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

Kurlin, V ORCID: 0000-0001-5328-5351 (2015) A one-dimensional homologically persistent skeleton of an unstructured point cloud in any metric space. Computer Graphics Forum, 34 (5). pp. 253-262.

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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).

Item Type:	Article
Depositing User:	Symplectic Admin
Date Deposited:	01 Mar 2017 07:42
Last Modified:	19 Jan 2023 07:15
DOI:	10.1111/cgf.12713
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3006107