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 R^n two real-valued functions c"X and h"X 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 c"P(t) entails that the Rips complex of P at scale r collapses to the 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 Cech complex of P or with a Rips complex of P. Regarding the reconstruction with 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 @m-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.
['Dominique Attali', 'André Lieutier', 'David Salinas']
Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes