Igor Yanovsky, Ph.D. - US grants

The investigators, their students and collaborators study mathematical formulations and efficient computational techniques for applications arising in image analysis, materials science, and atmospheric and climate modeling. This multidisciplinary research combines areas of computational mathematics, inverse problems, image analysis, interfaces and free boundaries, and atmospheric sciences. They study image restoration using cartoon and texture decompositions, restoration of images in the presence of a stochastic point spread function, implicit open curve evolution and applications to free boundary problems in materials sciences, and variational data fusion of atmospheric records acquired by multiple instruments. The research team develops novel variational approaches, iterative and numerical analysis techniques for solving these inverse problems.

The proposed activity provides the link between efficient mathematical formulations, imaging approaches, and applications in climate and materials sciences, where similar approaches have not yet been attempted. In particular, capability for merging data acquired by multiple sensors is a key part to our understanding of the Earth's climate system, and therefore, is of importance when making projections about climate change and climate impacts. Current atmospheric data fusion methods are largely ad hoc and establishing a firm mathematical foundation and computational methods for combining important records enhances their scientific credibility and further a wide range of scientific analyses. The investigators promote multidisciplinary teaching, training and learning. Mathematics students are exposed to a broad range of topics and techniques: (i) in applied and computational mathematics, image processing and analysis; and (ii) topics outside mathematics, including materials and atmospheric sciences.

The investigators, along wit their students and collaborators, will develop novel mathematical formulations and computational techniques for applications in data science, remote sensing, atmospheric sciences, and medical imaging. This multidisciplinary research includes advancement of discovery and understanding of many natural phenomena and the development of new imaging sciences methods for the medical field. Super-resolution of hurricane imagery will be of value to science, where many aspects of hurricane formation and strength prediction are still unknown, and to society, which could benefit from more accurate information being used in forecasts of storm strength and development. Improving the quality of images distorted by atmospheric turbulence will have applications in defense, while improving image registration algorithms will tremendously help research, diagnosis, and treatment decisions in the medical field. This project will provide support for one graduate student per year.

The project's activities will provide links between efficient mathematical formulations, imaging approaches and applications in remote sensing, atmospheric sciences, and medical imaging, where similar approaches have not yet been attempted. Novel variational approaches, iterative and numerical analysis techniques will be developed for solving these and related inverse problems. In particular, this investigation will study novel robust variational approaches and their numerical approximations, including: a new combined deconvolution and geometric correction variational model for restoration of atmospherically-distorted images; local and nonlocal total variation regularized super-resolution method and an efficient computational algorithm for space-time deconvolution of low-resolution sequences; novel applications of multiscale hierarchical decompositions to blind deconvolution and image registration. The investigators will promote multidisciplinary teaching, training and learning. Mathematics students will be exposed to a broad range of topics and techniques: (i) in applied and computational mathematics, image processing and analysis, and (ii) topics outside mathematics, including remote sensing, atmospheric sciences, and medical imaging.

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.