Convergent Primal-Dual Plug-and-Play Image Restoration: A General Algorithm and Applications

TL;DR

This paper introduces a convergent deep plug-and-play algorithm for image restoration that integrates primal-dual splitting, providing theoretical guarantees and broad applicability to various restoration tasks.

Contribution

It develops a general PnP framework with convergence guarantees by combining primal-dual splitting, addressing limitations of existing methods in handling complex data-fidelity terms.

Findings

The proposed algorithm guarantees convergence under reasonable conditions.

It effectively handles a broad class of image restoration problems.

Numerical experiments validate the method's practicality and effectiveness.

Abstract

We propose a general deep plug-and-play (PnP) algorithm with a theoretical convergence guarantee. PnP strategies have demonstrated outstanding performance in various image restoration tasks by exploiting the powerful priors underlying Gaussian denoisers. However, existing PnP methods often lack theoretical convergence guarantees under realistic assumptions due to their ad-hoc nature, resulting in inconsistent behavior. Moreover, even when convergence guarantees are provided, they are typically designed for specific settings or require a considerable computational cost in handling non-quadratic data-fidelity terms and additional constraints, which are key components in many image restoration scenarios. To tackle these challenges, we integrate the PnP paradigm with primal-dual splitting (PDS), an efficient proximal splitting methodology for solving a wide range of convex optimization problems, and develop a general convergent PnP framework. Specifically, we establish theoretical conditions for the convergence of the proposed PnP algorithm under a reasonable assumption. Furthermore, we show that the problem solved by the proposed PnP algorithm is not a standard convex optimization problem but a more general monotone inclusion problem, where we provide a mathematical representation of the solution set. Our approach efficiently handles a broad class of image restoration problems with guaranteed theoretical convergence. Numerical experiments on specific image restoration tasks validate the practicality and effectiveness of our theoretical results.