Uncertain Neighborhood Relations of Point Sets and Fuzzy Delaunay Triangulation

Voronoi diagrams are a classical tool for analyzing spatial neighborhood relations. For point elds 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 eld. 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. Deening a fuzzy Delaunay triangulation is taken as an example. 1 Motivation Voronoi Diagrams (VDs) are a classical tool for analyzing spatial neighborhood relations. For two dimensional point sets the spatial proximity easily can be visualized by the dual graph, the Delaunay Triangulation (DT), being extensible to higher dimensions Preparata and Shamos 1985] or to more general patterns Mehlhorn et al. 1991]. In image analysis VDs and DTs are commonly used to derive neighborhoods for grouping (e. g. Ahuja and Tuceryan 1989], Heuel and FF orstner 1998]) or for relational matching (e. g. Ogniewicz 1993]). No thresholds are required for establishing neighborhoods using VD which allows to postpone decisions on the adequateness of derived neighborhoods to a later stage. One of the primary criteria for grouping image features or other data is proximity, which can be established by a DT. Many procedures involving relational matching use neighborhood relations as a rst choice. Now, neighborhood relations derived from the VD are uncertain in case the common side of two Voronoi cells is comparably short or, equivalently, in case in FF orstner et al.