Decomposition of Interval-Valued Fuzzy Morphological Operations by Weak [alpha_1, alpha_2]-cuts

One of the extensions of binary morphology to greyscale images is given by the classical fuzzy mathematical morphology. Interval-valued fuzzy mathematical morphology further extends the latter theory by now also allowing uncertainty in the grey values of the image. In this paper, the decomposition of the intervalvalued fuzzy morphological operations into their binary counterparts is studied both in a general continuous framework and a discrete framework. It will be shown that some properties that do not hold in the continuous framework, do hold in the discrete framework, which is the framework that is used in practice. Keywords— Decomposition, interval-valued fuzzy sets, mathematical morphology.