On Decomposition Models in Imaging Sciences and Multi-time Hamilton--Jacobi Partial Differential Equations
Jérôme Darbon, Tingwei Meng · SIAM Journal on Imaging Sciences · 2020
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.