Title: Bayesian Image Restoration with Deep Learning-Based Priors

Abstract: Inverse problems are ubiquitous in signal and image processing. 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 achieved simultaneously by addressing the problem within the Bayesian statistical framework. This allows to include additional information by specifying a marginal distribution for the signal/image, known as the 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 include Plug-and-Play (PnP) and regularization by denoising (RED) methods.

This talk will discuss our recently proposed probabilistic approach to the RED method, which defines a new probability distribution based on a RED potential that can be used 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 the 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 rate of convergence. The effectiveness of this approach is demonstrated through its application to various restoration tasks, including image deblurring, inpainting and super-resolution. If time permits, I will also discuss an extension of the proposed framework to handle nonlinear Poisson inverse problems.