Geometry and Topology of Singular Structures with Applications to Computer Imaging

Damon教授一直在研究涉及奇异结构的几何、拓扑和变形性质的问题，包括分层集、映射和非孤立奇点，并将这些结果应用于开发计算机成像问题的几何方法。他建议进一步发展他的应用于计算机成像的工作，并根据最近的发现，继续他关于高度非孤立奇点性质的理论工作。他将扩展他引入的捕捉任何维度上物体形状的“骨架结构”的方法，以获得图像中多个物体的配置的“骨架/中间连接结构”。这种链接结构将提供一种数学结构，该结构可用于：对涉及多个生理特征的医学图像进行统计分析，改进图像中的分割，并提供用于分析图像中位置和形状信息的交互的严格统一的数学框架。此外，他的工作还将开发研究高度非孤立奇点的新方法，这些奇点的拓扑结构已经超出了以前为孤立奇点开发的方法的范围。这涉及到从可解线性代数群的表示理论中自然地产生的(高度奇异的)“自由因子”，确定奇点的“消失拓扑”以及补码和Milnor纤维的拓扑。在这项拨款下进行的拟议研究将涉及关于奇异空间的理论工作及其在医学成像问题中的应用。在光滑物体的研究中，不可避免地会出现奇异空间，它不同于我们通常所想象的光滑弯曲物体。在光滑对象的配置的情况下，调查者将开发适当的奇异结构，其允许同时分析各个对象的形状及其位置关系。这种结构可以用于各种成像问题，例如用于3D医学图像，其中为了治疗和诊断，必须确定多个生理特征的准确相对位置和特征。这将涉及到与几个计算机科学家小组的联合工作。研究人员还将研究这些奇异空间的性质，当它们的结构是高度奇异的，不能用传统方法理解时。他正在开发一种新的方法，既简化了这种奇点的定性结构，又提供了计算结构的定性数值不变量的显式代数方法。

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.