Vietoris–Rips complexes also provide topologically correct reconstructions of sampled shapes

Abstract

Given a point set that samples a shape, we formulate conditions under which the Rips complex of the point set at some scale reflects the homotopy type of the shape. For this, we associate with each compact set X of Rn two real-valued functions cX and hX defined on R+ 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 cP(t) entails that the Rips complex of P at scale r collapses to the Čech 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 hX over some interval guarantees a topologically correct reconstruction of the shape X either with a Čech complex of P or with a Rips complex of P. Regarding the reconstruction with Čech 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 μ-reach. Most importantly, our work shows that Rips complexes can also be used to provide shape reconstructions having the correct homotopy type. This may be of some computational interest in high dimensions.