Geodesic Delaunay triangulation and witness complex in the plane

We introduce a novel feature size for bounded planar domains endowed with an intrinsic metric. Given a point <i>x</i> in such a domain <i>X</i>, the <i>homotopy feature size</i> of <i>X</i> at <i>x</i>, or hfs(<i>x</i>) for short, measures half the length of the shortest loop through <i>x</i> that is not null-homotopic in <i>X</i>. The resort to an intrinsic metric makes hfs(<i>x</i>) rather insensitive to the local geometry of <i>X</i>, 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 <i>X.</i> Under reasonable sampling conditions involving hfs, we show that the geodesic Delaunay traingulation <i>D_X</i> (<i>L</i>) of a finite sampling <i>L</i> of <i>X</i> is homotopy equivalent to <i>X.</i> Moreover, <i>D_X</i> (<i>L</i>) is sandwiched between the geodesic witness complex <i>C^W_X</i> (<i>L</i>) and a relaxed version <i>C^W_{X, v}</i> (<i>L</i>), defined by a parameter <i>v.</i> Taking advantage of this fact, we prove that the homology of <i>D_X</i> (<i>L</i>) (and hence of <i>X</i>) can be retrieved by computing the persistent homology between <i>C^W_X</i> (<i>L</i>) and <i>C^W_{X, v}</i> (<i>L</i>). We propose algorithms for estimating hfs, selecting a landmark set of sufficient density, building its geodesic Delaunay triangulation, and computing the homology of <i>X</i> using <i>C^W_X</i> (<i>L</i>). We also present some simulation results in the context of sensor networks that corroborate our theoretical statements.