Diarra Fall - Bayesian formulation of Regularization by denoising. Application to image restoration

Inverse problems are ubiquitous in signal and image processing. Canonical examples include signal/image denoising (i.e, removing noise from a signal/image) and image reconstruction. As inverse problems are known to be ill-posed or at least, ill-conditioned, they require regularization by introducing additional constraints to mitigate the lack of information brought by the observations. A common difficulty is to select an appropriate regularizer, which has a decisive influence on the quality of the reconstruction. Another challenge is the confidence we may have in the reconstructed signal/image. To put it another way, it is desirable for a method to be able to quantify the uncertainty associated with the reconstructed image in order to encourage more principled decision-making. These two tasks (regularization and uncertainty quantification) can be overcome at the same time by addressing the problem within the Bayesian statistical framework. It allows to include additional information by specifying a marginal distribution for the signal/image, known as prior distribution. The traditional approach consists in defining the prior analytically, as a hand-crafted explicit function chosen to encourage specific desired properties of the recovered signal/image. Following the up-to-date surge of deep learning, data-driven regularization using priors specified by neural networks has become ubiquitous in signal and image inverse problems. Popular approaches within this methodology are Plug & Play (PnP) and regularization by denoising (RED) methods.

We have proposed a probabilistic approach to the RED method, defining a new probability distribution based on a RED potential, which can be chosen as the prior distribution in a Bayesian inversion task. We have also proposed a dedicated Markov chain Monte Carlo sampling algorithm that is particularly well suited to high-dimensional sampling of the resulting posterior distribution. In addition, we provide a theoretical analysis that guarantees convergence to the target distribution, and quantify the speed of convergence. The power of the proposed approach is illustrated in various restoration tasks such as image deblurring, inpainting, and super-resolution.