On Bayesian Posterior Mean Estimators in Imaging Sciences and Hamilton–Jacobi Partial Differential Equations

Abstract

Variational and Bayesian methods are two widely used set of approaches to solve image denoising problems. In a Bayesian setting, these approaches correspond, respectively, to using maximum a posteriori estimators and posterior mean estimators for reconstructing images. In this paper, we propose novel theoretical connections between Hamilton–Jacobi partial differential equations (HJ PDEs) and a broad class of posterior mean estimators with quadratic data fidelity term and log-concave prior. Where solutions to some first-order HJ PDEs with initial data describe maximum a posteriori estimators, here we show that solutions to some viscous HJ PDEs with initial data describe a broad class of posterior mean estimators. We use these connections to establish representation formulas and various properties of posterior mean estimators. In particular, we use these connections to show that some Bayesian posterior mean estimators can be expressed as proximal mappings of smooth functions and derive representation formulas for these functions.