The multi-cover persistence of Euclidean balls Conference Paper

Author(s): Edelsbrunner, Herbert; Osang, Georg
Title: The multi-cover persistence of Euclidean balls
Title Series: Leibniz International Proceedings in Information, LIPIcs
Affiliation IST Austria
Abstract: Given a locally finite X ⊆ ℝd and a radius r ≥ 0, the k-fold cover of X and r consists of all points in ℝd that have k or more points of X within distance r. We consider two filtrations - one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k - and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in ℝd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module from Delaunay mosaics that is isomorphic to the persistence module of the multi-covers.
Keywords: persistent homology; topology; Zigzag modules; Computation theory; discrete Morse theory; hyperplane arrangements; Delaunay mosaics; Delaunay
Conference Title: SoCG: Symposium on Computational Geometry
Volume: 99
Conference Dates: June 11 - 14, 2018
Conference Location: Budapest, Hungary
ISBN: 18688969 (ISSN); 9783959770668 (ISBN)
Publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik  
Date Published: 2018-06-11
Start Page: 341
End Page: 3414
Sponsor: This work is partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).
DOI: 10.4230/LIPIcs.SoCG.2018.34
Open access: yes (OA journal)
IST Austria Authors
  1. Georg Osang
    3 Osang
Related IST Austria Work