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.

达蒙教授的研究将关注奇异结构的几何，拓扑和变形性质，包括分层集，映射和非孤立奇点，以及这些结果的应用，以发展计算机成像问题的几何方法。这包括应用奇点理论的方法，从一个潜在的“骨架/中间结构”确定欧几里德空间中的一个区域及其边界的局部、相对和全局几何特性。第二，他正在开发，在联合工作，一种新的方法来寻找相交的样条曲面的几何造型。Damon教授将进一步发展他在骨骼和中间结构方面的研究成果，以：理解在一组互补区域或具有几何和物理相互作用的物体中，多个中间结构的行为。这将允许这些物体的时间演化，并扩大允许结构的类别，包括简并。这些想法将用于调查形状的几何特征的统计特性。第二，Damon教授将进一步开发跟踪样条曲面流下相交曲线演化的方法，并将其应用于计算具有样条定义的边界的区域的中轴的新方法。第三，他将应用他的方法来完成，在联合研究中，分析自然图像中的局部特征，允许阴影/阴影和镜面反射，几何特征，以及视点或光源的运动。他将进一步开始开发算法与成像科学家为他们的实施。教授达蒙？的研究将进一步研究用于建模对象和区域的空间的属性，以及提供几何空间属性及其表示的理解的方程系统。这些方法将应用于计算机成像中的几个问题。第一种方法将用于对多个对象及其交互进行建模，包括这些集合的统计属性。这种方法将允许退化的结构，这些得到的模型将用于分析在医学图像中的区域的相互作用。这些相同的方法也将用于理解变形对象的交叉点的演变，并应用于几何建模。第三，他将分析方程的属性，这些方程可用于对自然图像中的对象的图像进行建模，其中照明和视点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工作，以开发识别图像中物体的程序。