MATHEMATICAL MORPHOLOGY ON L-IMAGES

Abstract

In most applications, morphological operations are considered as unary operations, each associated with a structuring element. Due to the problem of grey-level overflow, the associated structuring elements of morphological operations on grey-level images (functions from R(n) or Z(n) to [0, m], where m is a fixed positive number) are limited to 'flat' or 'non-flat' structuring elements. However, neither flat or non-flat structuring elements are grey-level images. In this paper, we propose the notion of l-images, based on a clc-monoid (complete lattice-ordered commutative monoid), as a unified representation of binary images (functions from R(n) or Z(n) to {0, 1}), signals (functions from R(n) or Z(n) to the set of extended real numbers R*) and grey-level images. Then we generalize morphological operations to l-images such that the associated structuring elements can be arbitrary l-images. As consequences, we obtain several concrete expressions for dilations and erosions on grey-level images such that the associated structuring elements are also grey-level images. Therefore, the grey-level overflow problem is solved with no limitations on structuring elements. Furthermore, we show that the duality property of dilations and erosions on l-images holds if the underlying clc-monoid is self-dual.