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Gromov–Hausdorff Approximation of Filamentary Structures Using Reeb-Type Graphs

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Abstract

In many real-world applications, data appear to be sampled around 1-dimensional filamentary structures that can be seen as topological metric graphs. In this paper, we address the metric reconstruction problem of such filamentary structures from data sampled around them. We prove that they can be approximated with respect to the Gromov–Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and provide an efficient and easy-to-implement algorithm to compute such approximations in almost linear time. We illustrate the performance of our algorithm on a few datasets.

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Notes

  1. See [28], Chap. 1] for the definition of the length of a continuous curve in a general metric space.

  2. Recall that a minimizing geodesic in \(X\) is any curve \(\gamma : I \rightarrow X\), where \(I\) is a real interval, such that \(d_X(\gamma (t),\gamma (t')) = |t - t'|\) for any \(t,t' \in I\).

  3. Strictly speaking we should call it the \(\alpha \)-Reeb graph associated to the covering \(\mathcal {I}\), but we assume in the sequel that some covering \(\mathcal {I}\) has been chosen and we omit it in notations.

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Acknowledgments

The authors acknowledge Daniel Müllner and Gunnar Carlsson for fruitful discussions and for providing code for the Mapper algorithm. They acknowledge the European Project CG-Learning EC contract No. 255827; the ANR project TopData (ANR-13-BS01-0008); The National Basic Research Program of China (973 Program 2012CB825501); The NSF of China (11271011).

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Authors and Affiliations

  1. INRIA Saclay, Palaiseau, France

    Frédéric Chazal & Ruqi Huang

  2. Tsinghua University, Beijing, China

    Jian Sun

Authors
  1. Frédéric Chazal
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  2. Ruqi Huang
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Correspondence to Frédéric Chazal.

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Editors in charge: Siu-Wing Cheng and Olivier Devillers

Preliminary version in Proceedings of the 30th Annual Symposium on Computational Geometry, 2014.

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Chazal, F., Huang, R. & Sun, J. Gromov–Hausdorff Approximation of Filamentary Structures Using Reeb-Type Graphs. Discrete Comput Geom 53, 621–649 (2015). https://doi.org/10.1007/s00454-015-9674-1

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