弹性图形状分析的表示、统计和分类
Representations, Metrics and Statistics For Shape Analysis of Elastic Graphs
摘要：
过去用于对象的统计形状分析的方法主要集中在相同拓扑类内的对象，例如，标量函数、欧几里德曲线或曲面等。对于以更复杂的方式不同的对象，当前文献仅提供拓扑方法。本文介绍了一种影响深远的几何方法，用于分析图形对象的形状，如道路网络，血管，脑纤维束等，它表示这样的对象，表现出在几何形状和拓扑结构的差异，作为图形的曲线与任意形状（边缘）和连接在任意路口（节点）。为了进行统计分析，需要数学表示，度量和其他几何工具，如测地线，均值和协方差。本文利用商结构来开发有效的算法来计算这些数量，导致有用的统计工具，包括主成分分析和分析统计测试和建模的图形形状。使用各种模拟以及来自神经元和脑动脉网络的真实数据证明了该框架的有效性。
英文摘要：
Past approaches for statistical shape analysis of objects have focused mainly on objects within the same topological classes, e.g., scalar functions, Euclidean curves, or surfaces, etc. For objects that differ in more complex ways, the current literature offers only topological methods. This paper introduces a far-reaching geometric approach for analyzing shapes of graphical objects, such as road networks, blood vessels, brain fiber tracts, etc. It represents such objects, exhibiting differences in both geometries and topologies, as graphs made of curves with arbitrary shapes (edges) and connected at arbitrary junctions (nodes). To perform statistical analyses, one needs mathematical representations, metrics and other geometrical tools, such as geodesics, means, and covariances. This paper utilizes a quotient structure to develop efficient algorithms for computing these quantities, leading to useful statistical tools, including principal component analysis and analytical statistical testing and modeling of graphical shapes. The efficacy of this framework is demonstrated using various simulated as well as the real data from neurons and brain arterial networks.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
