A one-dimensional Homologically Persistent Skeleton of an unstructured point cloud in any metric space

Kurlin, Vitaliy

Abstract

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).

Citation

Kurlin, V. (2015). A one-dimensional Homologically Persistent Skeleton of an unstructured point cloud in any metric space. Computer Graphics Forum, 34(5), 253-262. https://doi.org/10.1111/cgf.12713

• Journal Article Type: Article
• Publication Date: Aug 10, 2015
• Deposit Date: Sep 18, 2015
• Publicly Available Date: Aug 10, 2016
• Journal: Computer Graphics Forum
• Print ISSN: 0167-7055
• Electronic ISSN: 1467-8659
• Publisher: Wiley
• Peer Reviewed: Peer Reviewed
• Volume: 34
• Issue: 5
• Pages: 253-262
• DOI: https://doi.org/10.1111/cgf.12713
• Keywords: Categories and Subject Descriptors (according to ACM CCS), I.5.1 [Pattern Recognition]: Models—Structural.
• Public URL: https://durham-repository.worktribe.com/output/1399753