Measuring the Distance Between Merge Trees

Beketayev, Kenes; Yeliussizov, Damir; Morozov, Dmitriy; Weber, Gunther H.; Hamann, Bernd

doi:10.1007/978-3-319-04099-8_10

Cited by 69 publications

(67 citation statements)

References 21 publications

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“…Another topological approach to analyzing time‐dependent data is due to [FOTT08]. A close work to ours is due to Beketayev et al [BYM*13] where two merge trees are compared by means of branch decompositions. They avoid instabilities by considering a large number of these decompositions, which on the other hand leads to very long computation times.…”

Section: Related Workmentioning

confidence: 99%

“…Beketayev et al [BYM*13] avoid these instabilities by considering branch decompositions with permuted branches, thereby explicitly allowing that parents have lower weights than their children. Since the number of these permutations grows exponentially with the number of extrema [BYM*13], this leads quickly to infeasible computation times. Therefore, we propose to use only the default branch decomposition and study the actual effect of these instabilities in real data sets in the next section.…”

Section: Evaluation Comparison and Applicationsmentioning

confidence: 99%

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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%

Section: Evaluation Comparison and Applicationsmentioning

confidence: 99%

Extended Branch Decomposition Graphs: Structural Comparison of Scalar Data

Saikia

Seidel

Weinkauf

2014

Computer Graphics Forum

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“…However, it is not clear how to generalize these results to Reeb graphs containing loops, an important family of features of the Reeb graph. Another distance based on the branch decomposition of merge trees was proposed in [6], together with a polynomial time algorithm to compute it. This distance, however, is not stable with respect to changes in the function and also does not generalize beyond trees.…”

Section: Introductionmentioning

confidence: 99%

Measuring Distance between Reeb Graphs

Bauer

Wang

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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120

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We propose a metric for Reeb graphs, called the functional distortion distance. Under this distance, the Reeb graph is stable against small changes of input functions. At the same time, it remains discriminative at differentiating input functions. In particular, the main result is that the functional distortion distance between two Reeb graphs is bounded from below by the bottleneck distance between both the ordinary and extended persistence diagrams for appropriate dimensions.As an application of our results, we analyze a natural simplification scheme for Reeb graphs, and show that persistent features in Reeb graph remains persistent under simplification. Understanding the stability of important features of the Reeb graph under simplification is an interesting problem on its own right, and critical to the practical usage of Reeb graphs.

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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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Measuring the Distance Between Merge Trees

Cited by 69 publications

References 21 publications

Extended Branch Decomposition Graphs: Structural Comparison of Scalar Data

Extended Branch Decomposition Graphs: Structural Comparison of Scalar Data

Measuring Distance between Reeb Graphs

Computing the Fréchet Distance between Real-Valued Surfaces

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