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Molecular graphs of a finite points set: a generalization of the Delaunay triangulation

The Delaunay triangulation is generated from a points set and a structuring element of type disc. In the Delaunay triangulation definition, replacing the disc by a planar-connected region (we call a molecule), which is a union of a fixed number of discs, allows construction of what we name the molecular graphs. In a finite points set, the molecular graphs record the empty regions which are identical to the molecule, independently of translation, rotation and scaling transforms. The molecular graphs are applied to pattern recognition problem. Knowing a template (an input pattern represented by a molecule), the addressed problem is to identify the existing patterns, whose shapes are similar to the template, in a given input points set. The proposed solution is based on a generalization of the αshapes ; the disc of radius α in the ordinary α-shape is replaced, in the generalized version, by a template of size depending on α.