Author(s):

Edelsbrunner, Herbert; Osang, Georg

Title: 
The multicover 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 kfold 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 orderk Delaunay mosaics of X, and construct a zigzag module from Delaunay mosaics that is isomorphic to the persistence module of the multicovers.

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  LeibnizZentrum für Informatik

Date Published:

20180611

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. I02979N35 of the Austrian Science Fund (FWF).

DOI: 
10.4230/LIPIcs.SoCG.2018.34

Open access: 
yes (OA journal) 