Representations, Metrics and Statistics For Shape Analysis of Elastic Graphs
This provides a method for analyzing complex shapes in domains like neuroscience and infrastructure, but it is incremental as it extends existing geometric approaches to graphical objects.
The paper tackles the problem of statistical shape analysis for graphical objects with varying geometries and topologies, such as road networks and brain fiber tracts, by introducing a geometric approach that uses a quotient structure to compute metrics and statistical tools, demonstrating efficacy on simulated and real data from neurons and brain arterial networks.
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.