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Title: Statistical shape analysis of complex networks of curves Authors:  Anuj Srivastava - Florida State University (United States) [presenting]
Abstract: Imaging data from many applications leads to geometrical structures resembling complex pathways or curvilinear networks. We will call them ``shape networks''. A prominent example of a shape network is the Brain Arterial Network or BAN in the human brain, which is a complex arrangement of individual arteries, branching patterns, and inter-connectivities. Another example is a road network. Shapes or structures of these objects play an essential role in characterizing and understanding the functionality of larger systems. One would like tools for statistically analyzing shape networks, i.e., quantifying shape differences, summarizing shapes, comparing populations, and studying the effects of covariates on these shapes. The purpose is to represent and statistically analyze shape networks as ``elastic shape graphs''. Each elastic shape graph consists of nodes, or points in 3D, connected by some 3D curves, or edges, with arbitrary shapes. We develop a mathematical representation, a Riemannian metric, and other geometrical tools, such as computations of geodesics, means, covariances, and PCA, for helping analyze elastic shape graphs. We apply this framework to analyzing shapes of BANs taken from 92 subjects. Specifically, we generate shape summaries of BANs, perform shape PCA, and study the effects of age and gender on their shapes. We conclude that age has a clear, quantifiable effect on BAN shapes. Specifically, we find an increased variance in BAN shapes as age increases.