g. the total variation (TV) norm. This paper proposes to use the recent split-and-augmented Gibbs sampler (SPA) to sample efficiently from an approximation of the initial target distribution when log-concave prior distributions are used. SPA embeds proximal Markov chain Monte Carlo (MCMC) algorithms to sample from possibly non-smooth log-concave full conditionals. The benefit of the proposed approach is illustrated on several experiments including different regularizers, intensity levels and with both analysis and synthesis approaches.