Performance Analysis of Plug-and-Play ADMM: A Graph Signal Processing Perspective

TL;DR

This paper analyzes the performance of Plug-and-Play ADMM in image restoration from a graph signal processing perspective, revealing its intrinsic pre-denoising property and providing new theoretical insights.

Contribution

It introduces a geometric interpretation, conditions for MAP equivalence, and a novel consensus equilibrium analysis for PnP ADMM with graph filter denoisers.

Findings

Performance gain linked to pre-denoising characteristic

Conditions for equivalent MAP optimization established

New analysis method via consensus equilibrium introduced

Abstract

The Plug-and-Play (PnP) ADMM algorithm is a powerful image restoration framework that allows advanced image denoising priors to be integrated into physical forward models to generate high quality image restoration results. However, despite the enormous number of applications and several theoretical studies trying to prove the convergence by leveraging tools in convex analysis, very little is known about why the algorithm is doing so well. The goal of this paper is to fill the gap by discussing the performance of PnP ADMM. By restricting the denoisers to the class of graph filters under a linearity assumption, or more specifically the symmetric smoothing filters, we offer three contributions: (1) We show conditions under which an equivalent maximum-a-posteriori (MAP) optimization exists, (2) we present a geometric interpretation and show that the performance gain is due to an intrinsic pre-denoising characteristic of the PnP prior, (3) we introduce a new analysis technique via the concept of consensus equilibrium, and provide interpretations to problems involving multiple priors.