Wasserstein Distances, Geodesics and Barycenters of Merge Trees

Pont, Mathieu; Vidal, Jules; Delon, Julie; Tierny, Julien

doi:10.1109/tvcg.2021.3114839

Cited by 22 publications

(140 citation statements)

References 101 publications

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“…The alignment distance is still NPhard [JWZ94] for trees of arbitrary degree, but can be computed in quadratic time if the trees have bounded degree [JWZ94], which is a reasonable assumption for merge trees (however, not true in general). Constrained and 1-degree edit distances are computable in polynomial time for both bounded degree trees and arbitrary degree trees, specifically quadratic time for bounded degree [Zha96,PVDT21], and in time…”

Section: Distance Measures For Treesmentioning

confidence: 99%

“…Pont et al [PVDT21] further restricted the constrained edit mappings specifically for the usecase of BDTs of merge trees. They introduced the constraint that if a node is deleted, i.e.…”

Section: Distance Measures For Treesmentioning

confidence: 99%

“…First, saddle swap instabilities are a key problem of all tractable edit distances on merge trees and are of course still present in the branch mapping distance. To handle such instabilities, [SMKN20] and [PVDT21] use a preprocessing step to collapse small edges (through an ε-parameter) such that branches in a parent-child relation that are probable to be influenced by such instabilities become siblings instead. Of course, this preprocessing is also applicable to our method and should yield the same improvements, however, we did not implement or test it.…”

Section: Algorithmmentioning

confidence: 99%

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Branch Decomposition‐Independent Edit Distances for Merge Trees

Wetzels

Leitte

Garth

2022

Computer Graphics Forum

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Edit distances between merge trees of scalar fields have many applications in scientific visualization, such as ensemble analysis, feature tracking or symmetry detection. In this paper, we propose branch mappings, a novel approach to the construction of edit mappings for merge trees. Classic edit mappings match nodes or edges of two trees onto each other, and therefore have to either rely on branch decompositions of both trees or have to use auxiliary node properties to determine a matching. In contrast, branch mappings employ branch properties instead of node similarity information, and are independent of predetermined branch decompositions. Especially for topological features, which are typically based on branch properties, this allows a more intuitive distance measure which is also less susceptible to instabilities from small‐scale perturbations. For trees with 𝒪(n) nodes, we describe an 𝒪(n4) algorithm for computing optimal branch mappings, which is faster than the only other branch decomposition‐independent method in the literature by more than a linear factor. Furthermore, we compare the results of our method on synthetic and real‐world examples to demonstrate its practicality and utility.

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Section: Distance Measures For Treesmentioning

confidence: 99%

“…Pont et al [PVDT21] further restricted the constrained edit mappings specifically for the usecase of BDTs of merge trees. They introduced the constraint that if a node is deleted, i.e.…”

Section: Distance Measures For Treesmentioning

confidence: 99%

Section: Algorithmmentioning

confidence: 99%

See 1 more Smart Citation

Branch Decomposition‐Independent Edit Distances for Merge Trees

Wetzels

Leitte

Garth

2022

Computer Graphics Forum

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“…Recent approaches aimed at summarizing an ensemble of topological descriptors by computing a notion of average descriptor, given a specific metric. This notion has been well studied for persistence diagrams [67,111,112], with direct applications to ensemble clustering [112], and analogies have been developed for merge trees [90,115].…”

Section: Related Workmentioning

confidence: 99%

“…The conciseness, stability [30] and expressiveness of this diagram made it a popular tool for data summarization tasks, as it provides visual hints about the number, ranges and salience of the features of interest. To compare two datasets f i and f j , persistence diagrams can be efficiently compared with the notion of L 2 -Wasserstein distance [23,61,111] (we leave the practical study of distances between more advanced topological descriptors [90,103] to future work). This distance is based on a bipartite assignment optimization problem (between the points of the two diagrams to compare), for which exact [80] and approximate [8] implementations are publicly available [11,106].…”

Section: Topological Data Analysismentioning

confidence: 99%

Topological Analysis of Ensembles of Hydrodynamic Turbulent Flows -- An Experimental Study

Nauleau¹,

Vivodtzev²,

Bridel-Bertomeu³

et al. 2022

Preprint

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1: Topological Data Analysis protocols applied on an ensemble dataset of a Kelvin-Helmholtz instability. (a) The 180 members of the ensemble obtained with variations of timesteps, interpolation schemes, orders, resolutions and Riemann solvers (Tab. 1). (b)The top cluster represents the time separation of t 0 and t 1 for the flows S 1 and S 2 with the Wasserstein distance and the bottom cluster with the L 2 -norm. Red lines show the timestep separation with our clustering method whereas the sphere colors are the ground truth, illustrating the limitation of the L 2 -norm. (c) Persistence curve protocol: Differences between integrals of persistence curves (gray area) of the enstrophy computed with a SLAU2 solver, an order 7 TENO scheme and a resolution of 1024 × 1024 for various configurations (S 1 at t 0 , S 2 and S 3 at t 1 ). These integral differences exhibit the appearance of vortices (critical points) as the time increases. (d) Outlier distance protocol: Wasserstein distance matrix for 5 configurations S 1 (t 0 , HLLC), S 2 (t 1 , Roe), S 3 (t 1 , HLLC), S 4 (t 2 , Roe), S 5 (t 2 , HLLC) computed with an order 7 WENO-Z interpolation scheme at 512 × 512. The sum of each row the configuration maximizing this distance between solvers and timesteps, here S 1 . (e) Unsupervised classification: Wasserstein distance matrix for the previous configurations with an order 7 WENO-Z interpolation scheme at 256 × 256. The clustering based on the Wasserstein distance and colored according to the Kmeans clustering method successfully segments the time steps.

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