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:
If the wavelet is the difference between two resolutions, we have:
and:
then the wavelet coefficients can be computed by .