Geometry and Topology of Singular Structures with Applications to Imaging

奇异结构的几何和拓扑及其在成像中的应用

• 批准号: 0706941
• 负责人: James Damon
• 金额: $ 10.51万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-08-15 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706941&HistoricalAwards=false
• 关键词: Geometry; Topology; Singular; Structures; Applications

Professor Damon?s research will concern the geometry, topology and deformation properties of singular structures, including stratified sets, mappings, and nonisolated singularities, and the application of these results to develop geometric methods for problems in computer imaging. This includes applying methods from singularity theory to determine from an underlying "skeletal/medial structure" the local, relative and global geometric properties of a region in Euclidean space and its boundary. Second, he is developing, in joint work, a new method for finding intersection of spline surfaces for geometric modeling. Third, he is developing methods to characterize local features of objects in natural images which include lighting, geometric features, and viewer movement.Professor Damon will further develop his results on skeletal and medial structures to:understand the behavior of multiple medial structures in a collection of complementary regions or objects with geometric and physical interaction. This will allow the time evolution of such objects, and enlarge the class of allowable structures to include degeneracies. These ideas will be used for investigating statistical properties of geometrical features for shape. Second, Professor Damon will further develop methods for following the evolution of intersection curves under flows of spline surfaces and apply them to give a new method for computing the medial axis of regions with boundaries defined by splines. Third, he will apply his methods to complete, in joint research, the analysis of local features in natural images allowing shade/shadow and specularity, geometric features, and movement of either viewpoint or light source. He will further begin developing algorithms with imaging scientists for their implementation.Professor Damon?s research will further investigate properties of spaces used to model objects and regions, and systems of equations which provide understanding of properties of geometric spaces and their representations. These methods will be applied to several problems in computer imaging. The first method will be used for modeling multiple objects and their interaction, including statistical properties of such collections. This method will allow for degeneracies of the structures, and these resulting models will be used for analyzing interaction of regions in medical images. These same methods will also be used for understanding the evolution of intersections of deforming objects with applications to geometric modeling. Third, he will analyze the properties of equations which can be used to model the images of objects in natural images, where lighting and viewpoint affect the appearance of geometric features of objects. These results will be used in joint work with computer scientists to develop procedures for identifying objects in images.

达蒙教授的研究将涉及奇异结构的几何形状，拓扑和变形特性，包括分层集，映射和非分离奇异性，并应用这些结果来开发计算机成像中问题的几何方法。这包括应用奇异理论的方法来确定欧几里得空间中一个区域及其边界的局部，相对和全局几何特性的基本“骨骼/内侧结构”。其次，他在联合工作中开发了一种新方法，用于寻找用于几何建模的样条表面的相交。第三，他正在开发方法来表征自然图像中对象的局部特征，包括照明，几何特征和观众运动。戴蒙（Damon）将进一步发展其在骨骼和内侧结构上的结果，以了解：了解互补区域中多个内侧结构的行为，或者是几何和物理相互作用。这将允许此类对象的时间演变，并扩大允许结构的类别包含归化性。这些想法将用于研究形状几何特征的统计特性。其次，达蒙教授将进一步开发用于遵循样条表面流下相交曲线演变的方法，并将其应用它们提供一种新的方法，以计算具有跨度定义的边界的区域内侧轴的新方法。第三，他将使用他的方法在联合研究中完成对自然图像中局部特征的分析，从而允许阴影/阴影和镜面，几何特征以及视点或光源的移动。他将进一步开始使用成像科学家的实施来开发算法。Damon的研究将进一步研究用于建模对象和区域的空间的特性，以及方程式的系统，这些方程式提供了对几何空间及其表示的特性的理解。这些方法将应用于计算机成像中的几个问题。第一种方法将用于建模多个对象及其相互作用，包括此类集合的统计属性。该方法将允许结构的变性，这些结果模型将用于分析医疗图像中区域的相互作用。这些相同的方法还将用于理解变形对象与几何建模应用程序的相互作用的演变。第三，他将分析方程的属性，这些方程式可用于模拟自然图像中对象的图像，其中照明和观点会影响对象的几何特征的外观。这些结果将用于与计算机科学家的联合合作，以开发用于识别图像中对象的程序。