On the construction of interval-valued fuzzy morphological operators

Classical fuzzy mathematical morphology is one of the extensions of original binary morphology to greyscale morphology. Recently, this theory was further extended to interval-valued fuzzy mathematical morphology by allowing uncertainty in the grey values of the image and the structuring element. In this paper, we investigate the construction of increasing interval-valued fuzzy operators from their binary counterparts and work this out in more detail for the morphological operators, which results in a nice theoretical link between binary and interval-valued fuzzy mathematical morphology. The investigation is done both in the general continuous and the practical discrete case. It will be seen that the characterization of the supremum in the discrete case leads to stronger relationships than in the continuous case.