Gromov-Hausdorff Approximation of Filament Structure Using Reeb-type Graph

Chazal, Frédéric; Sun, Jian

doi:10.1145/2582112.2582129

Cited by 20 publications

(20 citation statements)

References 36 publications

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“…An α-component at time t corresponds to a connected 3D-component in a horizontal slice of M with thickness α and centered at t (see Figure 11). This notion of α-component is similar to the α-Reeb graph as defined by Chazal and Sun [4].…”

Section: Robustnessmentioning

confidence: 90%

Trajectory Grouping Structure

Buchin

Kreveld³

et al. 2013

Lecture Notes in Computer Science

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Abstract.The collective motion of a set of moving entities like people, birds, or other animals, is characterized by groups arising, merging, splitting, and ending. Given the trajectories of these entities, we define and model a structure that captures all of such changes using the Reeb graph, a concept from topology. The trajectory grouping structure has three natural parameters that allow more global views of the data in group size, group duration, and entity inter-distance. We prove complexity bounds on the maximum number of maximal groups that can be present, and give algorithms to compute the grouping structure efficiently. We also study how the trajectory grouping structure can be made robust, that is, how brief interruptions of groups can be disregarded in the global structure, adding a notion of persistence to the structure. Furthermore, we showcase the results of experiments using data generated by the NetLogo flocking model and from the Starkey project. The Starkey data describe the movement of elk, deer, and cattle. Although there is no ground truth for the grouping structure in this data, the experiments show that the trajectory grouping structure is plausible and has the desired effects when changing the essential parameters. Our research provides the first complete study of trajectory group evolvement, including combinatorial, algorithmic, and experimental results.

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Section: Robustnessmentioning

confidence: 90%

Trajectory Grouping Structure

Buchin

Kreveld³

et al. 2013

Lecture Notes in Computer Science

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“…Part of the result (Theorem 3) shown in the paper also appears in [10]. The paper is organized as follows: The basic notions and definitions used throughout the paper are recalled in Sect.…”

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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“…Chazal et al [3] introduced a new α-Reeb graph for a graph reconstruction in different settings. The distance between points in a noisy sample C is measured geodesically within a given neighborhood graph on C, while we consider offsets of C with respect to the ambient distance in R 2 .…”

Section: Comparison With Related Past Skeletonization Workmentioning

confidence: 99%

“…• For a cloud C ⊂ R 2 of any n points, the skeleton HoPeS(C) with O(n) edges can be found in time O(n log n), which is comparable only with [3], [11], [18].…”

Section: Comparison With Related Past Skeletonization Workmentioning

confidence: 99%

A Homologically Persistent Skeleton is a Fast and Robust Descriptor of Interest Points in 2D Images

Kurlin

2015

Computer Analysis of Images and Patterns

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. 2D images often contain irregular salient features and interest points with non-integer coordinates. Our skeletonization problem for such a noisy sparse cloud is to summarize the topology of a given 2D cloud across all scales in the form of a graph, which can be used for combining local features into a more powerful object-wide descriptor.We extend a classical Minimum Spanning Tree of a cloud to a Homologically Persistent Skeleton, which is scale-and-rotation invariant and depends only on the cloud without extra parameters. This graph (1) is computable in time O(n log n) for any n points in the plane;(2) has the minimum total length among all graphs that span a 2D cloud at any scale and also have most persistent 1-dimensional cycles; (3) is geometrically stable for noisy samples around planar graphs.

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Gromov-Hausdorff Approximation of Filament Structure Using Reeb-type Graph

Cited by 20 publications

References 36 publications

Trajectory Grouping Structure

Trajectory Grouping Structure

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

A Homologically Persistent Skeleton is a Fast and Robust Descriptor of Interest Points in 2D Images

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