Professor Damon has been investigating problems involving the geometry, topology and deformation properties of singular structures, including stratified sets, mappings, and nonisolated singularities, and the application of these results for developing geometric methods for problems in computer imaging. He proposes to further develop his work which applies to computer imaging and also continue his theoretical work, based on recent discoveries, on the properties of highly nonisolated singularities. He will extend the methods involving "skeletal structures", which he introduced to capture the shape of objects in any dimension, to obtain "skeletal/medial linking structures" for configurations of multiple objects in images. Such a linking structure will provide a mathematical structure which could be used for: the statistical analysis of medical images involving multiple physiological features, improved segmentation in images, and providing a rigorous unified mathematical framework for analyzing the interaction of positional and shape information in images. As well his work will develop new methods for studying highly nonisolated singularities whose topology has been beyond the reach of previous methods developed for isolated singularities. This involves determining the "vanishing topology" of the singularities as well as the topology of the complements and Milnor fibers for (highly singular) "free divisors" which naturally arise from representation theory of solvable linear algebraic groups.
The proposed investigations to be carried out under this grant will concern both theoretical work on singular spaces and its application to problems in medical imaging. Singular spaces, which are different from our usual image of smoothly bending objects, inevitably arise in the study of smooth objects. In the case of configurations of smooth objects, the investigator will develop appropriate singular structures which allow for the simultaneous analysis of the shapes of the individual objects and their positional relations. Such structures can be used for various imaging problems, such as for 3D medical images, where for treatment and diagnosis, the exact relative position and features of multiple physiological features must be determined. This will involve joint work with several groups of computer scientists. The investigator will also investigate the properties of these singular spaces when their structure is highly singular and cannot be understood using traditional methods. He is developing a new method for both simplifying the qualitative structure of such singularities and providing explicit algebraic methods for computing qualitative numerical invariants of the structures.