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

arXiv: https://arxiv.org/abs/1906.09502 · HTML: https://mlsanity.com/paper/1906.09502/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2019.

## Built on by (most cited first, 8 of 8)

- [SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems](https://mlsanity.com/paper/2201.05475/) (2022)
- [Leveraging Hamilton-Jacobi PDEs with time-dependent Hamiltonians for continual scientific machine learning](https://mlsanity.com/paper/2311.07790/) (2023)
- [Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs](https://mlsanity.com/paper/2105.03336/) (2021)
- [Leveraging Multi-time Hamilton-Jacobi PDEs for Certain Scientific Machine Learning Problems](https://mlsanity.com/paper/2303.12928/) (2023)
- [Hamilton-Jacobi Based Policy-Iteration via Deep Operator Learning](https://mlsanity.com/paper/2406.10920/) (2024)
- [Connecting Hamilton--Jacobi partial differential equations with maximum a posteriori and posterior mean estimators for some non-convex priors](https://mlsanity.com/paper/2104.11285/) (2021)
- [On Hamilton-Jacobi PDEs and image denoising models with certain non-additive noise](https://mlsanity.com/paper/2105.13997/) (2021)
- [Deep learning methods for inverse problems using connections between proximal operators and Hamilton-Jacobi equations](https://mlsanity.com/paper/2512.23829/) (2025)

## Abstract (arXiv)

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.