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

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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 \(\mathbb{L}\)-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 \(\mathbb{L}\)-fuzzy sets forms a complete lattice whenever the underlying set \(\mathbb{L}\) constitutes a complete lattice. Based on these observations, we develop a general approach towards \(\mathbb{L}\)-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 \(\mathbb{L}\)-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.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Campinas, Campinas, SP, 13083-859, Brazil

    Peter Sussner & Estevão Esmi

  2. Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, 9000, Ghent, Belgium

    Mike Nachtegael, Tom Mélange, Glad Deschrijver & Etienne Kerre

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  1. Peter Sussner
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  2. Mike Nachtegael
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  4. Glad Deschrijver
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  6. Etienne Kerre
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Correspondence to Peter Sussner.

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This work was partially supported by CNPq under grant No. 309608/2009-0 and by FAPESP under grant No. 2009/16284-2.

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Sussner, P., Nachtegael, M., Mélange, T. et al. Interval-Valued and Intuitionistic Fuzzy Mathematical Morphologies as Special Cases of \(\mathbb{L}\)-Fuzzy Mathematical Morphology. J Math Imaging Vis 43, 50–71 (2012). https://doi.org/10.1007/s10851-011-0283-1

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