THE FUNDAMENTALS OF FUZZY MATHEMATICAL MORPHOLOGY .2. IDEMPOTENCE, CONVEXITY AND DECOMPOSITION

Fuzzy mathematical morphology is an alternative extension of binary ma thematical morphology to gray-scale images. This paper discusses some of the more advanced properties of the fuzzy morphological operations. The possible extensivity of the fuzzy closing, anti-extensivity of th e fuzzy opening and idempotence of the fuzzy closing and fuzzy opening are studied in detail. It is demonstrated that these properties only partially hold. On the other hand, it is shown that the fuzzy morpholo gical operations satisfy the same translation invariance and have the same convexity properties as the binary morphological operations. Fina lly, the paper investigates the possible decomposition, by taking (str ict) alpha-cuts, of the fuzzy morphological operations into binary mor phological operations.