Rips Complexes of Planar Point Sets
msra(2007)
摘要
Fix a finite set of points in Euclidean $n$-space $\euc^n$, thought of as a
point-cloud sampling of a certain domain $D\subset\euc^n$. The Rips 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 Rips complex to
$\euc^n$ 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 Rips complex accurately captures
connectivity and fundamental group data. This implies that the fundamental
group of a 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'-Rips complexes with nearly arbitrary fundamental
groups. This topological noise can be mitigated by examining a pair of
quasi-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.
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关键词
fundamental group,simplicial complex,higher order,point cloud,free group
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