On Decomposition Models in Imaging Sciences and Multi-time Hamilton-Jacobi Partial Differential Equations

Jérôme Darbon; Tingwei Meng

doi:10.1137/19m1266332

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arXiv · 1906.09502

On Decomposition Models in Imaging Sciences and Multi-time Hamilton-Jacobi Partial Differential Equations

Jérôme Darbon·Tingwei Meng

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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.

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BibTeX RIS

Jérôme Darbon, Tingwei Meng. 2019-06-22. On Decomposition Models in Imaging Sciences and Multi-time Hamilton-Jacobi Partial Differential Equations. https://doi.org/10.1137/19m1266332

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