Connecting Hamilton-Jacobi Partial Differential Equations with Maximum a Posteriori and Posterior Mean Estimators for Some Non-convex Priors

Jérôme Darbon (), Gabriel P. Langlois () and Tingwei Meng ()
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Jérôme Darbon: Brown University, Division of Applied Mathematics
Gabriel P. Langlois: Brown University, Division of Applied Mathematics
Tingwei Meng: Brown University, Division of Applied Mathematics

Chapter 6 in Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging, 2023, pp 209-233 from Springer

Abstract: Abstract 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 Darbon (SIAM J. Imag. Sci. 8:2268–2293, 2015), Darbon and Meng, (On decomposition models in imaging sciences and multi-time Hamilton-Jacobi partial differential equations, arXiv preprint arXiv:1906.09502, 2019), 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 Darbon and Langlois, (On Bayesian posterior mean estimators in imaging sciences and Hamilton-Jacobi partial differential equations, arXiv preprint arXiv:2003.05572, 2020). 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.

Keywords: Hamilton–Jacobi partial differential equation; Maximum a posteriori estimation; Bayesian posterior mean estimation; Min-plus algebra; Imaging inverse problems (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/978-3-030-98661-2_56

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