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#### stkwave1

Name

stkwave1 One-dimensional wavelet transform using Starck's algorithm (band-limited scaling function)

Command Synopsis

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

Function Summary

void stkwave1 (np , in , out )

Fsignal in , out ;

int np ;

Description

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.

wj+1(k) = < f (x), 2-(j+1)(2-(j+1)x - k) >

cj+1(k) = < f (x), 2-(j+1)(2-(j+1)x - k) >

In the frequency domain, these equations become:

() = ()(2j)

() = ()(2j)

with () =
and () =
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:

() = B3(4)

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:

(2) = () - (2)

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:
1. Compute by FFT, set (f )= and initialize j to 1.
2. Multiply (f ) to H gives the approximation for a resolution j : (f )
3. Multiply (f ) to G gives the details for a resolution j : (f )
4. 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.
The obtained details are ordonned in function of their arrival in a fsignal which is ended by the approximation .

In this module, the input signal is assumed to be already in the Fourier domain that is, = .

See Also

Version 1.0

Last Modification date : Thu Apr 15 08:03:52 2004

Author

Claire Jonchery, Amandine Robin

Next: Bibliography Up: Reference Previous: ridgthres   Contents   Index
mw 2004-05-05