Geometry-Aware Merge Tree Comparisons for Time-Varying Data With Interleaving Distances

Illustration of the improvements of unconstrained edit distances over constrained edit distances on the TOSCA ensemble: two embedded mappings of critical points are shown on the left, the same mappings in a typical merge tree layout on the right. In between, distance matrices for multiple members of the ensemble can be found. The merge trees on the right contain a saddle swap: the nesting of head-and hand subtrees is changed between the first and the second tree. Such saddle swaps cause semantically poor mappings (left and right, top row) as well as cluster being overshadowed by noise in the distance matrix(center) when using constrained edit distances. Such problems do not arise for the unconstrained distance (bottom row). The example instance shown in the two mappings can also be observed to contribute noise within the "david" cluster in the upper distance matrix, but not in the lower one.

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“…Lastly, there are alternative distance measures which focus on combining topological and geometrical similarity [16,26,32,37,39].…”

Section: Related Workmentioning

confidence: 99%

A Deformation-based Edit Distance for Merge Trees

Wetzels

Garth

2022

2022 Topological Data Analysis and Visualization (TopoInVis)

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“…Apart from merge trees, scalar field comparison is often done via distances on other topological descriptors such as contour trees [30,54], persistence diagrams [11][12][13][14], reeb graphs [2,3,19] or extremum graphs [38,55]. Some distances also combine geometric and topological measures [21,66,68]. A survey on the methods can be found in [67].…”

Section: Related Workmentioning

confidence: 99%

“…Examples are the works of Lukasczyk et al [32][33][34], Bremer et al [6,7,64] or others [39,[47][48][49]. A method by Yan et al [66] combines geometric and topological measures. Furthermore, topological methods can also be used for more advanced analysis of time series such as temporal merge tree maps [26] or geodesics [43,57].…”

Section: Related Workmentioning

confidence: 99%

Merge Tree Geodesics and Barycenters with Path Mappings

Wetzels,

Pont,

Tierny

et al. 2023

IEEE Trans. Visual. Comput. Graphics

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Fig. 1: Two merge tree geodesics with their corresponding mappings. The top row shows the mappings of critical points (restricted to maxima) embedded into the domain. In the bottom row the mapping between the two corresponding merge trees and their interpolated geodesic is shown. The most persistent mapped features are highlighted and colored to show the correspondence between embedding and planar layout. The Wasserstein geodesic can be seen on the left, the path mapping geodesic on the right. Clearly, the middle tree on the right is a more meaningful interpolation of the two input trees than the middle tree on the left; and yields a more intuitive correspondence between the nodes. Note that the used layout differs between the two figures. However, the two trees in front are indeed identical in both figures, as are the trees in the back.

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“…This includes several algorithms that calculate a tree edit distance [32,36]. While these methods are agnostic to the geometric embedding of the features, there are also attempts to combine topological distances with geometric properties [37]. In some examples, explicit temporal interpolation [35] or optical flow [33] was used to determine feature correspondence.…”

Section: Related Workmentioning

confidence: 99%

Towards Benchmark Data Generation for Feature Tracking in Scalar Fields

Nilsson

Lukasczyk

Masood

et al. 2022

2022 Topological Data Analysis and Visualization (TopoInVis)

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Feature tracking is a common task in visualization applications, where methods based on topological data analysis (TDA) have successfully been applied in the past for feature definition as well as tracking. In this work, we focus on tracking extrema of temporal scalar fields. A family of TDA approaches address this task by establishing one-to-one correspondences between extrema based on discrete gradient vector fields. More specifically, two extrema of subsequent time steps are matched if they fall into their respective ascending and descending manifolds. However, due to this oneto-one assignment, these approaches are prone to fail where, e.g., extrema are located in regions with low gradient magnitude, or are located close to boundaries of the manifolds. Therefore, we propose a probabilistic matching that captures a larger set of possible correspondences via neighborhood sampling, or by computing the overlap of the manifolds. We illustrate the usefulness of the approach with two application cases.

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Geometry-Aware Merge Tree Comparisons for Time-Varying Data With Interleaving Distances

Cited by 10 publications

References 52 publications

A Deformation-based Edit Distance for Merge Trees

A Deformation-based Edit Distance for Merge Trees

Merge Tree Geodesics and Barycenters with Path Mappings

Towards Benchmark Data Generation for Feature Tracking in Scalar Fields

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