Interval-Valued and Intuitionistic Fuzzy Mathematical Morphologies as Special Cases of $\mathbb{L}$ -Fuzzy Mathematical Morphology

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 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 L-fuzzy sets forms a complete lattice whenever the underlying set L constitutes a complete lattice. Based on these observations, we develop a general approach towards 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 L-fuzzy MM. As an application of these ideas, we generate a combination of some well-known medical image reconstruction Grants or other notes.Peter Sussner and Estevão Esmi University of Campinas, Department of Applied Mathematics, Campinas, SP, 13083-859, Brazil Tel.: +55-19-3521 5959 Fax: +55-19-3289-5766 E-mail: sussner,ra050652@ime.unicamp.br Mike Nachtegael, Tom Mélange, Glad Deschrijver, and Etienne Kerre Ghent University, Department of Applied Mathematics and Computer Science, Krijgslaan 281 -S9, 9000 Ghent, Belgium Tel.: +32-9-264 4765 Fax: +32-9-264 4995 E-mail: Mike.Nachtegael,Tom.Melange,Glad.Deschrijver,Etienne.Kerre @UGent.be techniques in terms of interval-valued fuzzy image processing.

Tunable equivalence fuzzy associative memories

“…Hence, -FAMs can be viewed as approaches towards lattice computing, "an evolving collection of tools and mathematical modeling methodologies with the capacity to process lattice ordered data per se including logic values, numbers, sets, symbols, graphs, etc." [69][70][71][72][73].…”

Mathematical morphology on bipolar fuzzy sets: general algebraic framework

“…Similar definitions can also be found e.g. in [34,42,40] in the case of interval-valued fuzzy sets of intuitionistic fuzzy sets, for a specific partial ordering (Pareto-like ordering), or more recently in [97] in the more general setting of L-fuzzy sets [55].…”

“…Let us finally comment about the choice of L. A more general setting could rely on L-fuzzy sets [55], by only assuming that L is a poset or a complete lattice. This line was followed in the recent work [97] for instance. Here we have chosen to keep L as presented above for simplifying the presentation and highlighting the bipolar nature of the information with its two components, but all what follows in this section and in the next one actually applies to more general forms of it.…”

Analyzing the R-Implications of Weber and Fodor in the Counting of Mycorrhizal Fungi Spores

“…Definition 6: [26] Let I be a residual implication (obtained from a left-continuous T-norm T) and the family , ; 0,1 of associated disjunctions.…”