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DOI: 10.4230/LIPIcs.SoCG.2024.80
Measure-Theoretic Reeb Graphs and Reeb Spaces

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

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

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.
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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)

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@InProceedings{wang_et_al:LIPIcs.SoCG.2024.80,
  author =	{Wang, Qingsong and Ma, Guanqun and Sridharamurthy, Raghavendra and Wang, Bei},
  title =	{{Measure-Theoretic Reeb Graphs and Reeb Spaces}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{80:1--80:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.80},
  URN =		{urn:nbn:de:0030-drops-200257},
  doi =		{10.4230/LIPIcs.SoCG.2024.80},
  annote =	{Keywords: Reeb graph, Reeb space, metric measure space, topological data analysis}
}
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