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Parametrizing compactly supported orthonormal wavelets by discrete moments

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Abstract

We discuss parametrizations of filter coefficients of scaling functions and compactly supported orthonormal wavelets with several vanishing moments. We introduce the first discrete moments of the filter coefficients as parameters. The discrete moments can be expressed in terms of the continuous moments of the related scaling function. To solve the resulting polynomial equations we use symbolic computation and in particular Gröbner bases. The cases of four to ten filter coefficients are discussed and explicit parametrizations are given.

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References

  1. Becker T. and Weispfenning V. (1993). Gröbner Bases, Graduate Texts in Mathematics, vol. 141. Springer, New York

    Google Scholar 

  2. Bourbaki N. (1990). Algebra. II. Chap. 4–7. Elements of Mathematics (Berlin). Springer, Berlin

    Google Scholar 

  3. Bratteli O. and Jorgensen P. (2002). Wavelets through a Looking Glass. Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc., Boston

    Google Scholar 

  4. Buchberger, B.: An algorithm for finding the bases elements of the residue class ring modulo a zero dimensional polynomial ideal (German). Ph.D. Thesis, University of Innsbruck (1965) (English translation published in [7])

  5. Buchberger B. (1970). Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems. Aequationes Math. 4: 374–383

    Article  MATH  MathSciNet  Google Scholar 

  6. Buchberger B. (1998). Introduction to Gröbner bases. In: Buchberger, B. and Winkler, F. (eds) Gröbner Bases and Applications (Linz, 1998), Lond Math. Soc. Lect Note Ser., vol. 251, pp 3–31. Cambridge University Press, Cambridge

    Google Scholar 

  7. Buchberger, B.: Bruno Buchberger’s Ph.D. thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. J. Symb. Comput. 41(3–4): 475–511 (2006) (Translated from the 1965 German original by Michael P. Abramson)

  8. Charoenlarpnopparut C. and Bose N. (1999). Multidimensional FIR filter bank design using Gröbner bases. IEEE Trans. Circuits Syst., II, Analog Digit. Signal Process. 46(12): 1475–1486

    Article  MATH  Google Scholar 

  9. Chyzak F., Paule P., Scherzer O., Schoisswohl A. and Zimmermann B. (2001). The construction of orthonormal wavelets using symbolic methods and a matrix analytical approach for wavelets on the interval. Exp. Math. 10(1): 67–86

    MATH  MathSciNet  Google Scholar 

  10. Cohen A. (1990). Ondelettes, analyses multirésolutions et filtres miroirs en quadrature. Ann. Inst. H. Poincaré Anal. Non Linéaire 7(5): 439–459

    MATH  Google Scholar 

  11. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 2nd edn. Undergraduate Texts in Mathematics. Springer, New York (1997). An introduction to computational algebraic geometry and commutative algebra

  12. Daubechies I. (1988). Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41(7): 909–996

    Article  MATH  MathSciNet  Google Scholar 

  13. Daubechies I. (1992). Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics (SIAM), Philadelphia

    MATH  Google Scholar 

  14. Daubechies I. (1993). Orthonormal bases of compactly supported wavelets. II. Variations on a theme. SIAM J. Math. Anal. 24(2): 499–519

    Article  MATH  MathSciNet  Google Scholar 

  15. Faugère J.C., Moreau de Saint-Martin F. and Rouillier F. (1998). Design of regular nonseparable bidimensional wavelets using Gröbner basis techniques. IEEE Trans. Signal Process. 46(4): 845–856

    Article  MathSciNet  Google Scholar 

  16. Haar A. (1910). Zur Theorie der orthogonalen Funktionensysteme. (Erste Mitteilung.). Math. Ann. 69: 331–371

    Article  MathSciNet  Google Scholar 

  17. Hereford, J.M., Roach, D.W., Pigford, R.: Image compression using parameterized wavelets with feedback. pp. 267–277. SPIE (2003)

  18. Jorgensen P.E.T. (2003). Matrix factorizations, algorithms, wavelets. Notices Am. Math. Soc. 50(8): 880–894

    MATH  MathSciNet  Google Scholar 

  19. Knuth, D.E.: The art of computer programming. Vol. 1: Fundamental algorithms, 3rd edn. Addison-Wesley, Reading. xx, p. 650 (1997)

  20. Lai, M.J., Roach, D.W.: Parameterizations of univariate orthogonal wavelets with short support. In: Approximation Theory, X, St Louis, MO, 2001, Innov. Appl. Math., pp. 369–384. Vanderbilt University Press, Nashville, TN (2002)

  21. Lawton W.M. (1990). Tight frames of compactly supported affine wavelets. J. Math. Phys. 31(8): 1898–1901

    Article  MATH  MathSciNet  Google Scholar 

  22. Lawton W.M. (1991). Necessary and sufficient conditions for constructing orthonormal wavelet bases. J. Math. Phys. 32(1): 57–61

    Article  MATH  MathSciNet  Google Scholar 

  23. Lebrun J. and Selesnick I. (2004). Gröbner bases and wavelet design. J. Symb. Comput. 37(2): 227–259

    Article  MATH  MathSciNet  Google Scholar 

  24. Lebrun J. and Vetterli M. (2001). High-order balanced multiwavelets: Theory, factorization and design. IEEE Trans. Signal Process. 49(9): 1918–1930

    Article  MathSciNet  Google Scholar 

  25. Lina J.M. and Mayrand M. (1993). Parametrizations for Daubechies wavelets. Phys. Rev. E (3) 48(6): R4160–R4163

    Article  MathSciNet  Google Scholar 

  26. Mallat S. (1998). A Wavelet Tour of Signal Processing. Academic, San Diego

    MATH  Google Scholar 

  27. Park H. (2004). Symbolic computation and signal processing. J. Symb. Comput. 37(2): 209–226

    Article  MATH  Google Scholar 

  28. Park H., Kalker T. and Vetterli M. (1997). Gröbner bases and multidimensional FIR multirate systems. Multidimensional Syst. Signal Process. 8(1–2): 11–30

    Article  MATH  Google Scholar 

  29. Pollen D. (1990). SU I (2,F[z,1/z]) for F a subfield of C. J. Am. Math. Soc. 3(3): 611–624

    Article  MATH  MathSciNet  Google Scholar 

  30. Regensburger G. and Scherzer O. (2005). Symbolic computation for moments and filter coefficients of scaling functions. Ann. Comb. 9(2): 223–243

    Article  MATH  MathSciNet  Google Scholar 

  31. Schneid J. and Pittner S. (1993). On the parametrization of the coefficients of dilation equations for compactly supported wavelets. Computing 51(2): 165–173

    Article  MATH  MathSciNet  Google Scholar 

  32. Selesnick I.W. and Burrus C.S. (1998). Maximally flat low-pass FIR filters with reduced delay. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 45(1): 53–68

    Article  Google Scholar 

  33. Shann W.C. and Yen C.C. (1999). On the exact values of orthonormal scaling coefficients of lengths 8 and 10. Appl. Comput. Harmon. Anal. 6(1): 109–112

    Article  MATH  MathSciNet  Google Scholar 

  34. Sherlock B.G. and Monro D.M. (1998). On the space of orthonormal wavelets. IEEE Trans. Signal Process. 46(6): 1716–1720

    Article  MATH  MathSciNet  Google Scholar 

  35. Strang G. and Nguyen T. (1996). Wavelets and filter banks. Wellesley–Cambridge Press, Wellesley

    Google Scholar 

  36. Unser M. and Blu T. (2003). Wavelet theory demystified. IEEE Trans. Signal Process. 51(2): 470–483

    Article  MathSciNet  Google Scholar 

  37. Wang S.H., Tewfik A.H. and Zou H. (1994). Correction to ‘parametrization of compactly supported orthonormal wavelets’. IEEE Trans. Signal Process. 42(1): 208–209

    Article  Google Scholar 

  38. Wells R.O. (1993). Parametrizing smooth compactly supported wavelets. Trans. Am. Math. Soc. 338(2): 919–931

    Article  MATH  MathSciNet  Google Scholar 

  39. Zou H. and Tewfik A.H. (1993). Parametrization of compactly supported orthonormal wavelets. IEEE Trans. Signal Process. 41(3): 1428–1431

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, 4040, Linz, Austria

    Georg Regensburger

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  1. Georg Regensburger
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Correspondence to Georg Regensburger.

Additional information

This work was supported by the Austrian Science Fund (FWF) under the SFB grant F1322.

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Regensburger, G. Parametrizing compactly supported orthonormal wavelets by discrete moments. AAECC 18, 583–601 (2007). https://doi.org/10.1007/s00200-007-0054-9

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  • DOI: https://doi.org/10.1007/s00200-007-0054-9

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