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Computing

, Volume 100, Issue 7, pp 715–739 | Cite as

Inverse formulas of parameterized orthogonal wavelets

  • Oscar Herrera-Alcántara
  • Miguel González-Mendoza
Article
  • 91 Downloads

Abstract

We review the parameterization of orthogonal wavelet based filters of length 4, 6, 8, and 10, and present their inverse formulas, which means to determine the parameter values from filter coefficients. Experimental results support the validity of these inverse formulas when parameters are restricted to \([0, 2\pi )\) for practical applications, such as image processing where parameters are optimized to maximize the number of negligible wavelet coefficients.

Keywords

Wavelets Filter parameterization Orthogonality Image processing 

Mathematics Subject Classification

65T60 94A12 94A08 

References

  1. 1.
    Mallat SG (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Anal Mach Intell 11(7):674–693CrossRefGoogle Scholar
  2. 2.
    Vetterli M, Herley C (1992) Wavelets and filter banks: theory and design. IEEE Trans Signal Process 40(9):2207–2232CrossRefGoogle Scholar
  3. 3.
    Hehong Z, Tewfik A (1993) Parametrization of compactly supported orthonormal wavelets. IEEE Trans Signal Process 41(3):1428–1431CrossRefGoogle Scholar
  4. 4.
    Daubechies I (1992) Ten lectures on wavelets. Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefGoogle Scholar
  5. 5.
    Pollen D (1990) SU1(2F(z,1/z) for F a subfield of C. J Am Math Soc 3:611–624MathSciNetzbMATHGoogle Scholar
  6. 6.
    Wells JRO (1993) Parameterizing smooth compactly supported wavelets. Trans Am Math Soc 338(2):919–931CrossRefGoogle Scholar
  7. 7.
    Schneid J, Pittner S (1993) On the parametrization of the coefficients of dilation equations for compactly supported wavelets. Computing 51:165–173MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lina JM, Mayrand M (1993) Parametrizations for Daubechies wavelets. Phys Rev E 48(6):R4160–R4163MathSciNetCrossRefGoogle Scholar
  9. 9.
    Sabah MA (2008) Optimal selection of threshold levels and wavelet filters for high quality ECG signal compression. J Eng Sci 36(5):1225–1243Google Scholar
  10. 10.
    Suhail MA, Dawoud MM (2001) Watermarking security enhancement using filter parametrization in feature domain. In: Proceedings of the 2001 workshop on multimedia and security: new challenges, MM 38; Sec 01, New York. ACM, pp 15–18Google Scholar
  11. 11.
    Zhang Z, Telesford QK, Giusti C, Lim KO, Bassett DS (2016) Choosing wavelet methods, filters, and lengths for functional brain network construction. PLOS ONE 11(6):e0157243CrossRefGoogle Scholar
  12. 12.
    Lai MJ, Roach DW (2002) Parameterizations of univariate orthogonal wavelets with short support. In: Chui CK, Schumaker LL, Stoeckler J (eds) Approximation theory X: splines, wavelets, and applications. Vanderbilt University Press, Nashville, pp 369–384Google Scholar
  13. 13.
    Roach DW (2008) The parameterization of the length eight orthogonal wavelets with no parameter constraints. In: Neamtu M, Schumaker LL (eds) Approximation theory XII: San Antonio 2007. Nashboro Press, Brentwood, pp 332–347Google Scholar
  14. 14.
    Roach DW (2010) Frequency selective parameterized wavelets of length ten. J Concr Appl Math 8(1):165–179MathSciNetzbMATHGoogle Scholar
  15. 15.
    Schneid J, Pittner S (1993) On the parametrization of the coefficients of dilation equations for compactly supported wavelets. Computing 51(2):165–173MathSciNetCrossRefGoogle Scholar
  16. 16.
    Herrera O, González M (2011) Optimization of parameterized compactly supported orthogonal wavelets for data compression. Springer, Berlin, pp 510–521Google Scholar
  17. 17.
    Soman KP, Ramachandran KI (2005) Insight into wavelets: from theory to practice. Prentice-Hall, New Delhi, p 71289710Google Scholar
  18. 18.
    Herrera O, Mora R (2011) Aplicación de algoritmos genéticos a la compresión de imágenes con evolets. Sociedad Mexicana de Inteligencia Artificial, Mexico, pp 157–165Google Scholar
  19. 19.
    Mallat S (1998) A wavelet tour of signal processing. Academic Press Inc., CambridgezbMATHGoogle Scholar
  20. 20.
    Herrera O (2010) On the best evolutionary wavelet based filter to compress a specific signal. Springer, Berlin, pp 394–405Google Scholar
  21. 21.
    Kuri A (1999) A comprehensive approach to genetic algorithms in optimization and learning. National Polytechnic Institute, MexicoGoogle Scholar
  22. 22.
    Weber M (1999) Frontal face dataset. California Institute of Technology. http://www.vision.caltech.edu/Image_Datasets/Caltech256

Copyright information

© The Author(s) 2018
corrected publication September 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Universidad Autónoma MetropolitanaAzcapotzalcoMéxico
  2. 2.Instituto Tecnológico y de Estudios Superiores de MonterreyAtizapán de ZaragozaMéxico

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