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

• Published in: Applicable algebra in engineering, communication and computing communication and computing, 1998, Vol.9 (1), p.45-89
• Main Authors: Ronse, Christian, Heijmans, Henk J.A.M.
• Format: Article
• Language: English
• ISSN: 0938-1279; 1432-0622
• DOI: 10.1007/s002000050095

We study the idempotence of operators of the form epsilon join operation id meet operation 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 join operation id and id meet operation delta . We consider also the conditions under which epsilon join operation id meet operation delta is the composition of epsilon join operation id and id meet operation 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 idqq delta is called an annular opening, epsilon join operation id is called an annular closing, and epsilon meet operation idqq delta is called an annular filter. Our theory can be applied to the design of idempotent morphological filters removing isolated spots in digital pictures.