Wavelet Transforms

  • Xiaoxia Yin
  • Brian W.-H. Ng
  • Derek Abbott


The history of the WT can be traced back to Fourier theory. Fourier decomposition expands a signal as an integral of sinusoidal oscillations over a range of frequencies (Ng 2003). A major limitation of Fourier theory is in the loss of temporal information in the Fourier transform. Wavelets were first introduced in 1987, as the foundation of a powerful new approach to signal processing, called multiresolution theory (Mallat 1999). Multiresolution theory incorporates and unifies techniques from a variety of disciplines, including subband coding from signal processing, quadrature mirror filtering from speech recognition, and pyramidal image processing (Gonzalez and Woods 2002). Formally, a multiresolution analysis (MRA) allows the representation of signals with their WT coefficients. The theory underlying MRA allows a systematic method for constructing (bi)orthogonal wavelets (Daubechies 1988) and leads to the fast DWT, also known as Mallat’s pyramid algorithm (Qian 2002; Mallat 1989). In practice, the DWT has been applied to many different problems (Meyer 1990; Strang and Nguyen 1996; Daubechies 1992).


Entropy Shrinkage Compaction Convolution Stein 


  1. Daubechies-I. (1988). Orthonormal bases of compactly supported wavelets, Communications on Pure & Applied Mathematics, 41(7), pp. 909–996.MathSciNetMATHCrossRefGoogle Scholar
  2. Daubechies-I. (1992). Ten lectures on wavelets, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.MATHCrossRefGoogle Scholar
  3. Divine-D. V., and Godtliebsen-F. (2007). Bayesian modeling and significant features exploration in wavelet power spectra, Nonlinear Processes in Geophysics, 14, pp. 79–88.CrossRefGoogle Scholar
  4. Donoho-D. L. (1995). De-noising by soft thresholding, IEEE Transactions on Information Theory, 41(3), pp. 613–627.MathSciNetMATHCrossRefGoogle Scholar
  5. Ferguson-B., and Abbott-D. (2001a). De-noising techniques for terahertz responses of biological samples, Microelectronics Journal (Elsevier), 32(12), pp. 943–953.Google Scholar
  6. Florida State University (2005). Web. http://micro.magnet.fsu.edu/index.html Last Checked: August 19 2005.
  7. Gonzalez-R. C., and Woods-R. E. (2002). Digital Image Processing, Prentice-Hall, Inc., New Jersey.Google Scholar
  8. Guo-Y., Zhang-H., Wang-X., and Cavallaro-R. (2001). VLSI implementation of Mallat fast discrete wavelet transform algorithm with reduced complexity, IEEE Global Telecommunications Conference, 1, pp. 25–29.Google Scholar
  9. Hadjiloucas-S., Galvõ-R. K. H., Becerra-V. M., Bowen-J. W., Martini-R., Brucherseifer-M., Pellemans-H. P. M., Haring Bolívar-P., Kurz-H., and Chamberlain-J. M. (2004). Comparison of state space and ARX models of a waveguide’s THz transient response after optimal wavelet filtering, IEEE Transactions on Microwave Theory and Techniques MTT, 52(10), pp. 2409–2419.CrossRefGoogle Scholar
  10. Hubbard-B. (1998). The World According to Wavelets, 2nd edn, A.K.Peters, Wellesley, Massachusetts.MATHGoogle Scholar
  11. Mallat-S. (1989). A theory for multiresolution signal decomposition: The wavelet representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(7), pp. 674–693.CrossRefGoogle Scholar
  12. Mallat-S. G. (1999). A Wavelet Tour of Signal Processing, San Diego : Academic Press, CA.MATHGoogle Scholar
  13. Meyer-Y. (1990). Ondelettes et Operateurs, Tome 1, Herrmann, Paris.Google Scholar
  14. Ng-B. W.-H. (2003). Wavelet Based Image Texture Segmentation using a Modified K-means Algorithm (PhD Thesis), University of Adelaide.Google Scholar
  15. Percival-D., and Walden-A. (2000). Wavelet Methods for Time Series Analysis, Cambridge University Press, Cambridge, England.MATHGoogle Scholar
  16. Qian-S. (2002). Time-Frequency and Wavelet Transforms, 1st edn, Prentice Hall, Inc., New Jersey, USAGoogle Scholar
  17. Sherlock-B. G., and Monro-D. M. (1998). On the space of orthonormal wavelets, IEEE Transactions on Signal Processing, 46(6), pp. 1716–1720.MathSciNetMATHCrossRefGoogle Scholar
  18. Strang-G., and Nguyen-T. (1996). Wavelets and Filter Banks, 1st edn, Wellesley-Cambridge Press, Wellesley, USA.Google Scholar
  19. Tuqun-J., and Vaidyanathan-P. P. (2000). A state-space approach to the design of globally optimal FIR energy compaction filters, IEEE Transaction Signal Processing, 48(10), pp. 2822–2838.CrossRefGoogle Scholar
  20. Vetterli-M., and Kovacevic-J. (1995). Wavelets and Subband Coding, Prentice-Hall PTR, New Jersey.MATHGoogle Scholar
  21. Walnut-D. F. (2001). An Introduction to Wavelet Analysis, 1st edn, Birkhäuser, Boston, USA.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Xiaoxia Yin
    • 1
  • Brian W.-H. Ng
    • 1
  • Derek Abbott
    • 1
  1. 1.School of Electrical and Electronic EngineeringUniversity of AdelaideAdelaideAustralia

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