Composing morphological filters

A morphological filter is an operator on a complete lattice which is increasing and idempotent. Two well-known classes of morphological filters are openings and closings. Furthermore, an interesting class of filters, the alternating sequential filters, is obtained if one composes openings and closings. This paper explains how to construct morphological filters, and derived notions such as overfilters, underfilters, inf-overfilters, and sup-underfilters by composition, the main ingredients being dilations, erosions, openings, and closings. The class of alternating sequential filters is extended by composing overfilters and underfilters. Finally, it is shown that any composition consisting of an equal number of dilations and erosions from an adjunction is a filter. The abstract approach is illustrated with some experimental results.