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

• Published in: Computer graphics forum Vol. 34; no. 5; pp. 253 - 262
• Main Author: Kurlin, V.
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.08.2015
• Subjects: 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; 1467-8659
• DOI: 10.1111/cgf.12713

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).