Generalized Idempotence in Fuzzy Mathematical Morphology
In binary mathematical morphology, the idempotence of the binary closing and binary opening has led to the study of objects that are closed or open w.r.t. a given structuring element. For these objects, an interesting generalized idempotence law holds. In recent years, fuzzy mathematical morphology has been proposed as an alternative theory to gray-scale morphology. In this paper, we consider the most general approach, the so-called logical approach, based on fuzzy logical operators. In particular, the role of t-norms and their residual operators is stressed. It is shown that the generalized idempotence laws still hold in fuzzy mathematical morphology when choosing a continuous t-norm and its residual implicator as underlying conjunctor and implicator, and in particular when choosing a nilpotent t-norm and its residual implicator. In the latter case, the fuzzy closing and fuzzy opening are dual operations, and hence results concerning the fuzzy closing are obtained by duality from the fuzzy opening. The unique role of nilpotent t-norms, and hence of the Lukasiewicz t-norm, is demonstrated.
KeywordsMathematical Morphology Fuzzy Opening Fuzzy Object Dual Operation Residual Operator
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