Higher Degree F-transforms Based on B-splines of Two Variables

  • Martins KokainisEmail author
  • Svetlana Asmuss
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 610)


The paper deals with the higher degree fuzzy transforms (F-transforms with polynomial components) for functions of two variables in the case when two-dimensional generalized fuzzy partition is given by B-splines of two variables. We investigate properties of the direct and inverse F-transform in this case and prove that using B-splines as basic functions of fuzzy partition allows us to improve the quality of approximation.


F-transform Two-dimensional fuzzy partition Approximation error B-splines 


  1. 1.
    Bede, B., Rudas, I.J.: Approximation properties of fuzzy transforms. Fuzzy Sets Syst. 180(1), 20–40 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dan̆ková, M., Hodáková, P., Perfilieva, I., Vajgl, M.: Edge detection using F-transform. In: Proceedings of ISDA 2011, pp. 672–677, Spain (2011)Google Scholar
  3. 3.
    Hodáková, P.: Fuzzy (F-)transform of functions of two variables and its applications in image processing. Ph.D. thesis, University of Ostrava, Ostrava, Czech Republic (2014)Google Scholar
  4. 4.
    Hodáková, P., Perfilieva, I.: F\(^{1}\)-transform of functions of two variables. In: Montero, J., Pasi, G., Ciucci, D. (eds.) Proceedings of EUSFLAT 2013. Advances in Intelligent Systems Research, Milan, Italy, vol. 32, pp. 547–553. Atlantis Press (2013)Google Scholar
  5. 5.
    Kodorane, I., Asmuss, S.: On approximation properties of spline based F-transform with respect to fuzzy m-partition. In: Montero, J., Pasi, G., Ciucci, D. (eds.) Proceedings of EUSFLAT 2013. Advances in Intelligent Systems Research, Milan, Italy, vol. 32, pp. 772–779. Atlantis Press (2013)Google Scholar
  6. 6.
    Kokainis, M., Asmuss, S.: Approximation properties of higher degree F-transforms based on B-splines. In: IEEE International Conference on Fuzzy Systems FUZZ-IEEE 2015, Istanbul, Turkey (2015)Google Scholar
  7. 7.
    Perfilieva, I.: Fuzzy approach to solution of differential equations with imprecise data: application to reef growth problem. In: Demicco, R., Klir, G. (eds.) Fuzzy Logic in Geology, pp. 275–300. Academic Press, Cambridge (2003)Google Scholar
  8. 8.
    Perfilieva, I.: Fuzzy transforms. In: Peters, J.F., Skowron, A., Dubois, D., Grzymała-Busse, J.W., Inuiguchi, M., Polkowski, L. (eds.) Transactions on Rough Sets II. LNCS, vol. 3135, pp. 63–81. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Perfilieva, I.: Fuzzy transforms: theory and applications. Fuzzy Sets Syst. 157(8), 993–1023 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Perfilieva, I.: F-transform. In: Kacprzyk, J., Pedrycz, W. (eds.) Handbook of Computational Intelligence, pp. 113–130. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  11. 11.
    Perfilieva, I., Chaldeeva, E.: Fuzzy transformation and its applications. In: Proceedings of the 4th Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainty, pp. 116–124. Czech Republic (2001)Google Scholar
  12. 12.
    Perfilieva, I., Danková, M., Bede, B.: Towards a higher degree F-transform. Fuzzy Sets Syst. 180(1), 3–19 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Perfilieva, I., Dan̆ková, M., Vajgl, M., Hodáková, P.: F-transform based image fusion. In: Ukimura, O. (ed.) Image Fusion, pp. 3–22. INTECH, Rijeka (2011)Google Scholar
  14. 14.
    Perfilieva, I., Hodáková, P., Hurtik, P.: Differentiation by the F-transform and application to edge detection. Fuzzy Sets Syst. 288, 96–114 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Schoenberg, I.J.: On interpolation by spline functions and its minimal properties. In: On Approximation Theory / Über Approximationstheorie. ISNM, vol. 5, pp. 109–129. Birkhäuser (1964)Google Scholar
  16. 16.
    Schoenberg, I.J.: Cardinal Spline Interpolation. CBMS, vol. 12. SIAM, Philadelphia (1973)CrossRefzbMATHGoogle Scholar
  17. 17.
    Štěpnička, M., Valášek, R.: Fuzzy transforms for function with two variables. In: Proceedings of the 6th Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainity, pp. 100–107. Czech Republic (2003)Google Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LatviaRigaLatvia
  2. 2.Institute of Mathematics and Computer ScienceUniversity of LatviaRigaLatvia

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