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Title: Measure-Theoretic Reeb Graphs and Reeb Spaces
A Reeb graph is a graphical representation of a scalar function on a topological space that encodes the topology of the level sets. A Reeb space is a generalization of the Reeb graph to a multiparameter function. In this paper, we propose novel constructions of Reeb graphs and Reeb spaces that incorporate the use of a measure. Specifically, we introduce measure-theoretic Reeb graphs and Reeb spaces when the domain or the range is modeled as a metric measure space (i.e., a metric space equipped with a measure). Our main goal is to enhance the robustness of the Reeb graph and Reeb space in representing the topological features of a scalar field while accounting for the distribution of the measure. We first introduce a Reeb graph with local smoothing and prove its stability with respect to the interleaving distance. We then prove the stability of a Reeb graph of a metric measure space with respect to the measure, defined using the distance to a measure or the kernel distance to a measure, respectively.  more » « less
Award ID(s):
2145499 2301361 1910733
PAR ID:
10515293
Author(s) / Creator(s):
; ; ;
Editor(s):
Mulzer, Wolfgang; Phillips, Jeff M
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Journal Name:
40th International Symposium on Computational Geometry (SoCG 2024), Leibniz International Proceedings in Informatics (LIPIcs)
Volume:
293
ISSN:
978-3-95977-316-4
ISBN:
978-3-95977-316-4
Page Range / eLocation ID:
80:1-80:18
Subject(s) / Keyword(s):
Reeb graph Reeb space metric measure space topological data analysis theory of computation design and analysis of algorithms mathematics of computing topology
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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