α-trimmed lexicographical extrema for pseudo-morphological image analysis

The extension of mathematical morphology to colour, and more generally to multivariate image data, continues to be an open problem. As its underlying theory is defined in terms of complete lattices, the main challenge lies in introducing a complete lattice structure on the image intensity range, hence vectorial extrema computation methods are necessary. In this paper, we circumvent the need for a multivariate ordering, and propose a method for directly computing the multivariate extrema of vector sets. To this end the α-trimming principle is employed in combination with lexicographical ordering. The resulting pseudo-morphological operators, although deprived of important properties, present the advantage of a “collective” calculation, taking into account the distribution of vectors within the structuring element. They are tested against state of the art methodologies in applications treating noise reduction and texture classification, where they are shown to exhibit superior performances.