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 cases our approach rests on tessellations whose classes are $$\mathcal {R}$$-open sets. In the discrete case, the passage from partitions to tessellations is expressed by 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 $$\mathbb {Z}^{2}$$ and the centered cubic grid in $$\mathbb {Z}^{3}$$, which are the only translation invariant grids which preserve connectivity and permit the use of saliency functions.