Digitization of Partitions and Tessellations

We study hierarchies of partitions in a topological space where the interiors of the classes and their frontiers are simultaneously represented. In both continuous and discrete spaces our approach rests on tessellations whose classes are R-open sets. In the discrete case, the passage from partitions to tessellations is expressed by the Alexandrov topology and yields double resolutions. A new topology is proposed to remove the ambiguities of the diagonal configurations. It leads to the triangular grid in Z2 and the centered cubic grid in Z3, which are the only translation invariant grids which preserve connectivity and permit the use of saliency functions.