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

## TL;DR

This paper explores the theoretical links between multi-time Hamilton-Jacobi PDEs and variational image decomposition models, revealing how solutions and minimizers relate and proposing methods for models with non-unique solutions.

## Contribution

It establishes new theoretical connections between Hamilton-Jacobi PDEs and image decomposition, including uniqueness proofs and regularization techniques for non-unique minimizers.

## Key findings

- Minimal values governed by multi-time Hamilton-Jacobi PDEs
- Minimizers represented via Hamilton-Jacobi momentum
- Regularization approach for non-unique minimizers

## 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.