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We study the idempotence of operators of the form &#108 &#136 id &#35 &#105 (where &#108 &#104 &#105 and both &#108 and &#105 are increasing) on a modular lattice &#147, in relation to the idempotence of the operators &#108 &#136 id and id &#35 &#105. We consider also the conditions under which &#108 &#136 id &#35 &#105 is the composition of &#108 &#136 id and id &#35 &#105. The case where &#105 is a dilation and &#108 an erosion is of special interest. When &#147 is a complete lattice on which Minkowski operations can be defined, we obtain very precise conditions for the idempotence of these operators. Here id &#35 &#105 is called an annular opening, &#108 &#136 id is called an annular closing, and &#108 &#136 id &#35 &#105 is called an annular filter. Our theory can be applied to the design of idempotent morphological filters removing isolated spots in digital pictures.

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