Measure-Theoretic Reeb Graphs and Reeb Spaces
40th International Symposium on Computational Geometry (SoCG 2024), 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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To cite: Qingsong Wang and Guanqun Ma and Raghavendra Sridharamurthy and Bei Wang "Measure-Theoretic Reeb Graphs and Reeb Spaces" SoCG 2024 10.4230/LIPIcs.SoCG.2024.80
BibTeX:
@InProceedings{WangetalSOCG2024, 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} }