The Wavelet transform using the Fourier transform

We start with the set of scalar products . If has a cut-off frequency [#starck1<#7555,#starck2<#7556,#starck3<#7557,#starck4<#7558], the data are correctly sampled. The data at the resolution j=1 are:

and we can compute the set c₁(k) from c₀(k) with a discrete filter :

and

where n is an integer. So:

The cut-off frequency is reduced by a factor 2 at each step, allowing a reduction of the number of samples by this factor.

The wavelet coefficients at the scale j+1 are:

and they can be computed directly from c_j(k) by:

where g is the following discrete filter:

and

The frequency band is also reduced by a factor 2 at each step. Applying the sampling theorem, we can build a pyramid of elements. For an image analysis the number of elements is . The overdetermination is not very high.

The B-spline functions are compact in this directe space. They correspond to the autoconvolution of a square function. In the Fourier space we have:

B₃(x) is a set of 4 polynomials of degree 3. We choose the scaling function which has a B₃(x) profile in the Fourier space:

In the direct space we get:

This function is quite similar to a Gaussian one and converges rapidly to . For 2-D the scaling function is defined by , with .It is an isotropic function.

The wavelet transform algorithm with n_p scales is the following one:

1.

We start with a B3-Spline scaling function and we derive , h and g numerically.

2.

We compute the corresponding image FFT. We name T₀ the resulting complex array;

3.

We set j to . We iterate:

4.

We multiply T_j by . We get the complex array W_j+1. The inverse FFT gives the wavelet coefficients at the scale 2^j;

5.

We multiply T_j by . We get the array T_j+1. Its inverse FFT gives the image at the scale 2^j+1. The frequency band is reduced by a factor 2.

6.

We increment j

7.

If , we go back to 4.

8.

The set describes the wavelet transform.

If the wavelet is the difference between two resolutions, we have:

and:

then the wavelet coefficients can be computed by .