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

10 pagesInternational audienceWe 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