Geometric Measure Theory, Image Processing, and Nonlinear Partial Differential Equations

该项目将为图像处理中使用的新数值方法提供数学基础。图像处理的应用无处不在。它们的范围从分析医学图像（如CAT或MRI图像）到法医科学、人脸识别、卫星图像、语音识别（声学成像）等，所有这些应用都是由快速数学算法来完成图像增强和恢复工作的。 为了测量图像之间的差异，本项目正在考虑的数值方法使用所谓的“地球移动器距离”（在更一般的情况下，它被称为Wasserstein度量）。这是推土机距离的一种新颖应用，有望大幅提高图像处理算法的速度。对研究生的博士学位的培训，咨询和指导与该项目的研究任务交织在一起。该项目将支持至少3名研究生，其中两名是女生。这项研究涉及几个不同的分析主题，数学分析是统一的几何测量理论的技术。计算推土机距离的数值方法是基于一类满足边界条件的向量场的正则化和极小化。然而，目前还没有理论证实极小值的存在或收敛的正则化问题的原始之一。这些问题将通过使用变分法、伽玛收敛和发散测度向量场迹理论来处理边界条件来解决。同时，本计画将考虑关于发散测度向量场理论的几个尚未解决的问题。图像处理领域的另一个重要方面是由Rudin，Osher和Fatemi（ROF模型）开创的评估图像噪声的方法的进步。该项目的研究还将包括使用非线性椭圆方程分析种群分离模型，目的是了解分离种群的自由边界。该奖项反映了NSF的法定使命，通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project will provide a mathematical foundation for the novel numerical methods used in image processing. Applications of image processing are ubiquitous. They range from analyzing medical images, such as CAT or MRI images, to forensic sciences, face recognition, satellite imagery, speech recognition (acoustic imaging), etc. All of these applications are underlied by fast mathematical algorithms that do the work of image enhancement and restoration. To measure the difference between images, numerical methods under consideration in this project use a so-called "earth mover's distance" (in a more general context it is known as the Wasserstein metric). This is a novel application of the earth mover's distances that promises a substantial speed-up of image processing algorithms. Training, advising, and mentoring of graduate students toward their doctoral degree is intertwined with the research tasks of the project. The project will support at least 3 graduate students, two of them female students. This research is concerned with several diverse analytical topics; the mathematical analysis is unified through techniques of geometric measure theory. The numerical methods to compute the earth mover's distance are based on a regularization and a minimization over a class of vector fields satisfying a boundary condition. However, there is currently no theory confirming either the existence of minimizers or the convergence of the regularized problem to the original one. These questions will be addressed by using techniques of calculus of variations, gamma convergence, and the theory of traces of divergence-measure vector fields to deal with the boundary condition. Simultaneously this project will consider several unresolved problems concerning the theory of divergence-measure vector fields. Another important aspect of the field of image processing to be investigated is the advancement of methods of evaluating the image noise pioneered by Rudin, Osher, and Fatemi (the ROF model). The research of this project will also include analysis of models for segregation of populations using non-linear elliptic equations, with the goal to understand the free boundaries that separate the populations.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.