Geometry and Topology of Singular Structures with Applications to Imaging

S教授的研究将涉及奇异结构的几何、拓扑和变形性质，包括分层集、映射和非孤立奇点，并应用这些结果来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，从潜在的“骨架/中间结构”确定欧几里德空间中区域及其边界的局部、相对和全局几何性质。第二，在联合工作中，他正在开发一种为几何建模寻找样条曲面交点的新方法。第三，他正在开发一种方法来表征自然图像中对象的局部特征，包括光照、几何特征和观察者移动。Damon教授将进一步发展他在骨骼和中间结构方面的结果：理解具有几何和物理相互作用的互补区域或对象集合中的多个中间结构的行为。这将允许此类对象的时间演化，并扩大允许结构的类别，以包括简并。这些思想将被用于研究形状几何特征的统计特性。其次，Damon教授将进一步发展在样条曲面流动下跟踪相交曲线演化的方法，并将其应用于计算具有由样条定义的边界的区域的中轴的新方法。第三，他将应用他的方法在联合研究中完成对自然图像中允许阴影/阴影和镜面反射的局部特征、几何特征以及视点或光源的运动的分析。达蒙？S教授的研究将进一步研究用于建模对象和区域的空间的性质，以及提供对几何空间的性质及其表示的理解的方程组。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，包括此类集合的统计属性。这种方法将允许结构的退化，这些得到的模型将被用于分析医学图像中区域的相互作用。这些相同的方法也将用于理解变形对象的交点的演变以及几何建模的应用。第三，他将分析可用于模拟自然图像中对象的图像的方程的性质，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。

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.