  Many imaging problems can be formulated as inverse problems expressed as
finite-dimensional optimization problems. These optimization problems generally
consist of minimizing the sum of a data fidelity and regularization terms. In
[23,26], connections between these optimization problems and (multi-time)
Hamilton--Jacobi partial differential equations have been proposed under the
convexity assumptions of both the data fidelity and regularization terms. In
particular, under these convexity assumptions, some representation formulas for
a minimizer can be obtained. From a Bayesian perspective, such a minimizer can
be seen as a maximum a posteriori estimator. In this chapter, we consider a
certain class of non-convex regularizations and show that similar
representation formulas for the minimizer can also be obtained. This is
achieved by leveraging min-plus algebra techniques that have been originally
developed for solving certain Hamilton--Jacobi partial differential equations
arising in optimal control. Note that connections between viscous
Hamilton--Jacobi partial differential equations and Bayesian posterior mean
estimators with Gaussian data fidelity terms and log-concave priors have been
highlighted in [25]. We also present similar results for certain Bayesian
posterior mean estimators with Gaussian data fidelity and certain
non-log-concave priors using an analogue of min-plus algebra techniques.
