The first filtering is then performed by a twice magnified scale
leading to the
set. The signal difference
contains the information between these two scales and is
the discrete set associated with the wavelet transform corresponding
The associated wavelet is therefore .
The distance between samples increasing by a factor 2 from the scale (i-1) (i > 0) to the next one, ci(k) is given by:
and the discrete wavelet transform wi(k) by:
|wi(k) = ci-1(k) - ci(k)||(14.31)|
derive from the scaling function :
The algorithm allowing one to rebuild the data frame is evident: the
last smoothed array cnp is added to all the differences wi.
If we choose the linear interpolation for the scaling function (see figure 14.5):
c1 is obtained by:
The figure 14.6 shows the wavelet associated to the scaling function.
The wavelet coefficients at the scale j are:
The above à trous algorithm is easily extensible to the two
dimensional space. This leads to a convolution with a mask of
pixels for the wavelet connected to linear interpolation. The coefficents
of the mask are:
At each scale j, we obtain a set (we will call it wavelet plane in the following), which has the same number of pixels as the image.
If we choose a B3-spline for the scaling function, the coefficients
of the convolution mask in one dimension are
in two dimensions: