Wavelets with Scale Dependent Properties

  • Peter Paule
  • Otmar Scherzer
  • Armin Schoisswohl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2630)


In this paper we revisite the constitutive equations for coefficients of orthonormal wavelets. We construct wavelets that satisfy alternatives to the vanishing moments conditions, giving orthonormal basis functions with scale dependent properties. Wavelets with scale dependent properties are applied for the compression of an oscillatory one-dimensional signal.


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  1. 1.
    I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Commun. Pure Appl. Math., vol. 41, No. 7, pp. 901–996, 1988.CrossRefMathSciNetGoogle Scholar
  2. 2.
    I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992.zbMATHGoogle Scholar
  3. 3.
    S. Roman, The Umbral Calculus, Academic Press, Orlando, 1984.zbMATHGoogle Scholar
  4. 4.
    J. Lebrun, “High order balanced multiwavelets: Theory, factorization and design,” IEEE Trans. Signal Processing, vol. 49, pp. 1918–1930, 2001.CrossRefMathSciNetGoogle Scholar
  5. 5.
    F. Chyzak, P. Paule, O. Scherzer, A. Schoisswohl, and B. Zimmermann, “The construction of orthonormal wavelets using symbolic methods and a matrix analytical approach for wavelets on the interval” Experimental Mathematics, vol. 10, pp. 67–86, 2001.zbMATHMathSciNetGoogle Scholar
  6. 6.
    J.R. Williams and K. Amaratunga, “Introduction to wavelets in engineering,” Int. J. Numer. Methods Eng., vol. 37, No. 14, pp. 2365–2388, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    I. Daubechies, “Orthonormal bases of compactly supported wavelets. II: Variations on a theme,” SIAM J. Math. Anal., vol. 24, No. 2, pp. 499–519, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Unser, “Splines-a perfect fit for signal and image processing,” IEEE Signal Processing Magazine, vol. 16, pp. 22–38, 1999.CrossRefGoogle Scholar
  9. 9.
    T. Blu and M. Unser, “The fractional spline wavelet transform: definition and implementation,” preprint, 1999.Google Scholar
  10. 10.
    M. Unser and T. Blu, “Fractional splines and wavelets,” SIAM Rev., vol. 42, pp. 43–67, 2000.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Peter Paule
    • 1
  • Otmar Scherzer
    • 2
  • Armin Schoisswohl
    • 3
  1. 1.Research Institute for Symbolic ComputationJohannes Kepler UniversityLinzAustria
  2. 2.Applied Mathematics, Department of Computer ScienceUniversity InnsbruckInnsbruckAustria
  3. 3.GE Medical Systems Kretz UltrasoundZipfAustria

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