On lexicographical ordering in multivariate mathematical morphology

Since mathematical morphology is based on complete lattice theory, a vector order-ing method becomes indispensable for its extension to multivariate images. Among the several approaches developed with this purpose, lexicographical orderings are by far the most frequent, as they possess certain desirable theoretical properties. However, their main drawback consists of the excessive priority attributed to the first vector dimension. In this paper, the existing solutions to solving this prob-lem are recalled and two new approaches are presented. First, a generalisation of α-modulus lexicographical ordering is introduced, making it possible to accommo-date any quantisation function. Additionally, an input specific method is suggested, based on the use of a marker image. Comparative application results on colour noise reduction and texture classification are also provided. Key words: Multivariate mathematical morphology, lexicographical ordering, colour morphology, dimension prioritisation.