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

Abstract

Fuzzy mathematical morphology is an alternative extension of binary mathematical morphology to grayscale 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 the fuzzy opening and idem-potence 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 morphological operations satisfy the same translation invariance and have the same convexity properties as the binary morphological operations. Finally, the paper investigates the possible decomposition, by taking (strict) α-cuts, of the fuzzy morphological operations into binary morphological operations.