Uncertain Neighborhood Relations of Point Sets and Fuzzy Delaunay Triangulation

Abstract

Voronoi diagrams are a classical tool for analyzing spatial neighborhood relations. For point fields the spatial proximity can be easily visualized by the dual graph, the Delaunay triangulation. In image analysis VDs and DTs are commonly used to derive neighborhoods for grouping or for relational matching. Neighborhood relations derived from the VD, however, are uncertain in case the common side of two Voronoi cells is comparably short or, equivalently, in case four points of two neighboring triangles in a DT are close to a circle. We propose a measure for characterizing the uncertainty of neighborhoods in a plane point field. As a side result we show the measure to be invariant to the numbering of the four points, though being dependent on the cross ratio of four points. Defining a fuzzy Delaunay triangulation is taken as an example.