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

Dominique Attali (1) , André Lieutier (2) , David Salinas (1)

1 GIPSA-AGPIG - GIPSA - Architecture, Géométrie, Perception, Images, Gestes
2 Dassault Systèmes

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.

Mots clés: collapses; homotopy equivalence; metric spaces; Euclidean spaces; sampling; Cech complexes; clique complexes; Vietoris-Rips complexes; Shape reconstruction

Domaines: Géométrie algorithmique [cs.CG]

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