From Scalar-Valued Images to Hypercomplex Representations and Derived Total Orderings for Morphological Operators

In classical mathematical morphology for scalar images, the natural ordering of grey levels is used to define the erosion/dilation and the derived operators. Various operators can be sequentially applied to the resulting images always using the same ordering. In this paper we propose to consider the result of a prior transformation to define the imaginary part of a complex image, where the real part is the initial image. Then, total orderings between complex numbers allow defining subsequent morphological operations between complex pixels. In this case, the operators take into account simultaneously the information of the initial image and the processed image. In addition, the approach can be generalised to the hypercomplex representation (i.e., real quaternion) by associating to each image three different operations, for instance a directional filter. Total orderings initially introduced for colour quaternions are used to define the derived morphological transformations. Effects of these new operators are illustrated with different examples of filtering.