The Wavelet transform using the Fourier transform

We start with the set of scalar products . If has a cut-off frequency [,,,], the data are correctly sampled. The data at the resolution j=1 are:

and we can compute the set from 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 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:

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

In the direct space we get:

This function is quite similar to a Gaussian one and converges rapidly to 0. For 2-D the scaling function is defined by , with . It is an isotropic function.

The wavelet transform algorithm with 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 the resulting complex array;
3. We set j to 0. We iterate:
4. We multiply by . We get the complex array . The inverse FFT gives the wavelet coefficients at the scale ;
5. We multiply by . We get the array . Its inverse FFT gives the image at the scale . 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 .