Research Seminar - April 23, 1999

Abstract

Abstract axioms for connectivity were introduced by Serra for sets in 1988, then for objects in an abstract complete lattice in 1998; the latter approach was applied in particular to the case of numerical functions.

Here we generalize 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.

The abstract theory can be applied to grey-level and colour images, without any assumption of translation-invariance of operators, and also to analyse connectivity in a metric space.