Inferring Local Homology from Sampled Stratified Spaces

We study the reconstruction of a stratified space from a possibly noisy point sample. Specifically, we use the vineyard of the distance function restricted to a 1-parameter family of neighborhoods of a point to assess the local homology of the stratified space at that point. We prove the correctness of this assessment under the assumption of a sufficiently dense sample. We also give an algorithm that constructs the vineyard and makes the local assessment in time at most cubic in the size of the Delaunay triangulation of the point sample. Keywords. Topological data analysis, local homology, persistence, stratified spaces, simplicial complexes, power diagrams, Delaunay triangulations, algorithms.

• Type: Conference
• Year: 2007
• Where: FOCS
• Authors: Paul Bendich, David Cohen-Steiner, Herbert Edelsbrunner, John Harer, Dmitriy Morozov
• Keywords: FOCS 2007, Local Homology, Point Sample, Stratified Spaces, Theoretical Computer Science