Geometric algebra colour image representations and derived total orderings for morphological operators – Part I: Colour quaternions

The definition of morphological operators for colour images requires a total ordering for colour points. A colour can be represented by different algebraic structures, in this paper we focus on real quaternions. The paper presents two main contributions. On the one hand, we have studied different alternatives to introduce the scalar part to obtain full colour quaternions. On the other hand, several total lexicographic orderings for quaternions have been defined, according to the various quaternion decompositions. The properties of these quaternionic orderings have been characterised to enable the identification of the most useful ones to define colour morphological operators. The theoretical results are illustrated with examples of processed images which show the usefulness of the proposed operators for real life complex problems.