A one‐dimensional homologically persistent skeleton of an unstructured point cloud in any metric space. (10th August 2015)

• Record Type: Journal Article
• Authors: Kurlin, V.
• Is Part Of: Computer graphics forum. Volume 34:Number 5(2015)
• Page Start: 253
• Page End: 262
• Publication Date: 2015-08-10
• Subjects: Computer graphics -- Periodicals; 006.605
• DOI: 10.1111/cgf.12713
• Languages: English
• ISSNs: 0167-7055

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>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).</p> </abstract>