Plug-and-Play ADMM for Image Restoration: Fixed-Point Convergence and Applications

Alternating direction method of multiplier (ADMM) is a widely used algorithm for solving constrained optimization problems in image restoration. Among many useful features, one critical feature of the ADMM algorithm is its modular structure, which allows one to plug in any off-the-shelf image denoising algorithm for a subproblem in the ADMM algorithm. Because of the plug-in nature, this type of ADMM algorithms is coined the name “Plug-and-Play ADMM.” Plug-and-Play ADMM has demonstrated promising empirical results in a number of recent papers. However, it is unclear under what conditions and by using what denoising algorithms would it guarantee convergence. Also, since Plug-and-Play ADMM uses a specific way to split the variables, it is unclear if fast implementation can be made for common Gaussian and Poissonian image restoration problems. In this paper, we propose a Plug-and-Play ADMM algorithm with provable fixed-point convergence. We show that for any denoising algorithm satisfying an asymptotic criteria, called bounded denoisers, Plug-and-Play ADMM converges to a fixed point under a continuation scheme. We also present fast implementations for two image restoration problems on superresolution and single-photon imaging. We compare Plug-and-Play ADMM with state-of-the-art algorithms in each problem type and demonstrate promising experimental results of the algorithm.

I. Introduction

Many image restoration tasks can be posted as the following inverse problem: Given an observed image corrupted according to some forward model and noise, find the underlying image which “best explains” the observation. In estimation, we often formulate this problem as a maximum-a-posteriori (MAP) estimation [1], where the goal is to maximize the posterior probability:

$$\begin{align} \boldsymbol{\widehat{x}}&= \mathop {\underset{\mathbf{x}}{\text{argmax}}} p(\mathbf{x}| \mathbf{y}) \nonumber \\ &= \mathop {\underset{\mathbf{x}}{\text{argmin}}} -\log p(\mathbf{y}| \mathbf{x}) - \log p(\mathbf{x}), \end{align}$$ for some conditional probability defining the forward imaging model, and a prior distribution defining the probability distribution of the latent image. Because of the explicit use of the forward and the prior models, MAP estimation is also a model-based image reconstruction (MBIR) method [2] which has many important applications in deblurring [3]– [5], interpolation [6]– [8], super-resolution [9] –[12] and computed tomography [13] , to name a few.