Singular Structures in Medial and Scale-Based Geometry

0405947詹姆斯·N·达蒙私家侦探将继续调查两个表面上无关的问题。一个是关于分层集和非孤立奇点的几何、拓扑和变形性质。另一个涉及计算机成像问题的几何方法的发展。这些不同的问题得益于通过等价群的奇点理论的方法，也得益于几何和拓扑性质的分析，这些性质是横截到Whitney分层集的结果。他应用这些方法来确定对象的局部和全局几何属性及其从Blum中轴(这是用于表征形状属性的对象的骨架)的边界。他还应用这些方法来确定不同的尺度概念，即灰度图像的一般“基于尺度的几何图形”。第三，他还用这些无穷小方法确定了一大类高度奇异的完全交集的拓扑和几何。该项目将通过专注于使用奇点理论作为计算机成像中的问题工具来产生更广泛的影响。这采取了几种形式，包括：调查员继续与几个地点的计算机科学家进行联合互动，特别是在大学。首先，PI建议在他关于中间结构的结果的基础上建立将在3D成像中使用的3D对象的可搜索结构的具体模型，扩大允许结构的类别以包括退化，并将这些想法应用于形状的几何特征的统计特性中。其次，他将进一步发展基于尺度的几何方法：在参数存在的情况下确定奇异几何；开发方法来验证用于特征检测的一般几何性质；以及将这些方法应用于张量图像。第三，他建议进一步加深对这种高度奇异空间的基本几何和拓扑结构的理解，以及它们作为泛奇异截面的表示，以及它们作为Whitney分层集的性质。

DMS-0405947 James N. DamonThe 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.