Geodesic delaunay triangulations in bounded planar domains

Abstract

We introduce a new feature size for bounded domains in the plane endowed with an intrinsic metric. Given a point x in a domain X , the systolic feature size of X at x measures half the length of the shortest loop through x that is not null-homotopic in X . The resort to an intrinsic metric makes the systolic feature size rather insensitive to the local geometry of the domain, in contrast with its predecessors (local feature size, weak feature size, homology feature size). This reduces the number of samples required to capture the topology of X , provided that a reliable approximation to the intrinsic metric of X is available. Under sufficient sampling conditions involving the systolic feature size, we show that the geodesic Delaunay triangulation D x ( L ) of a finite sampling L is homotopy equivalent to X . Under similar conditions, D x ( L ) is sandwiched between the geodesic witness complex C W X ( L ) and a relaxed version C W X,ν ( L ). In the conference version of the article, we took advantage of this fact and proved that the homology of D x ( L ) (and hence the one of X ) can be retrieved by computing the persistent homology between C W X ( L ) and C W X,ν ( L ). Here, we investigate further and show that the homology of X can also be recovered from the persistent homology associated with inclusions of type C W X,ν ( L )↪ C W X,ν′ ( L ), under some conditions on the parameters ν≤ν′. Similar results are obtained for Vietoris-Rips complexes in the intrinsic metric. The proofs draw some connections with recent advances on the front of homology inference from point cloud data, but also with several well-known concepts of Riemannian (and even metric) geometry. On the algorithmic front, we propose algorithms for estimating the systolic feature size of a bounded planar domain X , selecting a landmark set of sufficient density, and computing the homology of X using geodesic witness complexes or Rips complexes.