Transformée en ondelettes rapide

Fast wavelet transform

The discrete wavelet transform uses a succession of discrete scales a^J , for j<0 with a=2^1/v, where v is an integer, called a number of channels in the octave.

It is supposed that the support of the wavelet y is [-K/2, K/2 ]. For a signal of size N and 1<=a^J< = N/K, we can define a discrete wavelet by sampling at scale A^J on integers, that is to say :

The signal and the wavelet are periodic over one period N, the discrete wavelet transform of F is:

This is a circular convolution between F and y₁[ n]=y[-n ]. These circular convolutions are calculated by a FFT and requires O(N log₂(N)) operations.

The scalogramme is computed with the wavelet transform and we can make a parabolic interpolation on the scales for better localisation of the peaks of the wavelet.
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