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.

本文讨论了 Plug-and-Play ADMM 算法在图像还原中的表现，并通过将去噪器限制为图滤波器类下的线性假设，提出了等价 MAP 优化存在的条件、表明了性能增益是由于 PnP prior 的固有预去噪特性以及提出了通过共识平衡概念的新分析技术，并为涉及多个先验的问题提供了解释。

插拔式 ADMM 的性能分析：基于图信号处理角度的研究

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

Regularized inversion methods for image reconstruction are used widely due to
their tractability and ability to combine complex physical sensor models with
useful regularity criteria. Such methods motivated the recently developed
Plug-and-Play prior method, which provides a framework to use advanced
denoising algorithms as regularizers in inversion. However, the need to
formulate regularized inversion as the solution to an optimization problem
limits the possible regularity conditions and physical sensor models.
In this paper, we introduce Consensus Equilibrium (CE), which generalizes
regularized inversion to include a much wider variety of both forward
components and prior components without the need for either to be expressed
with a cost function. CE is based on the solution of a set of equilibrium
equations that balance data fit and regularity. In this framework, the problem
of MAP estimation in regularized inversion is replaced by the problem of
solving these equilibrium equations, which can be approached in multiple ways.
The key contribution of CE is to provide a novel framework for fusing
multiple heterogeneous models of physical sensors or models learned from data.
We describe the derivation of the CE equations and prove that the solution of
the CE equations generalizes the standard MAP estimate under appropriate
circumstances.
We also discuss algorithms for solving the CE equations, including ADMM with
a novel form of preconditioning and Newton's method. We give examples to
illustrate consensus equilibrium and the convergence properties of these
algorithms and demonstrate this method on some toy problems and on a denoising
example in which we use an array of convolutional neural network denoisers,
none of which is tuned to match the noise level in a noisy image but which in
consensus can achieve a better result than any of them individually.

Consensus Equilibrium provides a new framework to fuse a variety of physical sensor models and denoising algorithms without expressing them with a cost function, replacing MAP estimation with solving a set of equilibrium equations that balance data fit and regularity.