Variational and Bayesian methods are two approaches that have been widely used to solve image reconstruction problems. In a Bayesian setting, these approaches correspond, respectively, to using maximum a posteriori estimators and posterior mean estimators for reconstructing images. In this talk, I will describe original connections between Hamilton-Jacobi partial differential equations (HJ PDEs) and a broad class of Bayesian methods and posterior mean estimators with Gaussian data fidelity term and log-concave prior. Whereas solutions to certain first-order HJ PDEs with initial data describe maximum a posteriori estimators in a Bayesian setting, here I will illustrate how solutions to some viscous HJ PDEs with initial data describe a broad class of Bayesian posterior mean estimators. I will then show how these connections can be used to establish several representation formulas and optimal bounds involving the posterior mean estimate. Finally, I will discuss how these connections can be used, first, to show that some posterior mean estimators can be expressed as proximal mappings of twice continuously differentiable functions and, two, derive representation formulas for these functions.
This is a joint work with Jérôme Darbon.
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