CAREER: Partial Differential Equation-based Image Processing with Applications to Radiation Oncology

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In the past decade, new nonlinear partial differentialequations (PDEs) have been developed for various image processingapplications, such as noise reduction, edge detection, imagesegmentation and restoration. While the attention of thescientific community in this area predominantly focused oncreating the new PDEs, very little attention was paid todeveloping numerical algorithms that approximate their solutions.The few numerical algorithms that are currently used suffer froma variety of problems: they are not accurate enough, too slow,and not fault-free. In this project, the investigator developsaccurate, efficient, and robust numerical algorithms fornonlinear PDEs in image processing. The research activities arebased on the investigator's extensive work in the field ofhyperbolic conservation laws, and include numerical methods forthe Hamilton-Jacobi equations, fast algorithms for high-ordernonlinear PDEs, algorithms for computing steady-state solutions,numerical homogenization of Hamilton-Jacobi equations andmulti-resolution analysis, analysis of nonlinear diffusionequations, constrained morphing active contours and geodesicflows, and "non-blind" algorithms for image processing. Aportion of the research activities focuses on improving existingalgorithms in order to solve a specific imaging problem inradiation oncology treatment planning. The investigator develops novel mathematical techniques forimage processing and uses these techniques for solving problemsin the field of radiation oncology imaging. Radiation oncologytreats cancer by delivering relatively small doses of radiationto tumors in order to eliminate cancer without destroying orchronically damaging healthy tissues in and around the growth.CT and MRI scans are used as three-dimensional anatomical modelsto ensure that the treatments conform geometrically to the tumortarget. This process depends critically upon identifying thelocation of the tumor as well as the healthy organs (in order tominimize the dose of radiation in these areas). Despite extendedresearch, the existing mathematical tools for image processingare unsuitable for clinical medical applications. Thesegmentation of the CT and MRI scans is still carried out bymanual tools, and consumes about one-half of the time required toplan the treatments. The investigator designs accurate andreliable automated algorithms that would significantly shortenthis time and have a big impact on radiation oncology. Heintegrates into his work educational activities that demonstratethe importance of applied mathematics in a broad spectrum ofsciences. Special emphasis is given to applications ofcomputational mathematics in biology and cutting-edgetechnologies. The planned educational activities includeprograms for junior-high, high-school, undergraduate, andgraduate students. The investigator works to increase the genderand ethnic diversity in the mathematical sciences by encouragingunder-represented groups to study applied mathematics and chooseit as a future career.