The wavelet transform is a form of a frequency transform. As basis functions one uses wavelets. The big advantage over the Fourier transform is the temporal (or spatial) locality of the base functions (see also short time Fourier transform) and the smaller complexity (O(N) instead of O(N log N) for the fast Fourier transform (where N is the data size).

Important applications are:

Types of wavelet transforms:
  • continuous wavelet transform (CWT)
  • discrete wavelet transform (DWT)
  • fast wavelet transform (FWT)
  • wavelet packets
  • complex wavelet transform

Table of contents
1 Continuous wavelet transform (CWT)
2 History
3 External links

Continuous wavelet transform (CWT)

The continuous wavelet transform is defined as

where represents translation, represents scale and is the transforming function or mother wavelet.

The original function can be reconstructed with the inverse transform

where
is called the admissibility constant. For a succesful inverse transform, the admissibility constant has to satisfy the admissibility condition:

History

  • First wavelet (Haar wavelet) by Alfred Haar (1909)
  • Since the 1950s: Jean Morlet and Alex Grossman
  • Since the 1980s: Yves Meyer, Stephane Mallat, Ingrid Daubechies, Ronald Coifman, Victor Wickerhauser

External links