N-ary Mathematical Morphology

Abstract Mathematical morphology on binary images can be fully described by set theory. However, it is not sufficient to formulate mathematical morphology for grey scale images. This type of images requires the introduction of the notion of partial order of grey levels, together with the definition of sup and inf operators. More generally, mathematical morphology is now described within the context of the lattice theory. For a few decades, attempts are made to use mathematical morphology on multivariate images, such as color images, mainly based on the notion of vector order. However, none of these attempts has given fully satisfying results. Instead of aiming directly at the multivariate case we propose first an extension of binary mathematical morphology to an intermediary situation: images composed of a finite number of independent unordered labels. We propose then an second extension to a continuous case.