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Vietoris-Rips Complexes also Provide Topologically Correct Reconstructions of Sampled Shapes

Abstract : We associate with each compact set $X$ of a Euclidean $n$-space two real-valued functions $c_X$ and $h_X$ defined which provide two measures of how much the set $X$ fails to be convex at a given scale. First, we show that, when $P$ is a finite point set, an upper bound on $c_P(t)$ entails that the Rips complex of $P$ at scale $r$ collapses to the \v Cech complex of $P$ at scale $r$ for some suitable values of the parameters $t$ and $r$. Second, we prove that, when $P$ samples a compact set $X$, an upper bound on $h_X$ over some interval guarantees a topologically correct reconstruction of the shape $X$ either with a \v Cech complex of $P$ or with a Rips complex of $P$. Regarding the reconstruction with \v Cech complexes, our work compares well with previous approaches when $X$ is a smooth set and surprisingly enough, even improves constants when $X$ has a positive $\mu$-reach. Most importantly, our work shows that Rips complexes can also be used to provide topologically correct reconstruction of shapes. This may be of some computational interest in high dimensions.
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https://hal.archives-ouvertes.fr/hal-00579864
Contributor : Dominique Attali <>
Submitted on : Friday, March 25, 2011 - 11:39:38 AM
Last modification on : Thursday, November 19, 2020 - 1:01:02 PM
Long-term archiving on: : Thursday, November 8, 2012 - 12:31:57 PM

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Dominique Attali, André Lieutier, David Salinas. Vietoris-Rips Complexes also Provide Topologically Correct Reconstructions of Sampled Shapes. 27th Annual Symposium on Computational Geometry (SoCG 2011), Jun 2011, Paris, France. pp.s/n. ⟨hal-00579864⟩

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