The discrete approach of the wavelet transform can be done with the
special version of the so-called à trous algorithm (with
holes) [,]. One assumes that the sampled data
are the scalar products at pixels k of the function
with a scaling function 
which corresponds to a low
pass filter.
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 
.
The distance between samples increasing by a factor
2 from the scale 
( i > 0 ) to the next one, 
is given by:
and the discrete wavelet transform 
by:
The coefficients 
derive from the scaling function 
:
The algorithm allowing one to rebuild the data frame is evident: the
last smoothed array 
is added to all the differences 
.
If we choose the linear interpolation for the scaling function 
(see figure 
):
we have:
is obtained by:
and 
is obtained from 
by:
The figure 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 
-spline for the scaling function, the coefficients
of the convolution mask in one dimension are
(
), and
in two dimensions:
