Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems

TL;DR

This paper introduces a novel Bregman Proximal Gradient-based Plug-and-Play framework with a specialized denoiser for Poisson inverse problems, improving convergence and restoration quality over traditional methods.

Contribution

It generalizes PnP methods to Poisson problems using Bregman divergence and develops a new Bregman Score Denoiser, with proven convergence in nonconvex settings.

Findings

The proposed methods converge to stationary points.

Experimental results show improved image restoration performance.

The Bregman Score Denoiser effectively captures Poisson noise characteristics.

Abstract

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.