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

Entropy Shrinkage Compaction Convolution Stein 

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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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