Plug-and-Play Unplugged: Optimization Free Reconstruction using Consensus Equilibrium

Regularized inversion methods for image reconstruction are used widely due to their tractability and their ability to combine complex physical sensor models with useful regularity criteria. Such methods were used in 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 severely limits both the expressiveness of possible regularity conditions and the variety of provably convergent Plug-and-Play denoising operators. In this paper, we introduce the concept of consensus equilibrium (CE), which generalizes regularized inversion to include a much wider variety of regularity operators without the need for an optimization formulation. Consensus equilibrium 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, including as a fixed point problem that generalizes the ADMM approach used in the Plug-and-Play method. We present the Douglas-Rachford (DR) algorithm for computing the CE solution as a fixed point and prove the convergence of this algorithm under conditions that include denoising operators that do not arise from optimization problems and that may not be nonexpansive. We give several examples to illustrate the idea of consensus equilibrium and the convergence properties of the DR algorithm and demonstrate this method on a sparse interpolation problem using electron microscopy data.