Variational Decomposition Models in Imaging Sciences and High Dimensional Multi-Time Hamilton-Jacobi Equations

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Many imaging science problems can be formulated as a variational inverse problem that can be solved using optimization techniques. Imaging models used to process data in order to extract the relevant information for applications are becoming more and more complex and involve many terms. The main objectives of the project consist of establishing new and unique connections between complex variational imaging models and partial differential equations in high dimensions. This connection will be used to better understand, create and learn robust models for processing data to extract the relevant information for applications. The investigator will also use this connection to create new numerical algorithms for solving these optimization problems.The fundamental research in this project will exhibit new and unique connections between image processing and computer vision techniques based on convex optimization and multi-time Hamilton-Jacobi partial differential equations in very high dimensions. The research will generate new mathematical results about the robustness of solutions of variational inverse problems in imaging sciences and new formulas that describe the impact of the estimation under perturbations of models. New representation formulas, existence and uniqueness results for a large class of multi-time Hamilton-Jacobi equations will be rigorously obtained.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.