Advances in Geometrical Analysis of Topologically-Varying Shapes

Statistical shape analysis using geometrical approaches provides comprehensive tools – such as geodesic deformations, shape averages, and principal modes of variability – all in the original object space. While geometrical methods have been limited to objects with fixed topologies (e.g. functions, closed curves, surfaces of genus zero, etc) in the past, this paper summarizes recent progress where geometrical approaches are beginning to handle topologically different objects – trees, graphs, etc – that exhibit arbitrary branching and connectivity patterns. The key idea is to “divide-and-conquer”, i.e. break complex objects into simpler parts and help register these parts across objects. Such matching and quantification require invariant metrics from Riemannian geometry and provide foundational tools for statistical shape analysis.