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Measure Theoretic Reeb Graphs and Reeb Spaces

Qingsong Wang, Guanqun Ma, Raghavendra Sridharamurthy, Bei Wang

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.

Measure Theoretic Reeb Graphs and Reeb Spaces

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.
Paper Structure (14 sections, 18 theorems, 19 equations, 4 figures)

This paper contains 14 sections, 18 theorems, 19 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background on Reeb Graphs and Reeb Spaces
  4. Background on Measure-Theoretic Concepts
  5. Reeb Graphs for Metric Measure Spaces
  6. Reeb Spaces for Metric Measure Spaces
  7. Range-Integrated Reeb Graphs
  8. Range-Integrated Reeb Spaces
  9. Conclusion and Discussion
  10. Detailed Proofs
  11. Detailed Proofs for Section \ref{['sec:reeb-graph']}
  12. Detailed Proofs for Section \ref{['sec:reeb-space']}
  13. Detailed Proofs for Section \ref{['sec:range-integrated']}
  14. Detailed Proofs for Section \ref{['sec:range-reeb-space']}

Key Result

Proposition 4

Let $(G, f)$ and $(H, h)$ be two Reeb graphs. Then if and only if $(G, f)$ is isomorphic to $(H, h)$.

Figures (4)

  • Figure 1: An example of a Reeb graph.
  • Figure 2: From left to right: a Reeb graph $G$, its $\varepsilon$-thickening with a function $f_{\varepsilon}$, and the Reeb graph of the $\varepsilon$-thickening.
  • Figure 3: Smoothed Reeb graphs based on distance to a measure (top) and kernel distance to a measure $D_{\mu, K}$ (bottom) with Gaussian kernel function. From left to right: (a) the original topological space ${X}{\hbox{${X}$}}$ colored by a bounded positive function (e.g., $d_{\mu,m}$ or $D_{\mu, K}$) on ${X}{\hbox{${X}$}}$; (b) the (locally) thickened spaces with a small $\mu$ value together with (c) the $d_{\mu,m}$-smoothed Reeb graph (top) and the $D_{\mu,K}$-smoothed Reeb graph (bottom); (d)-(e): similar to (b)-(c) with a large $\mu$ value.
  • Figure 4: Visualization of a Reeb graph $R(X, f)$ (left) and a range-integrated Reeb Graph $R(X, F_\mu \circ f)$ (right) respectively.

Theorems & Definitions (36)

  • Definition 1: Reeb graph
  • Definition 2: Smoothing of Reeb graph DeSilvaMunchPatel2016
  • Definition 3: Interleaving distance DeSilvaMunchPatel2016
  • Proposition 4: DeSilvaMunchPatel2016
  • Proposition 5
  • Theorem 6: DeSilvaMunchPatel2016
  • Definition 7: Reeb space
  • Definition 8: Metric measure space Sturm2006
  • Definition 9: 2-Wasserstein distance
  • Definition 10: Distance to a measure BuchetChazalOudot2016
  • ...and 26 more