  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.
