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The Multi-cover Persistence of Euclidean Balls

Authors Herbert Edelsbrunner, Georg Osang

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Herbert Edelsbrunner
Georg Osang

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Herbert Edelsbrunner and Georg Osang. The Multi-cover Persistence of Euclidean Balls. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SoCG.2018.34

Abstract

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.
Keywords
  • Delaunay mosaics
  • hyperplane arrangements
  • discrete Morse theory
  • zigzag modules
  • persistent homology

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