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. Â 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.
