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2010

Vietoris-Rips Complexes of Planar Point Sets

11 years 3 months ago
Vietoris-Rips Complexes of Planar Point Sets
Fix a finite set of points in Euclidean n-space En , thought of as a point-cloud sampling of a certain domain D En . The VietorisRips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. There is a natural "shadow" projection map from the Vietoris-Rips complex to En that has as its image 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 accurately 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'-VietorisRips complexes with nearly arbitrary fundamental groups. This topological no...
Erin W. Chambers, Vin de Silva, Jeff Erickson, Rob
Added 01 Mar 2011
Updated 01 Mar 2011
Type Journal
Year 2010
Where DCG
Authors Erin W. Chambers, Vin de Silva, Jeff Erickson, Robert Ghrist
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