L-fuzzy Sets and Intuitionistic Fuzzy Sets

DOI: 10.1007/978-3-540-72687-6_16 In book: Computational Intelligence Based on Lattice Theory, Publisher: Springer, pp.325-339

ABSTRACT Summary. In this article we firstly summarize some notions on L-fuzzy sets, where L denotes a complete lattice. We then study a special case of L-fuzzy sets, namely the “intuitionistic fuzzy sets”. The importance of these sets comes from the fact that the negation is
being defined independently from the fuzzy membership function. The latter implies both flexibility and e.ectiveness in fuzzy
inference applications. We additionally show several practical applications on intuitionistic fuzzy sets, in the context of
computational intelligence.

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    ABSTRACT: Mathematical morphology (MM) offers a wide range of tools for image processing and computer vision. MM was originally conceived for the processing of binary images and later extended to gray-scale morphology. Extensions of classical binary morphology to gray-scale morphology include approaches based on fuzzy set theory that give rise to fuzzy mathematical morphology (FMM). From a mathematical point of view, FMM relies on the fact that the class of all fuzzy sets over a certain universe forms a complete lattice. Recall that complete lattices provide for the most general framework in which MM can be conducted. The concept of \mathbbL\mathbb{L}-fuzzy set generalizes not only the concept of fuzzy set but also the concepts of interval-valued fuzzy set and Atanassov’s intuitionistic fuzzy set. In addition, the class of \mathbbL\mathbb{L}-fuzzy sets forms a complete lattice whenever the underlying set \mathbbL\mathbb{L} constitutes a complete lattice. Based on these observations, we develop a general approach towards \mathbbL\mathbb{L}-fuzzy mathematical morphology in this paper. Our focus is in particular on the construction of connectives for interval-valued and intuitionistic fuzzy mathematical morphologies that arise as special, isomorphic cases of \mathbbL\mathbb{L}-fuzzy MM. As an application of these ideas, we generate a combination of some well-known medical image reconstruction techniques in terms of interval-valued fuzzy image processing. KeywordsMathematical morphology–Complete lattice– \mathbbL\mathbb{L}-fuzzy sets–Interval-valued fuzzy sets–Atanassov’s intuitionistic fuzzy sets– \mathbbL\mathbb{L}-fuzzy mathematical morphology– \mathbbL\mathbb{L}-fuzzy connectives–Inclusion measure–Duality–Negation–Adjunction
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