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Measure-Theoretic Reeb Graphs and Reeb Spaces

Authors Qingsong Wang , Guanqun Ma , Raghavendra Sridharamurthy , Bei Wang

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Author Details

Qingsong Wang
  • University of Utah, Salt Lake City, UT, USA
Guanqun Ma
  • University of Utah, Salt Lake City, UT, USA
Raghavendra Sridharamurthy
  • University of Utah, Salt Lake City, UT, USA
Bei Wang
  • University of Utah, Salt Lake City, UT, USA

Funding

This project was partially supported by NSF DMS 2301361, NSF IIS 2145499, NSF IIS 1910733, DOE DE SC0021015, and DOE DE SC0023157.

Cite AsGet BibTex

Qingsong Wang, Guanqun Ma, Raghavendra Sridharamurthy, and Bei Wang. Measure-Theoretic Reeb Graphs and Reeb Spaces. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 80:1-80:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.80

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Topology
Keywords
  • Reeb graph
  • Reeb space
  • metric measure space
  • topological data analysis

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