A plug and play Bayesian algorithm for solving myope inverse problems

The emergence of efficient algorithms in variational and Bayesian frameworks braught significant advances to the field of inverse problems. However, such problems remain challenging when the observation operator is not perfectly known. In this paper we propose a Bayesian Plug-and-Play (PP) algorithm for solving a wide range of inverse problems where the signal/image is sparse in the original domain and the observation operator has to be estimated. The principle consists of plugging the prior considered for the target observation operator and keep using the same algorithm. The proposed method relies on a generic proximal non-smooth sampling scheme. This genericity makes the proposed algorithm novel in the sense that it can be used to solve a wide range or inverse problems. Our method is illustrated on a deblurring problem with unknown blur operator where promising results are obtained.