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

许多成像科学问题可以被公式化为可以使用优化技术来解决的变分逆问题。用于处理数据以提取应用相关信息的成像模型变得越来越复杂，并且涉及许多术语。该项目的主要目标包括建立新的和独特的复杂的变分成像模型和偏微分方程之间的连接在高维。这种联系将用于更好地理解，创建和学习用于处理数据的强大模型，以提取应用程序的相关信息。研究人员还将利用这种联系来创建新的数值算法来解决这些优化问题。该项目的基础研究将展示图像处理和计算机视觉技术之间的新的和独特的联系，这些联系基于凸优化和非常高维的多时间Hamilton-Jacobi偏微分方程。这项研究将产生新的数学结果的鲁棒性的变分反问题的解决方案在成像科学和新的公式，描述的影响下的模型扰动的估计。将严格获得一大类多时间Hamilton-Jacobi方程的新表示公式、存在性和唯一性结果。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响力进行评估而被认为值得支持。审查标准。

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.