In the past decade, new nonlinear partial differential equations (PDEs) have been developed for various image processing applications, such as noise reduction, edge detection, image segmentation and restoration. While the attention of the scientific community in this area predominantly focused on creating the new PDEs, very little attention was paid to developing numerical algorithms that approximate their solutions. The few numerical algorithms that are currently used suffer from a variety of problems: they are not accurate enough, too slow, and not fault-free. In this project, the investigator develops accurate, efficient, and robust numerical algorithms for nonlinear PDEs in image processing. The research activities are based on the investigator's extensive work in the field of hyperbolic conservation laws, and include numerical methods for the Hamilton-Jacobi equations, fast algorithms for high-order nonlinear PDEs, algorithms for computing steady-state solutions, numerical homogenization of Hamilton-Jacobi equations and multi-resolution analysis, analysis of nonlinear diffusion equations, constrained morphing active contours and geodesic flows, and "non-blind" algorithms for image processing. A portion of the research activities focuses on improving existing algorithms in order to solve a specific imaging problem in radiation oncology treatment planning. The investigator develops novel mathematical techniques for image processing and uses these techniques for solving problems in the field of radiation oncology imaging. Radiation oncology treats cancer by delivering relatively small doses of radiation to tumors in order to eliminate cancer without destroying or chronically damaging healthy tissues in and around the growth. CT and MRI scans are used as three-dimensional anatomical models to ensure that the treatments conform geometrically to the tumor target. This process depends critically upon identifying the location of the tumor as well as the healthy organs (in order to minimize the dose of radiation in these areas). Despite extended research, the existing mathematical tools for image processing are unsuitable for clinical medical applications. The segmentation of the CT and MRI scans is still carried out by manual tools, and consumes about one-half of the time required to plan the treatments. The investigator designs accurate and reliable automated algorithms that would significantly shorten this time and have a big impact on radiation oncology. He integrates into his work educational activities that demonstrate the importance of applied mathematics in a broad spectrum of sciences. Special emphasis is given to applications of computational mathematics in biology and cutting-edge technologies. The planned educational activities include programs for junior-high, high-school, undergraduate, and graduate students. The investigator works to increase the gender and ethnic diversity in the mathematical sciences by encouraging under-represented groups to study applied mathematics and choose it as a future career.
