2015/09/01 by Niccolò Cavazza, Massimo Ferri, Claudia Landi · 1 citation
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics #Betti number #Mathematics #Submanifold #Persistent homology #Euclidean space #Ball (mathematics) #Euclidean geometry #Combinatorics #Discrete mathematics #Pure mathematics #Mathematical analysis #Algorithm #Geometry
paper · open access · doi:10.1142/s0218195915500119
published in International Journal of Computational Geometry & Applications 25(03), 187-205 (World Scientific)
openalex publication_date 2015/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02
An exact computation of the persistent Betti numbers of a submanifold [Formula: see text] of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of [Formula: see text] is available. We show that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of [Formula: see text] from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain. Using these inequalities we improve a previous lower bound for the natural pseudodistance to assess dissimilarity between the shapes of two objects from a sampling of them. Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it.