Next: The Reconstruction Up: Multiresolution with scaling functions Previous: Multiresolution with scaling functions

### The Wavelet transform using the Fourier transform

We start with the set of scalar products . If has a cut-off frequency [#starck1<#14704,#starck2<#14705,#starck3<#14706,#starck4<#14707], the data are correctly sampled. The data at the resolution j=1 are:
 (14.47)

and we can compute the set c1(k) from c0(k) with a discrete filter :
 (14.48)

and
 (14.49)

where n is an integer. So:
 (14.50)

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:

 (14.51)

and they can be computed directly from cj(k) by:
 (14.52)

where g is the following discrete filter:
 (14.53)

and
 (14.54)

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:

 (14.55)

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:
 (14.56)

In the direct space we get:
 (14.57)

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 np 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 0. We iterate:
4.
We multiply Tj by . We get the complex array Wj+1. The inverse FFT gives the wavelet coefficients at the scale 2j;
5.
We multiply Tj by . We get the array Tj+1. Its inverse FFT gives the image at the scale 2j+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:

 (14.58)

and:
 (14.59)

then the wavelet coefficients can be computed by .

Next: The Reconstruction Up: Multiresolution with scaling functions Previous: Multiresolution with scaling functions
Petra Nass
1999-06-15