Geodesy and connectivity in lattices

Article type: Research Article

Authors: Ronse, Christian | Serra, Jean

Abstract: This paper generalizes the notion of symmetrical neighbourhoods, which have been used to define connectivity in the case of sets, to the wider framework of complete lattices having a sup-generating family. Two versions (weak and strong) of the notion of a symmetrical dilation are introduced, and they are applied to the generation of "connected components" from the so-called "geodesic dilations". It turns out that any "climbing" "weakly symmetrical" extensive dilation induces a "geodesic" connectivity. When the lattice is the one of subsets of a metric space, the connectivities which are obtained in this way may coincide with the usual ones under some conditions, which are clarified. The abstract theory can be applied to grey-level and colour images, without any assumption of translation-invariance of operators.

Keywords: neighbourhood, symmetry, connectivity, geodesy, complete, lattices, mathematical morphology, metrics

Journal: Fundamenta Informaticae, vol. 46, no. 4, pp. 349-395, 2001