A Lattice-Theoretical Framework for Annular Filters in Morphological Image Processing

We study the idempotence of operators of the form ɛ∨id∧δ (where ɛ≤δ and both ɛ and δ are increasing) on a modular lattice ℒ, in relation to the idempotence of the operators ɛ∨id and id∧δ. We consider also the conditions under which ɛ∨id∧δ is the composition of ɛ∨id and id∧δ. The case where δ is a dilation and ɛ an erosion is of special interest. When ℒ is a complete lattice on which Minkowski operations can be defined, we obtain very precise conditions for the idempotence of these operators. Here id∧δ is called an annular opening, ɛ∨id is called an annular closing, and ɛ∨id∧δ is called an annular filter. Our theory can be applied to the design of idempotent morphological filters removing isolated spots in digital pictures.