2026-07-25T06:57:56Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:19960 2024-06-06T06:21:13Z ddc:004 open_access

Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory Bauer, Ulrich Roll, Fabian persistent homology discrete Morse theory apparent pairs Wrap complex lexicographic optimal chains shape reconstruction We study the connection between discrete Morse theory and persistent homology in the context of shape reconstruction methods. Specifically, we consider the construction of Wrap complexes, introduced by Edelsbrunner as a subcomplex of the Delaunay complex, and the construction of lexicographic optimal homologous cycles, also considered by Cohen–Steiner, Lieutier, and Vuillamy in a similar setting. We show that for any cycle in a Delaunay complex for a given radius parameter, the lexicographically optimal homologous cycle is supported on the Wrap complex for the same parameter, thereby establishing a close connection between the two methods. We obtain this result by establishing a fundamental connection between reduction of cycles in the computation of persistent homology and gradient flows in the algebraic generalization of discrete Morse theory. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Ulrich Bauer and Fabian Roll 2024 Is Part Of LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2024.15 urn:nbn:de:0030-drops-199600 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.15 eng https://creativecommons.org/licenses/by/4.0/legalcode