Grain building ordering

Abstract

Given a set E, the partitions of E are usually ordered by merging of classes. In segmentation procedures, this ordering often generates small parasite classes. A new ordering, called "grain building ordering", or GBO, is proposed. It requires a connection over E and states that A ≤ B, with A,B ⊆ E, when each connected component of B contains a connected component of A. TheGBO applies to sets, partitions, and numerical functions. Thickenings ψ with respect to the GBO are introduced as extensive idempotent operators that do not create connected components. The composition product ψγ of a connected opening by a thickening is still a thickening. Moreover, when {γi, i ∈ I} is a granulometric family, then the two sequences {ψγi, ∈ I} and {γi ψ,i ∈ I} generate hierarchies, from which semi-groups can be derived. In addition, the approach allows us to combine any set of partitions or of tessellations into a synthetic one. © 2011 Springer-Verlag.