Decomposing and constructing fuzzy morphological operations over α-cuts: continuous and discrete case

Fuzzy mathematical morphology is an extension of binary morphology to gray-scale morphology, using techniques from fuzzy set theory. In this paper, we will study the decomposition and construction of fuzzy morphological operations based on α-cuts. First, we will investigate the relationship between α-cuts of the fuzzy morphological operations and the corresponding binary operations. Next, we will review several ways to obtain fuzzy morphological operations starting from binary operations and α-cuts. The investigation is carried out in both the continuous and the discrete case. It is interesting to observe that several properties that do not hold in the continuous case do hold in the discrete case. This is quite important since in practice we only work with discrete objects