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Дата изменения: Fri Feb 23 12:58:04 1996 Дата индексирования: Tue Oct 2 17:58:44 2012 Кодировка: |
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
.