Singular Structures in Medial and Scale-Based Geometry

内侧和基于尺度的几何中的奇异结构

• 批准号: 0405947
• 负责人: James Damon
• 金额: --
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405947&HistoricalAwards=false
• 关键词: Singular; Structures; Medial; Scale; Based

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.

James N.达蒙PI将继续调查两个表面上无关的问题。 其中之一涉及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的方法，通过组的等价，也从分析的几何和拓扑性质，这是横向的后果惠特尼分层集。 他应用这些方法来确定局部和全局的几何属性的对象和它的边界从它的Blum中轴（这是一个骨架的对象用于表征形状属性）。 他还应用这些方法来确定，为各种概念的规模，通用的“基于规模的几何”的灰度图像。第三，他还使用这些无穷小的方法来确定拓扑和几何的一大类高度奇异的完整的交叉。该项目将有一个更广泛的影响，通过它的重点是使用奇点理论作为工具的问题，在计算机成像。这采取了几种形式，包括：继续联合互动的调查人员与计算机科学家在几个地点，但特别是在大学的北卡罗来纳州;工具的发展，从奇点理论，这直接适用于问题的兴趣成像;具体的考虑，在与几位计算机科学家的联合工作中，当前感兴趣的几个具体成像问题，涉及特征统计和张量成像。首先，PI建议建立在他的中间结构结果的基础上：发展具体的模型，在拓扑结构方面的可搜索的结构的3D对象，这将是在3D成像中使用，扩大类允许的结构，包括退化，并应用这些想法，用于统计性质的几何特征的形状。其次，他将进一步开发基于尺度的几何方法：确定参数存在时的奇异几何;开发验证特征检测的通用几何属性的方法;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基本几何和拓扑结构的理解，以及它们作为泛奇点的部分的表示，以及它们作为惠特尼分层集的性质。