research-article

Wasserstein Distances, Geodesics and Barycenters of Merge Trees

Authors: Mathieu Pont, Jules Vidal, Julie Delon, Julien TiernyAuthors Info & Claims

IEEE Transactions on Visualization and Computer Graphics, Volume 28, Issue 1

Pages 291 - 301

https://doi.org/10.1109/TVCG.2021.3114839

Published: 01 January 2022 Publication History

Abstract

This paper presents a unified computational framework for the estimation of distances, geodesics and barycenters of merge trees. We extend recent work on the edit distance [104] and introduce a new metric, called the Wasserstein distance between merge trees, which is purposely designed to enable efficient computations of geodesics and barycenters. Specifically, our new distance is strictly equivalent to the $L$2-Wasserstein distance between extremum persistence diagrams, but it is restricted to a smaller solution space, namely, the space of rooted partial isomorphisms between branch decomposition trees. This enables a simple extension of existing optimization frameworks [110] for geodesics and barycenters from persistence diagrams to merge trees. We introduce a task-based algorithm which can be generically applied to distance, geodesic, barycenter or cluster computation. The task-based nature of our approach enables further accelerations with shared-memory parallelism. Extensive experiments on public ensembles and SciVis contest benchmarks demonstrate the efficiency of our approach - with barycenter computations in the orders of minutes for the largest examples - as well as its qualitative ability to generate representative barycenter merge trees, visually summarizing the features of interest found in the ensemble. We show the utility of our contributions with dedicated visualization applications: feature tracking, temporal reduction and ensemble clustering. We provide a lightweight C++ implementation that can be used to reproduce our results.

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IEEE Transactions on Visualization and Computer Graphics Volume 28, Issue 1

Jan. 2022

1190 pages

ISSN:1077-2626

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Lyu WSridharamurthy RPhillips JWang B(2024)Fast Comparative Analysis of Merge Trees Using Locality Sensitive HashingIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2024.345638331:1(141-151)Online publication date: 12-Sep-2024
https://dl.acm.org/doi/10.1109/TVCG.2024.3456383
Sisouk KDelon JTierny J(2024)Wasserstein Dictionaries of Persistence DiagramsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2023.333026230:2(1638-1651)Online publication date: 1-Feb-2024
https://dl.acm.org/doi/10.1109/TVCG.2023.3330262
Liu GIuricich F(2024)A Task-Parallel Approach for Localized Topological Data StructuresIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2023.332718230:1(1271-1281)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1109/TVCG.2023.3327182
Wetzels FPont MTierny JGarth C(2024)Merge Tree Geodesics and Barycenters with Path MappingsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2023.332660130:1(1095-1105)Online publication date: 1-Jan-2024
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Athawale TTriana BKotha TPugmire DRosen P(2024)A Comparative Study of the Perceptual Sensitivity of Topological Visualizations to Feature VariationsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2023.332659230:1(1074-1084)Online publication date: 1-Jan-2024
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Pont MTierny J(2023)Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)IEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2023.333475530:9(6390-6406)Online publication date: 28-Nov-2023
https://dl.acm.org/doi/10.1109/TVCG.2023.3334755

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