Abstract
In this work we are going to set up a new relationship between the L-fuzzy Concept Analysis and the Fuzzy Mathematical Morphology. Specifically we prove that the problem of finding fuzzy images or signals that remain invariant under a fuzzy morphological opening or under a fuzzy morphological closing, is equal to the problem of finding the L-fuzzy concepts of some L-fuzzy context. Moreover, since the Formal Concept Analysis and the Mathematical Morphology are the particular cases of the fuzzy ones, the showed result has also an interpretation for binary images or signals.
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References
Alcalde, C., Burusco, A., Fuentes-González, R., Zubia, I.: Treatment of L-fuzzy contexts with absent values. Inf. Sci. 179(1–2), 1–15 (2009)
Alcalde, C., Burusco, A., Fuentes-González, R.: Contextos con conjuntos de objetos y atributos iguales. In: Proceedings of ESTYLF08, pp. 85–89. Mieres—Langreo (2008)
Alcalde, C., Burusco, A., Fuentes-González, R.: Analysis of some L-fuzzy relational equations and the study of its solutions by means of the L-fuzzy Concept Theory. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 20(1), 21–40 (2012)
Baczynski, M., Jayaram, B.: Fuzzy Implications: studies in fuzziness and soft computing, vol. 231. Springer (2008)
De Baets, B.: Fuzzy morphology: a logical approach. In: Ayyub, B., Gupta, M. (eds.) Uncertainty Analysis, Engineering and Science: Fuzzy Logic, Statistics and neural network Approach, pp. 53–67, Kluwer Academic Publishers (1997)
Belohlavek, R.: Fuzzy closure operators. J. Math. Anal. Appl. 262, 473–491 (2001)
Belohlavek, R.: Fuzzy relational systems. Foundations and Principles. Kluwer Academic (2002)
Belohlavek, R.: Concept lattices and order in fuzzy logic. Ann. Pur. Appl. Logic 128(1–3), 277–298 (2004)
Belohlavek, R., Konecny, J.: Concept lattices of isotone vs. antitone Galois connections in graded setting: mutual reducibility revisited. Inf. Sci. 199, 133–137 (2012)
Bloch, I., Maître, H.: Fuzzy Mathematical Morphologies: a comparative study. Télécom Paris 94D001 (1994)
Bloch, I.: Duality vs. adjunction for fuzzy mathematical morphology and general form of fuzzy erosions and dilations. Fuzzy Sets Syst. 160, 1858–1867 (2009)
Burillo, P., Fuentes-González, R., Frago, N.: Inclusion grade and fuzzy implication operators. Fuzzy Sets Syst. 114, 417–429 (2000)
Burillo, P., Frago, N., Fuentes-González, R.: Generation of fuzzy mathematical morphologies. Mathware Soft. Comput. 8(1), 31–46 (2001)
Burusco, A., Fuentes-González, R.: Concept lattices defined from implication operators. Fuzzy Sets Syst. 114, 431–436 (2000)
Burusco, A., Fuentes-González, R.: The Study of the L-fuzzy Concept Lattice. Mathware Soft. Comput. I(3), 209–218 (1994)
Burusco, A., Fuentes-González, R.: Construction of the L-fuzzy Concept Lattice. Fuzzy Sets Syst. 97(1), 109–114 (1998)
Djouadi, Y., Prade, H.: Interval-valued fuzzy galois connections: algebraic requirements and concept lattice construction. Fundamenta Informaticae 99(2), 169–186 (2010)
Frago, N., Fuentes-González, R.: Descubrimiento de Conocimiento en Bases de Datos Utilizando Técnicas de Morfología Matemática Borrosa. Información tecnológica 18(6), 39–50 (2007)
Frago, N., Fuentes-González, R.: Procesos de descubrimiento de conocimiento en bases de datos usando grafos asociados a filtros morfológicos. In: Proceedings of ESTYLF08, pp. 85–89. Mieres - Langreo (2008)
Fuentes-González, R.: The incorporation of the mathematical morphology, the formal concept analysis and the fuzzy logic techniques and tools to the data cleaning, aggregating, reducing and mining stages of the knowledge discovery process. In: Troya, J.M., Ossowski, S. (eds.) Proceedings of CEDI, pp. 147–154. Granada (2005)
Ganter, B., Wille, R.: Formal concept analysis: mathematical foundations. Springer, Berlin (1999)
Goguen, J.A.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145–174 (1967)
Goutsias, J., Heijmans, H.J.A.M.: Fundamenta morphologicae mathematicae. Fundamenta Informaticae 41, 1–31 (2000)
Heijmans, H.J.A.M.: Morphological image operators. Academic Press Inc (1994)
Horak, Z., Kudelka, M.: Snasel, V.: FCA as a tool for inaccuracy detection in content-based image analysis. In: 2010 IEEE international conference on granular computing, pp. 223–228 (2010)
Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Theory and Applications. Prentice Hall PTR, New Jersey (1995)
Maragos, P.: Lattice image precessing: a unification of morphological and fuzzy algebric systems. J. Math. Imaging Vis. 22, 333–353 (2005)
Mas, M., Monserrat, M., Torrens, J.: S-implications and R-implications on a finite chain. Kybernetika 40(1), 3–20 (2004)
Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)
Matheron, G.: Random sets and integral geometry. Wiley, New York (1975)
Medina, J.: Multi-adjoint property-oriented concept lattices. Inf. Sci. 190, 95–106 (2012)
Medina, J., Ojeda-Aciego, M.: Multi-adjoint t-concept lattices. Inf. Sci. 180(5), 712–725 (2010)
Polland, S.: Fuzzy Begriffe. Springer (1997)
Serra, J.: Image analysis and mathematical morphology. Academic Press. I (fourth printing 1990) and II (second printing 1992)
Soille, P.: Morphological image analysis. Principles and Applications, 2nd edn. Springer (2004)
Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered sets, pp. 445–470. Reidel, Dordrecht (1982)
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Alcalde, C., Burusco, A. & Fuentes-González, R. Application of the L-fuzzy concept analysis in the morphological image and signal processing. Ann Math Artif Intell 72, 115–128 (2014). https://doi.org/10.1007/s10472-014-9397-7
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DOI: https://doi.org/10.1007/s10472-014-9397-7