The PI will continue investigating two problems that superficially are unrelated. One concerns the geometry, topology and deformation properties of stratified sets and nonisolated singularities. The other involves the development of geometric methods for problems in computer imaging. These different problems benefit from the approach of singularity theory via groups of equivalences and also from the analysis of geometric and topological properties, which are consequences of transversality to Whitney stratified sets. He has applied these methods to determine local and global geometric properties of an object and its boundary from its Blum medial axis (which is a skeleton of the object used to characterize shape properties). He has also applied these methods to determine, for various notions of scale, the generic "scale-based geometry" of grayscale images. Third, he has also used these infinitesimal methods to determine the topology and geometry of a large class of highly singular complete intersections.
The project will have a broader impact via its focus on the use of singularity theory as a tool for questions in computer imaging. This takes several forms including: continued joint interaction of the investigator with computer scientists at several locations, but especially at the Univ. of North Carolina; the development of tools from singularity theory, which directly apply to problems of interest in imaging; and the specific consideration, in joint work with several computer scientists, of several concrete imaging problems of current interest involving feature statistics and tensor imaging. First, the PI proposes to build on his results for medial structures to: develop concrete models in terms of topology for searchable structures for 3D objects which will be of use in 3D-imaging, to enlarge the class of allowable structures to include degeneracies, and to apply these ideas for uses in statistical properties of geometrical features for shape. Second, he will further develop methods of scale-based geometry to: determine the singular geometry in the presence of parameters; develop methods to verify generic geometric properties for feature detection; and apply the methods to tensor images. Third, he proposes to further advance the understanding of underlying geometric and topological structure of such highly singular spaces and their representations as sections of universal singularities, and their properties as Whitney stratified sets.
