On Decomposition Models in Imaging Sciences and Multi-time Hamilton-Jacobi Partial Differential Equations
Abstract
This paper provides new theoretical connections between multi-time Hamilton-Jacobi partial differential equations and variational image decomposition models in imaging sciences. We show that the minimal values of these optimization problems are governed by multi-time Hamilton-Jacobi partial differential equations. The minimizers of these optimization problems can be represented using the momentum in the corresponding Hamilton-Jacobi partial differential equation. Moreover, variational behaviors of both the minimizers and the momentum are investigated as the regularization parameters approach zero. In addition, we provide a new perspective from convex analysis to prove the uniqueness of convex solutions to Hamilton-Jacobi equations. Finally we consider image decomposition models that do not have unique minimizers and we propose a regularization approach to perform the analysis using multi-time Hamilton-Jacobi partial differential equations.
Cite
@article{arxiv.1906.09502,
title = {On Decomposition Models in Imaging Sciences and Multi-time Hamilton-Jacobi Partial Differential Equations},
author = {Jérôme Darbon and Tingwei Meng},
journal= {arXiv preprint arXiv:1906.09502},
year = {2020}
}