Measuring Distance between Reeb Graphs

Bauer, Ulrich; Ge, Xinna; Wang, Yusu

doi:10.1145/2582112.2582169

Cited by 97 publications

(123 citation statements)

References 35 publications

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“…Several methods deal with the comparison of topological structures. Bauer et al [BGW13] introduce a distance metric for Reeb graphs under which they are stable to small perturbations of the input. The algorithm has exponential complexity.…”

Section: Related Workmentioning

confidence: 99%

Extended Branch Decomposition Graphs: Structural Comparison of Scalar Data

Saikia

Seidel

Weinkauf

2014

Computer Graphics Forum

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We present a method to find repeating topological structures in scalar data sets. More precisely, we compare all subtrees of two merge trees against each other -in an efficient manner exploiting redundancy. This provides pair-wise distances between the topological structures defined by sub/superlevel sets, which can be exploited in several applications such as finding similar structures in the same data set, assessing periodic behavior in time-dependent data, and comparing the topology of two different data sets. To do so, we introduce a novel data structure called the extended branch decomposition graph, which is composed of the branch decompositions of all subtrees of the merge tree. Based on dynamic programming, we provide two highly efficient algorithms for computing and comparing extended branch decomposition graphs. Several applications attest to the utility of our method and its robustness against noise.

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Section: Related Workmentioning

confidence: 99%

Extended Branch Decomposition Graphs: Structural Comparison of Scalar Data

Saikia

Seidel

Weinkauf

2014

Computer Graphics Forum

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“…Recently, other measures [6,7,8,20,28] such as the interleaving distance or functional distortion distance have been investigated for contour trees. In particular, it is expected that the interleaving distance and the functional distortion distance are equal, but so far it has been proven only that they differ by at most a constant factor.…”

Section: Definitions Background and Resultsmentioning

confidence: 99%

Computing the Fréchet Distance between Real-Valued Surfaces

Buchin

Ophelders

Speckmann

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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The Fréchet distance is a well-studied measure for the similarity of shapes. While efficient algorithms for computing the Fréchet distance between curves exist, there are only few results on the Fréchet distance between surfaces. Recent work has shown that the Fréchet distance is computable between piecewise linear functions f and g : M → R k with M a triangulated surface of genus zero. We focus on the case k = 1 and M being a topological sphere or disk with constant boundary. Intuitively, we measure the distance between terrains based solely on the height function. Our main result is that in this case computing the Fréchet distance between f and g is in NP.We additionally show that already for k = 1, computing a factor 2 − ε approximation of the Fréchet distance is NP-hard, showing that this problem is in fact NP-complete. We also define an intermediate distance, between contour trees, which we also show to be NPcomplete to compute. Finally, we discuss how our and other distance measures between contour trees relate to each other.

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“…The algorithm of [24] also always outputs a graph but it does not come with metric guarantees. Recently, Bauer et al [6] define a metric based on the function for Reeb graph and show that it is stable under Gromov-Hausdorff distance. The implementation of our algorithm relies on the Mapper algorithm [34], which provides a way to visualize datasets endowed with a real-valued function as a graph, where the considered function is the distance to the chosen root point.…”

Section: Related Workmentioning

confidence: 99%

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

Chazal

Huang

Sun

2015

Discrete Comput Geom

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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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Measuring Distance between Reeb Graphs

Cited by 97 publications

References 35 publications

Extended Branch Decomposition Graphs: Structural Comparison of Scalar Data

Extended Branch Decomposition Graphs: Structural Comparison of Scalar Data

Computing the Fréchet Distance between Real-Valued Surfaces

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

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