Use of the wavelets: theory


There are many very precise works on the wavelets, we describe here only one part of the theory.
The theory of the wavelets can be seen in comparison of the theory of Fourier.

Wavelet transform:

 
The wavelet transform remove the sinusoid of the Fourier transform by a family of translated and dilated functions of a same source, the wavelet.

The parameters of translation and dilation are the two arguments of the wavelet transform.
The wavelet transform is defined by :






where the basic atom y has a null average, is centered in 0 and has a defined energy. The vectors are obtained by translation and dilation of the basic atom:

The preceding function is centered in the vicinity of u, like the atoms of the windowed Fourier transform.

The standard deviation in time is proportional to S. The standard deviation in frequency is inversely proportional to S. Here is an example of  the "boxs" of Heisenberg of atoms of wavelets:
 
 







On finer scales, one can " pile up " more boxes of Heisenberg coast at coast because the temporal resolution is better.
 

The transform in ondelettes has a time-frequency resolution which depends on the scale S. With the condition:

it is a complete, stable and redundant representation of the signal ; in particular, the wavelet transform is invertible. The redundancy results in the existence of a reproducing core.
  The transform in ondelettes is calculated by a fast wavelet transform. This one carries out a discrete transform by circular convolutions, themselves calculated by FFT.

In order to speed up the transform, we often use dyadic wavelet. The dyadic wavelet transform is implemented using filter banks.
 
 

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