Interval-Valued and Intuitionistic Fuzzy Mathematical Morphologies as Special Cases of L-Fuzzy Mathematical Morphology

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,[email protected] 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.