Set Operator Decomposition and Conditionally Translation Invariant Elementary Operators

Abstract

In the first part, we recall the axiomatic definition of the elementary morphological operators (dilations, erosions, anti-dilations and anti-erosions) and their characterization in the case of Boolean lattices. This characterization is used to derive the set operator decompositions from the general decompositions of operators between complete lattices. In the second part, we define the notions of “conditionally translation invariant” (c.t.i.) and of “locally c.t.i.” elementary operators. These operators are those usually implemented on digital computers. We show how any c.t.i. elementary operator can be decomposed in terms of locally ones.