Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse
  Problems
Samuel Hurault Ulugbek Kamilov Arthur Leclaire Nicolas Papadakis
Plug-and-Play (PnP) methods are efficient iterative algorithms for solving
ill-posed image inverse problems. PnP methods are obtained by using deep
Gaussian denoisers instead of the proximal operator or the gradient-descent
step within proximal algorithms. Current PnP schemes rely on data-fidelity
terms that have either Lipschitz gradients or closed-form proximal operators,
which is not applicable to Poisson inverse problems. Based on the observation
that the Gaussian noise is not the adequate noise model in this setting, we
propose to generalize PnP using theBregman Proximal Gradient (BPG) method. BPG
replaces the Euclidean distance with a Bregman divergence that can better
capture the smoothness properties of the problem. We introduce the Bregman
Score Denoiser specifically parametrized and trained for the new Bregman
geometry and prove that it corresponds to the proximal operator of a nonconvex
potential. We propose two PnP algorithms based on the Bregman Score Denoiser
for solving Poisson inverse problems. Extending the convergence results of BPG
in the nonconvex settings, we show that the proposed methods converge,
targeting stationary points of an explicit global functional. Experimental
evaluations conducted on various Poisson inverse problems validate the
convergence results and showcase effective restoration performance.