Measure Theoretic Reeb Graphs and Reeb Spaces
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912073626157056 |
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| author | Wang, Qingsong Ma, Guanqun Sridharamurthy, Raghavendra Wang, Bei |
| author_facet | Wang, Qingsong Ma, Guanqun Sridharamurthy, Raghavendra Wang, Bei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06748 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Measure Theoretic Reeb Graphs and Reeb Spaces Wang, Qingsong Ma, Guanqun Sridharamurthy, Raghavendra Wang, Bei Computational Geometry 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. |
| title | Measure Theoretic Reeb Graphs and Reeb Spaces |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2401.06748 |