  This paper introduces a Bayesian framework for image inversion by deriving a
probabilistic counterpart to the regularization-by-denoising (RED) paradigm. It
additionally implements a Monte Carlo algorithm specifically tailored for
sampling from the resulting posterior distribution, based on an asymptotically
exact data augmentation (AXDA). The proposed algorithm is an approximate
instance of split Gibbs sampling (SGS) which embeds one Langevin Monte Carlo
step. The proposed method is applied to common imaging tasks such as
deblurring, inpainting and super-resolution, demonstrating its efficacy through
extensive numerical experiments. These contributions advance Bayesian inference
in imaging by leveraging data-driven regularization strategies within a
probabilistic framework.
