A One-dimensional Homologically Persistent Skeleton of an Unstructured Point Cloud in any Metric Space
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Date
2015
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
The Eurographics Association and John Wiley & Sons Ltd.
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).
Description
@article{10.1111:cgf.12713,
journal = {Computer Graphics Forum},
title = {{A One-dimensional Homologically Persistent Skeleton of an Unstructured Point Cloud in any Metric Space}},
author = {Kurlin, Vitaliy},
year = {2015},
publisher = {The Eurographics Association and John Wiley & Sons Ltd.},
DOI = {10.1111/cgf.12713}
}