A Lattice-Theoretical Framework for Annular Filters in Morphological Image Processing. - Ronse, Christian (Applicable algebra in engineering, communication and computing.)
We study the idempotence of operators of the form l ? id # i (where l h i and both l and i are increasing) on a modular lattice ?, in relation to the idempotence of the operators l ? id and id # i. We consider also the conditions under which l ? id # i is the composition of l ? id and id # i. The case where i is a dilation and l an erosion is of special interest. When ? is a complete lattice on which Minkowski operations can be defined, we obtain very precise conditions for the idempotence of these operators. Here id # i is called an annular opening, l ? id is called an annular closing, and l ? id # i is called an annular filter. Our theory can be applied to the design of idempotent morphological filters removing isolated spots in digital pictures.