Advertisement

Central European Journal of Mathematics

, Volume 6, Issue 1, pp 159–169 | Cite as

On the exact values of coefficients of coiflets

  • Dana ČernáEmail author
  • Václav Finěk
  • Karel Najzar
Research Article
  • 155 Downloads

Abstract

In 1989, R. Coifman suggested the design of orthonormal wavelet systems with vanishing moments for both scaling and wavelet functions. They were first constructed by I. Daubechies [15, 16], and she named them coiflets. In this paper, we propose a system of necessary conditions which is redundant free and simpler than the known system due to the elimination of some quadratic conditions, thus the construction of coiflets is simplified and enables us to find the exact values of the scaling coefficients of coiflets up to length 8 and two further with length 12. Furthermore for scaling coefficients of coiflets up to length 14 we obtain two quadratic equations, which can be transformed into a polynomial of degree 4 for which there is an algebraic formula to solve them.

Keywords

orthonormal wavelet coiflet exact value of filter coefficients 

MSC

65T60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Adams W.W., Loustaunau P., An Introduction to Gröbner Bases, American Mathematical Society, 1994Google Scholar
  2. [2]
    Antonini M., Barlaud M., Mathieu P., Daubechies I., Image coding using wavelet transforms, IEEE Trans. Image Process., 1992, 1, 205–220CrossRefGoogle Scholar
  3. [3]
    Beylkin G., On the representation of operators in bases of compactly supported wavelets, SIAM J. Numer. Anal., 1992, 29, 1716–1740zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Beylkin G., Coifman R.R., Rokhlin V., Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math., 1991, 44, 141–183zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Bittner K., Urban K., Adaptive wavelet methods using semiorthogonal spline wavelets: Sparse evaluation of nonlinear functions, preprintGoogle Scholar
  6. [6]
    Buchberger B., An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal, PhD thesis, University of Inssbruck, Austria, 1965 (in German)Google Scholar
  7. [7]
    Burrus C.S., Odegard J.E., Coiflet Systems and Zero Moments, IEEE Trans. Signal Process., 1998, 46, 761–766CrossRefMathSciNetGoogle Scholar
  8. [8]
    Burrus C.S., Gopinath R.A., On the moments of the scaling function ψ 0, Proceedings of the ISCAS-92, 1992, 963–966Google Scholar
  9. [9]
    Černá D., Finěk V., On the computation of scaling coefficients of Daubechies wavelets, Cent. Eur. J. Math., 2004, 2, 399–419CrossRefMathSciNetGoogle Scholar
  10. [10]
    Chyzak F., Paule P., Scherzer O., Schoisswohl A., Zimmermann B., The construction of orthonormal wavelets using symbolic methods and a matrix analytical approach for wavelets on the interval, Experiment. Math., 2001, 10, 67–86zbMATHMathSciNetGoogle Scholar
  11. [11]
    Cohen A., Ondelettes analyses multirésolutions et filtres miroir en quadrature, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1990, 7, 439–459zbMATHGoogle Scholar
  12. [12]
    Cohen A., Daubechies I., Feauveau J.C., Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 1992, 45, 485–500zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Dahmen W., Kunoth A., Urban K., Biorthogonal spline wavelets on the interval — stability and moment conditions, Appl. Comput. Harmon. Anal., 1999, 6, 132–196zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Daubechies I., Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 1988, 41, 909–996zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Daubechies I., Orthonormal bases of compactly supported wavelets II Variations on a theme, SIAM J. Math. Anal., 1993, 24, 499–519zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Daubechies I., Ten lectures on wavelets, SIAM, Philadelphia, 1992zbMATHGoogle Scholar
  17. [17]
    Eirola T., Sobolev characterization of compactly supported wavelets, SIAM J. Math. Anal., 1992, 23, 1015–1030zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Finěk V., Approximation properties of wavelets and relations among scaling moments, Numer. Funct. Anal. Optim., 2004, 25, 503–513CrossRefMathSciNetGoogle Scholar
  19. [19]
    Finěk V., Approximation properties of wavelets and relations among scaling moments II, Cent. Eur. J. Math., 2004, 2, 605–613CrossRefMathSciNetGoogle Scholar
  20. [20]
    Lawton W.M., Necessary and sufficient conditions for constructing orthonormal wavelet bases, J. Math. Phys., 1991, 32, 57–61zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Lebrun J., Selesnick I., Grobner bases and wavelet design, J. Symb. Comp., 2004, 37, 227–259zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Monzón L., Beylkin G., Hereman W., Compactly supported wavelets based on almost interpolating and nearly linear phase filters (coiflets), Appl. Comput. Harmon. Anal., 1999, 7, 184–210zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    Regensburger G., Scherzer O., Symbolic computation for moments and filter coefficients of scaling functions, Ann. Comb., 2005, 9, 223–243zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    Regensburger G., Parametrizing compactly supported orthonormal wavelets by discrete moments, Applicable Algebra in Engineering, Communication and Computing, 2007, 18, 583–601CrossRefMathSciNetGoogle Scholar
  25. [25]
    Resnikoff H.L., Wells R.O., Wavelet analysis. The scalable structure of information, Springer-Verlag, New York, 1998zbMATHGoogle Scholar
  26. [26]
    Shann W.C., Yen C.C., On the exact values of orthonormal scaling coefficients of length 8 and 10, Appl. Comput. Harmon. Anal., 1999, 6, 109–112zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    Tian J., The mathematical theory and applications of biorthogonal Coifman wavelet systems, Ph.D. thesis, Rice University, Houston, TX, 1996Google Scholar
  28. [28]
    Tian J., Wells R.O. Jr., Vanishing moments and biorthogonal Coifman wavelet systems, Proceedings of 4th International Conference on Mathematics in Signal Processing, University of Warwick, England, 1997Google Scholar
  29. [29]
    Tian J., Wells R.O. Jr., Vanishing moments and wavelet approximation, Technical Report, CML TR95-01, Rice University, January 1995Google Scholar
  30. [30]
    Unser M., Approximation power of biorthogonal wavelet expansions, IEEE Transactions on Signal Processing, 1996, 44, 519–527CrossRefGoogle Scholar
  31. [31]
    Villemoes L.F., Energy moments in time and frequency for two-scale difference equation solutions and wavelets, SIAM J. Math. Anal., 1992, 23, 1519–1543zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© © Versita Warsaw and Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  1. 1.Department of Mathematics and Didactics of MathematicsTechnical University of LiberecLiberec 1Czech Republic
  2. 2.Department of Numerical MathematicsCharles UniversityPrague 2Czech Republic

Personalised recommendations