Geodesic Delaunay triangulation and witness complex in the plane

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 homotopy 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 homotopy 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 leads to a reduced number of samples that still capture the topology of X. Under reasonable sampling conditions, we show that the geodesic Delaunay triangulation DX(L) of a finite sampling L of a bounded planar domain X is homotopy equivalent to X. Moreover, under similar conditions, DX(L) is sandwiched between the geodesic witness complex CW X (L) and a relaxed version CW X,(L). Taking advantage of this fact, we prove that the homology of DX(L) (and hence the one of X) can be retrieved by computing the persistent...

• Type: Conference
• Year: 2008
• Where: SODA
• Authors: Jie Gao, Leonidas J. Guibas, Steve Oudot, Yue Wang
• Keywords: Algorithms, Feature Size, Geodesic Delaunay Triangulation, Homotopy Feature Size, SODA 2008