Pareto-dominated Hypervolume Measure: An Alternative Approach to Color Morphology

In this paper, an alternative approach to the non-linear filtering of mathematical morphology for color and multi-spectral images based on the Pareto-dominated hypervolume measure is presented. Parteo-set theory is studied in this context, and among others, successfully implemented in the morphological filtering of color images. The demand to assess the quality of a multi-objective optimization algorithms, in particular, has put forth several kind of measures. One of these measures, the hypervolume, bases on the Lebesque measure of all points dominated by a set of points and maps a set of Pareto-optimal points to a non-negative scalar. By considering the hypervolume in the color-image domain, it can be shown that this hypervolume corresponds to another kind of color morphology, were each pixel in the filtered image represents the hypervolume of its set of neighbours in the original color image. In the following, some properties of the hypervolume, as used as an image-processing filter will be derived, and some potential applications of this approach to color morphology will be suggested.