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arXiv · 2401.06748

Measure Theoretic Reeb Graphs and Reeb Spaces

Qingsong Wang·Guanqun Ma·Raghavendra Sridharamurthy·Bei Wang
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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, Bei Wang. 2024-01-12. Measure Theoretic Reeb Graphs and Reeb Spaces. https://arxiv.org/abs/2401.06748

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