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Approximation by multivariate higher degree F-transform based on B-splines

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Abstract

The paper deals with the integral and discrete versions of the direct and inverse higher degree fuzzy transforms (F-transforms) of multivariate functions. The aim is to generalize to the multidimensional case the results known for the univariate F-transforms with respect to a generalized fuzzy partition given by B-splines. We prove that using multivariate B-splines as the generating functions of multidimensional fuzzy partition allows to improve the quality of approximation of multivariate functions and their derivatives.

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Notes

  1. More precisely, of \( 2^d k_1 \ldots k_d \cdot \left( {\begin{array}{c}d+m\\ d\end{array}}\right) = O(1) \) such summands.

References

  • Alon N (1999) Combinatorial nullstellensatz. Comb Probab Comput 8(1–2):7–29

    Article  MathSciNet  MATH  Google Scholar 

  • Bede B, Rudas IJ (2011) Approximation properties of fuzzy transforms. Fuzzy Sets Syst 180(1):20–40

    Article  MathSciNet  MATH  Google Scholar 

  • Bhattacharya Halder S, Das P (2014) A method for approximating solution of ordinary differential equations by using fuzzy transforms. J Nonlinear Anal Optim 5(1):81–87

    MathSciNet  Google Scholar 

  • Dan̆ková M, Valášek R (2006) Full fuzzy transform and the problem of image fusion. J Electr Eng 57(7/S):82–84

    MATH  Google Scholar 

  • Di Martino F, Loia V, Sessa S (2010) A segmentation method for images compressed by fuzzy transforms. Fuzzy Sets Syst 161(1):56–74

    Article  MathSciNet  MATH  Google Scholar 

  • Gaeta M, Loia V, Tomasiello S (2016) Cubic B-spline fuzzy transforms for an efficient and secure compression in wireless sensor networks. Inf Sci 339:19–30

    Article  MathSciNet  Google Scholar 

  • Gautschi W (2004) Orthogonal polynomials. Computation and approximation. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Hodáková P (2014) Fuzzy (F-) transform of functions of two variables and its applications in image processing. Ph.D. dissertation, University of Ostrava

  • Holčapek M, Tichý T (2014) Discrete fuzzy transform of higher degree. In: IEEE international conference on fuzzy systems, FUZZ-IEEE 2014, Beijing, China, 6–11 July 2014, pp 604–611. IEEE

  • Holčapek M, Tichý T (2015) Discrete multivariate F-transform of higher degree. In: IEEE international conference on fuzzy systems, FUZZ-IEEE 2015, Istanbul, Turkey, 2–5 Aug 2015. IEEE

  • Khastan A, Perfilieva I, Alijani Z (2016) A new fuzzy approximation method to Cauchy problems by fuzzy transform. Fuzzy Sets Syst 288:75–95

    Article  MathSciNet  Google Scholar 

  • Kodorane I, Asmuss S (2013) 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, Milan, Italy, Advances in intelligent systems research, vol 32. Atlantis Press, pp 772–779

  • Kokainis M, Asmuss S (2015) Approximation properties of higher degree F-transforms based on B-splines. In: IEEE international conference on fuzzy systems, FUZZ-IEEE 2015, Istanbul, Turkey, 2–5 Aug 2015. IEEE

  • Kokainis M, Asmuss S (2016) Discrete higher degree F-transforms based on bivariate B-splains. In: Proceedings of the 12th international FLINS conference on uncertainty modelling in knowledge engineering and decision making, Roubaix, France, 24–26 Aug 2016, pp 264–269. World Scientific

  • Kokainis M, Asmuss S (2016) Higher degree F-transforms based on B-splines of two variables. In: Carvalho JP, Lesot M, Kaymak U, Vieira SM, Bouchon-Meunier B, Yager RR (eds) Proceedings of the 16th international conference IPMU, Eindhoven, Netherlands, Communications in computer and information science, vol. 610, pp 648–659. Springer Verlag

  • Novák V, Pavliska V, Perfilieva I, Štěpnička M (2013) F-transform and fuzzy natural logic in time series analysis. In: Montero J, Pasi G, Ciucci D (eds) Proceedings of EUSFLAT 2013, Milan, Italy, Advances in intelligent systems research, vol 32. Atlantis Press, pp 40–47

  • Novák V, Perfilieva I, Holčapek M, Kreinovich V (2014) Filtering out high frequencies in time series using F-transform. Inf Sci 274:192–209

    Article  MathSciNet  MATH  Google Scholar 

  • Novák V, Štěpnička M, Dvoák A, Perfilieva I, Pavliska V, Vavíčková L (2010) Analysis of seasonal time series using fuzzy approach. Int J Gener Syst 39(3):305–328

    Article  MathSciNet  MATH  Google Scholar 

  • Perfilieva I (2006) Fuzzy transforms and their applications to image compression. In: Bloch I, Petrosino A, Tettamanzi AGB (eds) Fuzzy logic and applications. 6th international workshop, WILF 2005, Crema, Italy, 15–17 Sept 2005, LNCS (LNAI), vol 3849. Springer, Berlin Heidelberg, pp 19–31

  • Perfilieva I (2006) Fuzzy transforms: Theory and applications. Fuzzy Sets Syst 157(8):993–1023

    Article  MathSciNet  MATH  Google Scholar 

  • Perfilieva I (2015) F-transform. In: Kacprzyk J, Pedrycz W (eds) Springer handbook of computational intelligence. Springer, Berlin, Heidelberg, pp 113–130

    Chapter  Google Scholar 

  • Perfilieva I, Dan̆ková M, Bede B (2011) Towards a higher degree F-transform. Fuzzy Sets Syst 180(1):3–19

    Article  MathSciNet  MATH  Google Scholar 

  • Perfilieva I, Haldeeva E (2001) Fuzzy transformation and its applications. In: Proceedings of the 4th Czech-Japan seminar on data analysis and decision making under uncertainty, Czech Republic, 2001, pp 116–124

  • Perfilieva I, Hodáková P (2013) \(\text{F}^1\)-transform of functions of two variables. In: Montero J, Pasi G, Ciucci D (eds.) Proceedings of EUSFLAT 2013, Milan, Italy, Advances in intelligent systems research, vol. 32, pp 547–553. Atlantis Press

  • Schoenberg IJ (1964) On interpolation by spline functions and its minimal properties. In: Butzer PL, Korevaar J (eds) On Approximation Theory / Über Approximationstheorie: Proceedings of the Conference held in the mathematical research institute at Oberwolfach, Black Forest, 4–10 Aug 1963, ISNM, vol 5. Springer, Basel, pp 109–129

  • Schoenberg IJ (1973) Cardinal spline interpolation, vol 12. CBMS-NSF regional conference series in applied mathematics. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Štěpnička M, Valášek R (2003) Fuzzy transforms for functions with two variables. In: Proceedings of the 6th Czech-Japan seminar on data analysis and decision making under uncertainity, pp 100–107

  • Štěpnička M, Valášek R (2005) Numerical solution of partial differential equations with help of fuzzy transform. In: Proceedings of the 14th IEEE international conference on fuzzy systems, pp 1104–1109

  • Tomasiello S (2016) An alternative use of fuzzy transform with application to a class of delay differential equations. Int J Comput Math 1–8. doi:10.1080/00207160.2016.1227436

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Authors and Affiliations

  1. Department of Mathematics, University of Latvia, Zellu 25, Riga, 1002, Latvia

    Martins Kokainis & Svetlana Asmuss

  2. Institute of Mathematics and Computer Science, University of Latvia, Raina bulvaris 29, Riga, 1459, Latvia

    Svetlana Asmuss

Authors
  1. Martins Kokainis
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  2. Svetlana Asmuss
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Correspondence to Martins Kokainis.

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Author Martins Kokainis declares that he has no conflict of interest. Author Svetlana Asmuss declares that she has no conflict of interest.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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Communicated by F. Di Martino, V. Novák.

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Kokainis, M., Asmuss, S. Approximation by multivariate higher degree F-transform based on B-splines. Soft Comput 21, 3587–3614 (2017). https://doi.org/10.1007/s00500-017-2654-z

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  • DOI: https://doi.org/10.1007/s00500-017-2654-z

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