stkwave1 One-dimensional wavelet transform using Starck's algorithm (band-limited scaling function)
stkwave1 np in out
np : resolution np
in : input in Fourier domain (the size of signal must be a power of 2)
out : result in Fourier domain, from left to right : details and approximation
void stkwave1 (np , in , out )
Fsignal in , out ;
int np ;
The module stkwave1 computes the wavelet transform of a one-dimensional signal according to the work
of Starck et al. [SBLP94] who use an overcomplete frequency-domain approach (band-limited wavelet).
As seen in owave1, multi-resolution analysis corresponds to considering a scale function and a wavelet used to compute details and approximations of a signal.
) = < f
In the frequency domain, these equations become:
) = < f
The frequency band is reduced by a factor of 2 while the resolution scales up.
We go from a resolution to the following resolution multiplying the filter H and the frequential signal.
The details are obtained filtering the same signal by G.
J.L. Starck uses a B3 spline for in the Fourier domain:
that is to say
(x) = ()4
The first difference with the Mallat's algorithm [Mal89] stands in the relation between and : here, corresponds to the difference between two resolutions:
The second difference with the Mallat's algorithm is the non-decimation of the details. Which implies, for a size N of the signal, that the obtained coefficients are ordonned in a 2N-signal.
The wavelet transformation algorithm for a resolution np is the following:
The obtained details are ordonned in function of their arrival in a fsignal which is ended by the approximation
- Compute by FFT, set
(f )= and initialize j to 1.
(f ) to H gives the approximation for a resolution j :
(f ) to G gives the details for a resolution j :
- If j < np, the frequency band of
(f ) is reduced by a factor 2 which corresponds to keep one coefficient out of two in the time space, j is then incremented and we go back to point 2.
In this module, the input signal is assumed to be already in the Fourier domain that is,
Last Modification date : Thu Apr 15 08:03:52 2004
Claire Jonchery, Amandine Robin