Geometry and Topology of Singular Structures with Applications to Computer Imaging

达蒙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，以及这些结果在计算机成像问题中的应用。他建议进一步发展他的工作，适用于计算机成像，并继续他的理论工作，根据最近的发现，对性质的高度非孤立的奇点。他将扩展涉及“骨架结构”的方法，他介绍了捕捉任何维度的对象的形状，以获得“骨架/中间连接结构”的图像中多个对象的配置。这样的链接结构将提供一种数学结构，其可以用于：涉及多个生理特征的医学图像的统计分析，图像中的改进的分割，以及提供用于分析图像中的位置和形状信息的相互作用的严格的统一数学框架。以及他的工作将开发新的方法来研究高度非孤立奇点的拓扑结构已经超出了以前的方法孤立奇点。这涉及到确定的“消失拓扑”的奇点以及拓扑的补充和米尔诺纤维（高度奇异）“自由因子”，这自然产生的表示理论的可解线性代数groups.The拟议的调查下进行的授权将关注两个奇异空间的理论工作和它的应用问题，在医学成像。奇异空间是在研究光滑物体的过程中不可避免地产生的，它不同于我们通常所想象的光滑弯曲物体。在光滑物体的配置的情况下，研究者将开发适当的奇异结构，允许同时分析单个物体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，其中对于治疗和诊断，必须确定多个生理特征的精确相对位置和特征。这将涉及与几组计算机科学家的联合工作。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，不能使用传统的方法来理解。他正在开发一种新的方法，既简化了定性结构的这种奇异性，并提供明确的代数方法计算定性数值不变量的结构。

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.