Convergent plug-and-play methods for image inverse problems with explicit and nonconvex deep regularization

Plug-and-play methods constitute a class of iterative algorithms for imaging inverse problems where regularization is performed by an off-the-shelf Gaussian denoiser. These methods have demonstrated impressive visual results, particularly when the denoiser is parameterized by deep neural networks. However, the theoretical convergence of PnP methods has yet to be fully established.This talk begins with an overview of the literature on PnP algorithms, followed by the introduction of new convergence results for these methods when paired with a specific denoiser, known as the Gradient-Step Denoiser. This denoiser writes as a gradient descent step on an explicit, nonconvex function parameterized by a deep neural network. The analysis shows that the resulting PnP algorithms converge to stationary points of explicit functionals. These algorithms are then applied to various ill-posed inverse problems, including deblurring, super-resolution, and inpainting. Finally, to address inverse problems corrupted by Poisson noise, we will introduce a novel Bregman version of PnP based on the Bregman Proximal Gradient (BPG) optimization algorithm.