Singular Structures in Medial and Scale-Based Geometry

詹姆斯·n·戴蒙私家侦探将继续调查两个表面上毫不相关的问题。一类研究了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。这些不同的问题受益于奇点理论通过等价群的方法，也受益于几何和拓扑性质的分析，这是惠特尼分层集的横向性的结果。他将这些方法应用于确定对象的局部和全局几何属性及其从Blum中轴线（用于表征形状属性的对象的骨架）的边界。他还将这些方法应用于确定各种尺度概念的灰度图像的通用“基于尺度的几何”。第三，他还使用这些无穷小方法确定了一大类高度奇异的完全交的拓扑和几何形状。该项目将通过其重点使用奇点理论作为计算机成像问题的工具而产生更广泛的影响。这有几种形式，包括：研究者与几个地方的计算机科学家，特别是在北卡罗莱纳大学的继续联合互动；奇点理论中工具的发展，直接应用于成像中的问题；在与几位计算机科学家的联合工作中，具体考虑了当前感兴趣的几个具体成像问题，包括特征统计和张量成像。首先，PI建议以他的中间结构的结果为基础：根据可搜索的3D物体结构的拓扑开发具体模型，这将用于3D成像，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几何特征的统计特性。其次，他将进一步发展基于尺度的几何方法，以确定存在参数的奇异几何；开发用于特征检测的通用几何属性验证方法；并将这些方法应用于张量图像。第三，他建议进一步推进对这种高度奇异空间的潜在几何和拓扑结构的理解，以及它们作为普遍奇点部分的表示，以及它们作为惠特尼分层集的性质。

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.