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
to .
The associated wavelet is therefore .
(14.29) |
The distance between samples increasing by a factor 2 from the scale (i-1) (i > 0) to the next one, c_{i}(k) is given by:
(14.30) |
and the discrete wavelet transform w_{i}(k) by:
w_{i}(k) = c_{i-1}(k) - c_{i}(k) | (14.31) |
The coefficients
derive from the scaling function :
(14.32) |
The algorithm allowing one to rebuild the data frame is evident: the
last smoothed array c_{np} is added to all the differences w_{i}.
If we choose the linear interpolation for the scaling function (see figure 14.5):
(14.34) |
c_{1} is obtained by:
(14.35) |
(14.36) |
The figure 14.6 shows the wavelet associated to the scaling function.
The wavelet coefficients at the scale j are:
(14.37) |
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 B_{3}-spline for the scaling function, the coefficients
of the convolution mask in one dimension are
(
), and
in two dimensions: