A lattice-theoretical framework for annular filters in morphological image processing

We study the idempotence of operators of the form epsilon boolean OR id boolean AND delta (where epsilon less than or equal to delta and both epsilon and delta are increasing) on a modular lattice L, in relation to the idempotence of the operators epsilon boolean OR id and id boolean AND delta, We consider also the conditions under which epsilon boolean OR id boolean AND delta is the composition of epsilon boolean OR id and id boolean AND delta. The case where delta is a dilation and epsilon 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 boolean AND delta is called an annular opening, epsilon boolean AND id is called an annular closing, and epsilon boolean OR id boolean AND delta is called an annular filter. Our theory can be applied to the design of idempotent morphological filters removing isolated spots in digital pictures.