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Summary: From Scalar-Valued Images to Hypercomplex
Representations and Derived Total Orderings for
Morphological Operators
Jesús Angulo
CMM-Centre de Morphologie Mathématique, Mathématiques et Systèmes, MINES
Paristech; 35, rue Saint Honoré, 77305 Fontainebleau Cedex, France
{jesus.angulo}@ensmp.fr
Abstract. 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 im-
age. Then, total orderings between complex numbers allow defining sub-
sequent 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 direc-
tional filter. Total orderings initially introduced for colour quaternions
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