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Lattices of fuzzy sets and bipolar fuzzy sets, and mathematical morphology

Lattices of fuzzy sets and bipolar fuzzy sets, and mathematical morphology,10.1016/j.ins.2010.03.019,Information Sciences,Isabelle Bloch

Lattices of fuzzy sets and bipolar fuzzy sets, and mathematical morphology   (Citations: 4)
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Mathematical morphology is based on the algebraic framework of complete lattices and adjunctions, which endows it with strong properties and allows for multiple extensions. In particular, extensions to fuzzy sets of the main morphological operators, such as dilation and erosion, can be done while preserving all properties of these operators. Another extension concerns bipolar fuzzy sets, where both positive information and negative information are handled, along with their imprecision. We detail these extensions from the point of view of the underlying lattice structure. In the case of bipolarity, its two-components nature raises the question of defining a proper partial ordering. In this paper, we consider Pareto (component-wise) and lexicographic orderings.
Journal: Information Sciences - ISCI , vol. 181, no. 10, pp. 2002-2015, 2011
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    • ...The unifying potential of LT for disparate data unification/fusion has been, partly, recognized in MM. For instance, a number of authors have established connections between mathematical morphology and fuzzy set theory [7, 9, 11, 13, 39, 53] including recent extensions of fuzzy MM such as interval-valued fuzzy, intuitionistic (bipolar) fuzzy and L-fuzzy MM [8, 40, 56, 60, 61]...

    Vassilis G. Kaburlasoset al. Binary Image 2D Shape Learning and Recognition Based on Lattice-Comput...

    • ...In some recent papers, Bloch has established links between these two approaches for the special cases of fuzzy sets and bipolar fuzzy sets [12, 13]...
    • ...2 and 3 are also valid for other partial ordering schemes that lead to complete lattices such as the lexicographical partial order on bipolar (intuitionistic) fuzzy sets [13]...
    • ...Another interpretation of bipolar FMM (i.e., intuitionistic FMM) was provided by I. Bloch who has presented applications to spatial reasoning in image processing [11, 13]...

    Peter Sussneret al. Interval-Valued and Intuitionistic Fuzzy Mathematical Morphologies as ...

    • ...The lexicographic ordering was considered too in [5]...
    • ...Based on these concepts, we can now propose a general definition for morphological erosions and dilations, thus extending our previous work in [2,3,5]...
    • ...This group has then proposed some extensions in [7], still for the specific case of Pareto ordering, which closely follow our previous results in [2,3,5]...
    • ...Examples of connectives and derived morphological operators, along with their properties, can be found for the Pareto ordering and for the lexicographic ordering in our previous work [2,3,5]...
    • ...The case of Pareto ordering and lexicographic ordering have been detailed in [2,3,5], showing different properties, behaviors and interpretations...

    Isabelle Bloch. Fuzzy Bipolar Mathematical Morphology: A General Algebraic Setting

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