TY - GEN
T1 - Computing the multicover bifiltration
AU - Corbet, René
AU - Kerber, Michael
AU - Lesnick, Michael
AU - Osang, Georg
N1 - Funding Information:
Funding The first two authors were supported by the Austrian Science Fund (FWF) grant number P 29984-N35 and W1230. The first author was partly supported by an Austrian Marshall Plan Scholarship, and by the Brummer & Partners MathDataLab.
Publisher Copyright:
© René Corbet, Michael Kerber, Michael Lesnick, and Georg Osang; licensed under Creative Commons License CC-BY 4.0 37th International Symposium on Computational Geometry (SoCG 2021).
PY - 2021/6/1
Y1 - 2021/6/1
N2 - Given a finite set A ⊂ ℝd, let Covr,k denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness.
AB - Given a finite set A ⊂ ℝd, let Covr,k denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness.
KW - Bifiltrations
KW - Denoising
KW - Higher-order Delaunay complexes
KW - Higher-order Voronoi diagrams
KW - Multiparameter persistent homology
KW - Nerves
KW - Rhomboid tiling
UR - http://www.scopus.com/inward/record.url?scp=85108226630&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SoCG.2021.27
DO - 10.4230/LIPIcs.SoCG.2021.27
M3 - Conference paper
AN - SCOPUS:85108226630
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 37th International Symposium on Computational Geometry, SoCG 2021
A2 - Buchin, Kevin
A2 - de Verdiere, Eric Colin
PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik
T2 - 37th International Symposium on Computational Geometry, SoCG 2021
Y2 - 7 June 2021 through 11 June 2021
ER -