We associate with each compact set X of R n 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 theČech 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 aČech complex of P or with a Rips complex of P . Regarding the reconstruction withČech 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 topologically correct reconstruction of shapes. This may be of some computational interest in high dimensions.
If given points are densely sampled from a compact set in R n , the VietorisRips complex at a certain scale correctly represents the topology of the set REF .
Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes
SoCG '11
Mathematics Computer Science