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DOI: 10.4230/LIPIcs.SoCG.2024.44

Stability and Approximations for Decorated Reeb Spaces

Authors: Justin Curry, Washington Mio, Tom Needham, Osman Berat Okutan, and Florian Russold

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

Abstract

Given a map f:X → M from a topological space X to a metric space M, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when M = ℝ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of f. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data.

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Justin Curry, Washington Mio, Tom Needham, Osman Berat Okutan, and Florian Russold. Stability and Approximations for Decorated Reeb Spaces. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 44:1-44:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{curry_et_al:LIPIcs.SoCG.2024.44,
  author =	{Curry, Justin and Mio, Washington and Needham, Tom and Okutan, Osman Berat and Russold, Florian},
  title =	{{Stability and Approximations for Decorated Reeb Spaces}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{44:1--44:17},
  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.2024.44},
  URN =		{urn:nbn:de:0030-drops-199891},
  doi =		{10.4230/LIPIcs.SoCG.2024.44},
  annote =	{Keywords: Reeb spaces, Gromov-Hausdorff distance, Persistent homology}
}

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Stability and Approximations for Decorated Reeb Spaces

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