Computing the Multicover Bifiltration

• Published in: Discrete & computational geometry Vol. 70; no. 2; pp. 376 - 405
• Main Authors: Corbet, René, Kerber, Michael, Lesnick, Michael, Osang, Georg
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2023; Springer Nature B.V
• Subjects: Algorithms; Bar codes; Bifiltrations; Combinatorial analysis; Combinatorics; Computation; Computational Mathematics and Numerical Analysis; Construction; Denoising; Geometry; Higher-order Delaunay complexes; Homology; Mathematics; Mathematics and Statistics; Multiparameter persistent homology; Nerves; Rhomboid tiling; Tiling; Higher-order Delaunay complexes; Bifiltrations; Nerves; 55N31; Rhomboid tiling; Multiparameter persistent homology; Denoising
• ISSN: 0179-5376; 1432-0444; 1432-0444
• DOI: 10.1007/s00454-022-00476-8

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