2026-08-18T20:42:00Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:8747 2024-03-06T10:42:33Z ddc:004 open_access

The Multi-cover Persistence of Euclidean Balls Edelsbrunner, Herbert Osang, Georg Delaunay mosaics hyperplane arrangements discrete Morse theory zigzag modules persistent homology Given a locally finite X subseteq R^d and a radius r >= 0, the k-fold cover of X and r consists of all points in R^d 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 R^{d+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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Herbert Edelsbrunner and Georg Osang 2018 Is Part Of LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2018.34 urn:nbn:de:0030-drops-87471 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.34 eng https://creativecommons.org/licenses/by/3.0/legalcode