2026-07-25T06:29:59Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20025 2024-06-06T06:21:17Z ddc:004 open_access

Measure-Theoretic Reeb Graphs and Reeb Spaces Wang, Qingsong Ma, Guanqun Sridharamurthy, Raghavendra Wang, Bei Reeb graph Reeb space metric measure space topological data analysis 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Qingsong Wang and Guanqun Ma and Raghavendra Sridharamurthy and Bei Wang 2024 Is Part Of LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2024.80 urn:nbn:de:0030-drops-200257 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.80 eng https://creativecommons.org/licenses/by/4.0/legalcode