Vietoris-Rips Complexes of Planar Point Sets

Vietoris-Rips Complexes of Planar Point Sets,10.1007/s00454-009-9209-8,Discrete & Computational Geometry,Erin W. Chambers,Vin de Silva,Jeff Erickson,R

Vietoris-Rips Complexes of Planar Point Sets   (Citations: 3)
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Fix a finite set of points in Euclidean n-space En, thought of as a point-cloud sampling of a certain domain DEn. The Vietoris- Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional ap- proximation to the homotopy type of D. There is a natural "shadow" projection map from the Vietoris-Rips complex to En that has as its im- age a more accurate n-dimensional approximation to the homotopy type of D. We demonstrate that this projection map is 1-connected for the planar case n = 2. That is, for planar domains, the Vietoris-Rips complex accu- rately captures connectivity and fundamental group data. This implies that the fundamental group of a Vietoris-Rips complex for a planar point set is a free group. We show that, in contrast, introducing even a small amount of uncertainty in proximity detection leads to 'quasi'-Vietoris- Rips complexes with nearly arbitrary fundamental groups. This topo- logical noise can be mitigated by examining a pair of quasi-Vietoris-Rips complexes and using ideas from persistent topology. Finally, we show that the projection map does not preserve higher-order topological data for planar sets, nor does it preserve fundamental group data for point sets in dimension larger than three.
Journal: Discrete & Computational Geometry - DCG , vol. 44, no. 1, pp. 75-90, 2010
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