Computing the Multicover Bifiltration

• DOI: 10.1007/s00454-022-00476-8
• Publication Date: 2023-02-21T19:23:50Z
• Authors: René Corbet; Michael Kerber; Michael Lesnick; Georg Osang
• Keywords: Computational Geometry (cs.CG); FOS: Computer and information sciences; higher-order Delaunay complexes; nerves; multiparameter persistent homology; higher-order Voronoi diagrams; 01 natural sciences; Article; 004; rhomboid tiling; 516; denoising; FOS: Mathematics; Computer Science - Computational Geometry; Algebraic Topology (math.AT); Mathematics - Algebraic Topology; ddc:004; 0101 mathematics; Bifiltrations

Given a finite set A⊂Rd, let Covr,k denote the of all points within distance r to at least k A. Allowing and vary, we obtain 2-parameter family spaces that grow larger when increases or decreases, called multicover bifiltration. Motivated by problem computing homology this bifiltration, introduce two closely related combinatorial bifiltrations, one polyhedral other simplicial, which are both topologically equivalent bifiltration far smaller than Čech-based model considered in prior work Sheehy. Our construction is rhomboid tiling Edelsbrunner Osang, can be efficiently computed using variant an algorithm given these authors. Using implementation for dimension 2 3, provide experimental results. simplicial useful understanding proving its correctness.