Summary
In binary mathematical morphology, the idempotence of the binary closing and binary opening has led to the study of objects that are closed or open w.r.t. a given structuring element. For these objects, an interesting generalized idempotence law holds. In recent years, fuzzy mathematical morphology has been proposed as an alternative theory to gray-scale morphology. In this paper, we consider the most general approach, the so-called logical approach, based on fuzzy logical operators. In particular, the role of t-norms and their residual operators is stressed. It is shown that the generalized idempotence laws still hold in fuzzy mathematical morphology when choosing a continuous t-norm and its residual implicator as underlying conjunctor and implicator, and in particular when choosing a nilpotent t-norm and its residual implicator. In the latter case, the fuzzy closing and fuzzy opening are dual operations, and hence results concerning the fuzzy closing are obtained by duality from the fuzzy opening. The unique role of nilpotent t-norms, and hence of the Lukasiewicz t-norm, is demonstrated.
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References
Bandemer H. and Näther W., “Fuzzy Data Analysis”, Theory and Decision Library, Series B: Mathematical and statistical methods, Kluwer Academic Publishers, Dordrecht, 1992
Bloch I., Normes et conormes triangulaires: un outil général pour la génération de morphologies mathématiques floues, Telecom Paris 93D001, 1993
Bloch I. and Maitre H., Constructing fuzzy mathematical morphology: alternative ways, in: “Proceedings of the Second IEEE Conference on Fuzzy Systems” (San Francisco, CA, USA), pp. 1303–1308, 1993
Bloch I. and Maître H., Fuzzy mathematical morphologies: a comparative study, Pattern Recognition, Vol. 28, pp. 1341–1387, 1995
De Baets B., “Oplossen van vaagrelationele vergelijkingen: een ordetheoretische benadering”, Ph.D. thesis, University of,Gent, 1995 (in Dutch)
De Baets B., An order-theoretic approach to solving sup-T equations, in: “Fuzzy Set Theory and Advanced Mathematical Applications (Ruan D., ed.)”, Kluwer Academic Publishers, Norwell, pp. 67–87, 1995
De Baets B., Model implicators and their characterization, in: “Proceedings of the First ICSC International Symposium on Fuzzy Logic (Steele N., ed.)” (Zürich, Switzerland), ICSC Academic Press, pp. A42 — A49, 1995
De Baets B., Idempotent closing and opening operations in fuzzy mathematical morphology, in: “Proceedings of the Joint Third International Symposium on Uncertainty Modeling and Analysis and Annual Conference of the North American Fuzzy Information Processing Society (Ayyub B., ed.)” (College Park, MD, USA), Los Alamitos, IEEE Computer Society Press, pp. 228–233, 1995
De Baets B., Fuzzy morphology: a logical approach, in: “Uncertainty Analysis in Engineering and the Sciences: Fuzzy Logic, Statistics and Neural Network Approach (Ayyub B. and Gupta M., eds.)”, Kluwer Academic Publishers, Norwell, pp. 53–67, 1997
De Baets B., Sup-T equations: state of the art, in: “Computational Intelligence: Soft Computing and Fuzzy-Neural Integration with Applications (Kaynak O., Zadeh L., Turksen B. and Rudas I., eds.)”, NATO ASI Series F: Computer and Systems Sciences, Vol. 162, Springer Verlag, pp. 80–93, 1998
De Baets B. and Kerre E., A primer on solving fuzzy relational equations on the unit interval, Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 2, pp. 205–225, 1994
De Baets B., Kerre E. and Gupta M., The fundamentals of fuzzy mathematical morphology. Part 1: Basic concepts, Int. J. General Systems, Vol. 23, pp. 155–171, 1994
De Baets B., Kerre E. and Gupta M., The fundamentals of fuzzy mathematical morphology. Part 2: Idempotence, convexity and decomposition, Int. J. General Systems, Vol. 23, pp. 307–322, 1995
De Baets B., Kwasnikowska N. and Kerre E., Fuzzy morphology based on conjunctive uninorms, in: “Proceedings of the Seventh International Fuzzy Systems Association Congress” (Prague, Czech Republic), Vol. I, pp. 215–220, 1997
De Baets B. and Mesiar R., Residual implicators of continuous t-norms, in: “Proceedings of the Fourth European Congress on Intelligent Techniques and Soft Computing (Zimmermann H.-J., ed.)” (Aachen, Germany), ELITE, Vol. 1, pp. 27–31, 1996
De Baets B. and Mesiar R., Metrics and T-equalities, Journal of Mathematical Analysis and Applications, submitted
De Cooman G. and Kerre E., Order norms on bounded partially ordered sets, The Journal of Fuzzy Mathematics, Vol. 2, pp. 281–310, 1994
Dougherty E. and Giardina C., “Image Processing - Continuous to Discrete, Volume 1”, Prentice-Hall, 1987
Fodor J., Contrapositive symmetry of fuzzy implications Fuzzy Sets and Systems, Vol. 69, pp. 141–156, 1995
Haralick R., Sternberg S. and Zhuang X., Image analysis using mathematical morphology, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 9, pp. 532–550, 1987
Schweizer B. and Sklar A., A.sociative functions and abstract semi-groups, Publ. Math. Debrecen, Vol. 10, pp. 69–84, 1963
Serra J., “Image Analysis and Mathematical Morphology”, Academic Press, London, 1982
Sinha D. and Dougherty E., Fuzzification of set inclusion: theory and applications, Fuzzy Sets and Systems, Vol. 55, pp. 15–42, 1993
Trillas E., Sobre funciones de negación en la teorió de conjuntos diffusos, Stochastica, Vol. 3, pp. 47–59, 1979
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De Baets, B. (2000). Generalized Idempotence in Fuzzy Mathematical Morphology. In: Kerre, E.E., Nachtegael, M. (eds) Fuzzy Techniques in Image Processing. Studies in Fuzziness and Soft Computing, vol 52. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1847-5_2
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DOI: https://doi.org/10.1007/978-3-7908-1847-5_2
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