Hierarchical Shape Segmentation and Registration via Topological Features of Laplace-Beltrami Eigenfunctions

This work introduces a method to hierarchically segment articulated shapes into meaningful parts and to register these parts across populations of near-isometric shapes (e.g. head, arms, legs and fingers of humans in different body postures). The method exploits the isometry invariance of eigenfunctions of the Laplace-Beltrami operator and uses topological features (level sets at important saddles) for the segmentation. Concepts from persistent homology are employed for a hierarchical representation, for the elimination of topological noise and for the comparison of eigenfunctions. The obtained parts can be registered via their spectral embedding across a population of near isometric shapes. This work also presents the highly accurate computation of eigenfunctions and eigenvalues with cubic finite elements on triangle meshes and discusses the construction of persistence diagrams from the Morse-Smale complex as well as the relation to size functions.