A Subclass of the Length 12 Parameterized Wavelets

Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 13)


In this paper, a subclass of the length 12 parameterized wavelets is given. This subclass is a parameterization of the coefficients of a subset of the trigonometric polynomials, m(ω), that satisfy the necessary conditions for orthogonality, that is m(0)=1 and \(\vert m(\omega ){\vert }^{2} + \vert m(\omega + \pi ){\vert }^{2} = 1\), but is not sufficient to represent all possible trigonometric polynomials satisfying these constraints. This parameterization has three free parameters whereas the general parameterization would have five free parameters. Finally, we graph some example scaling functions from the parameterization and conclude with a numerical experiment.


Image Compression Trigonometric Polynomial Biorthogonal Wavelet Orthogonal Wavelet Boundary Extension 
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  1. 1.
    D. Colella and C. Heil, The characterization of continuous, four-coefficient scaling functions and wavelets, IEEE Trans. Inf. Th., Special Issue on Wavelet Transforms and Multiresolution Signal Analysis, 38 (1992), pp. 876-881.Google Scholar
  2. 2.
    I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.MATHCrossRefGoogle Scholar
  3. 3.
    Q.T. Jiang, Paramterization of m-channel orthogonal multifilter banks, Advances in Computational Mathematics 12 (2000), 189–211.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    M. J. Lai and D. W. Roach, Parameterizations of univariate orthogonal wavelets with short support,Approximation Theory X: Splines, Wavelets, and Applications, C. K. Chui, L. L. Schumaker, and J. Stockler (eds.), Vanderbilt University Press, Nashville, 2002, 369–384.Google Scholar
  5. 5.
    W. Lawton, Necessary and sufficient conditions for constructing orthonormal wavelet bases, J. Math. Phys.32 (1991), 57–61.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    G. Regensburger, Parametrizing compactly supported orthonormal wavelets by discrete moments. Appl. Algebra Eng., Commun. Comput. 18, 6 (Nov. 2007), 583-601.Google Scholar
  7. 7.
    H. L. Resnikoff, J. Tian, R. O. Wells, Jr., Biorthogonal wavelet space: parametrization and factorization, SIAM J. Math. Anal. 33 (2001), no. 1, 194–215.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    D. W. Roach, The Parameterization of the Length Eight Orthogonal Wavelets with No Parameter Constraints, Approximation Theory XII: San Antonio 2007, M. Neamtu and L. Schumaker (eds.), Nashboro Press, pp. 332-347, 2008.Google Scholar
  9. 9.
    D. W. Roach, Frequency selective parameterized wavelets of length ten, Journal of Concrete and Applicable Mathematics, vol. 8, no. 1, pp. 1675-179, 2010.MathSciNetGoogle Scholar
  10. 10.
    A. Said and W. A. Pearlman, A new fast and efficient image codec based on set partitioning in hierarchical trees, IEEE Transactions on Circuits and Systems for Video Technology 6 (1996), 243–250.CrossRefGoogle Scholar
  11. 11.
    J. Schneid and S. Pittner, On the parametrization of the coefficients of dilation equations for compactly supported wavelets, Computing 51 (1993), 165–173.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    J. M. Shapiro, Embedded image coding using zerotrees of wavelet coefficients, IEEE Transactions Signal Processing 41 (1993), 3445–3462.MATHCrossRefGoogle Scholar
  13. 13.
    R. O. Wells, Jr., Parameterizing smooth compactly supported wavelets, Trans. Amer. Math. Soc. 338 (1993), 919–931.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    M. V. Wickerhauser, Comparison of picture compression methods: wavelet, wavelet packet, and local cosine transform coding, Wavelets: theory, algorithms, and applications (Taormina, 1993), Academic Press, San Diego, CA, 1994, 585–621.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Murray State UniversityMurrayUSA

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