A one-dimensional homologically persistent skeleton of an unstructured point cloud in any metric space

Real data are often given as a noisy unstructured point cloud, which is hard to visualize. The important problem is to represent topological structures hidden in a cloud by using skeletons with cycles. All past skeletonization methods require extra parameters such as a scale or a noise bound. We def...

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Veröffentlicht in:Computer graphics forum Jg. 34; H. 5; S. 253 - 262
1. Verfasser: Kurlin, V.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Oxford Blackwell Publishing Ltd 01.08.2015
Schlagworte:
Analysis
Approximation
Categories and Subject Descriptors (according to ACM CCS)
Clouds
Graph theory
Graphs
Homology
I.5.1 [Pattern Recognition]: Models-Structural
Metric space
Optimization
Studies
Three dimensional models
Topological manifolds
Topology
Visualization
ISSN:0167-7055, 1467-8659
Online-Zugang:Volltext
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Beschreibung
Zusammenfassung:Real data are often given as a noisy unstructured point cloud, which is hard to visualize. The important problem is to represent topological structures hidden in a cloud by using skeletons with cycles. All past skeletonization methods require extra parameters such as a scale or a noise bound. We define a homologically persistent skeleton, which depends only on a cloud of points and contains optimal subgraphs representing 1‐dimensional cycles in the cloud across all scales. The full skeleton is a universal structure encoding topological persistence of cycles directly on the cloud. Hence a 1‐dimensional shape of a cloud can be now easily predicted by visualizing our skeleton instead of guessing a scale for the original unstructured cloud. We derive more subgraphs to reconstruct provably close approximations to an unknown graph given only by a noisy sample in any metric space. For a cloud of n points in the plane, the full skeleton and all its important subgraphs can be computed in time O(n log n).
Bibliographie:ark:/67375/WNG-699WQVRM-Q
istex:98091B9BEAE2D8D29C81423B34827489A675A340
ArticleID:CGF12713
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ISSN:0167-7055
1467-8659
DOI:10.1111/cgf.12713