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Wavelet Transforms

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

Abstract

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).

Keywords

Filter Bank Wavelet Packet Wavelet Function Biorthogonal Wavelet Multiresolution Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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