Geometry and Topology of Singular Structures with Applications to Imaging

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Professor Damon?s research will concern the geometry, topology and deformation properties of singular structures, including stratified sets, mappings, and nonisolated singularities, and the application of these results to develop geometric methods for problems in computer imaging. This includes applying methods from singularity theory to determine from an underlying "skeletal/medial structure" the local, relative and global geometric properties of a region in Euclidean space and its boundary. Second, he is developing, in joint work, a new method for finding intersection of spline surfaces for geometric modeling. Third, he is developing methods to characterize local features of objects in natural images which include lighting, geometric features, and viewer movement.Professor Damon will further develop his results on skeletal and medial structures to:understand the behavior of multiple medial structures in a collection of complementary regions or objects with geometric and physical interaction. This will allow the time evolution of such objects, and enlarge the class of allowable structures to include degeneracies. These ideas will be used for investigating statistical properties of geometrical features for shape. Second, Professor Damon will further develop methods for following the evolution of intersection curves under flows of spline surfaces and apply them to give a new method for computing the medial axis of regions with boundaries defined by splines. Third, he will apply his methods to complete, in joint research, the analysis of local features in natural images allowing shade/shadow and specularity, geometric features, and movement of either viewpoint or light source. He will further begin developing algorithms with imaging scientists for their implementation.Professor Damon?s research will further investigate properties of spaces used to model objects and regions, and systems of equations which provide understanding of properties of geometric spaces and their representations. These methods will be applied to several problems in computer imaging. The first method will be used for modeling multiple objects and their interaction, including statistical properties of such collections. This method will allow for degeneracies of the structures, and these resulting models will be used for analyzing interaction of regions in medical images. These same methods will also be used for understanding the evolution of intersections of deforming objects with applications to geometric modeling. Third, he will analyze the properties of equations which can be used to model the images of objects in natural images, where lighting and viewpoint affect the appearance of geometric features of objects. These results will be used in joint work with computer scientists to develop procedures for identifying objects in images.