N-ary Mathematical Morphology

Abstract

Abstract. Mathematical morphology on binary images can be fully described by set theory. However, it is not su cient 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 de nition 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 an extension of mathematical morphology to an intermediary situation: images composed of a nite number of independent unordered categories.

Generalized Morphology using Sponges

“…This means we again have a sponge that is isomorphic to the inner product sponge. We expect this sponge might be useful in the context of barycentric coordinates encoding probabilities [22] (translated to a hypersphere through the mapping discussed by Gromov [30, p. 14], also see [20]), possibly through an extension of n-ary mathematical morphology [16].…”