Wavelet transform
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:
- image compression and video compression: wavelet compression
- Solving differential equations
- signal processing
- continuous wavelet transform (CWT)
- discrete wavelet transform (DWT)
- fast wavelet transform (FWT)
- wavelet packets
- complex wavelet transform
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The continuous wavelet transform is defined as
The original function can be reconstructed with the inverse transform
Continuous wavelet transform (CWT)
where represents translation, represents scale and is the transforming function or mother wavelet.
where
is called the admissibility constant. For a succesful inverse transform, the admissibility constant has to satisfy the admissibility condition:
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