Multiresolution analysis [25] results from the embedded subsets generated by the interpolations at different scales.
A function 
is projected at each step j onto the subset
. This projection is defined by the scalar product 
of
with the scaling function 
which is dilated and
translated:
As 
is a scaling function which has the property:
or
where 
is the Fourier transform of the function 
. We get:
Equation 
permits to
compute directly the set 
from 
.
If we start from the set 
we compute all the sets
, with j>0, without directly computing any other scalar
product:
At each step, the number of scalar products is divided by 2. Step by step
the signal is smoothed and information is lost. The remaining
information can be restored using the complementary subspace 
of
in 
.
This subspace can be generated by a suitable wavelet function
with translation and dilation.
or
We compute the scalar products 
with:
With this analysis, we have built the first part of a filter bank
[34]. In order to restore the original data, Mallat uses
the properties of orthogonal wavelets, but the theory has been
generalized to a large class of filters [8] by introducing two
other filters 
and 
named conjugated to h and
g. The restoration is performed with:
In order to get an exact restoration, two conditions are required for the conjugate filters:
Figure: The filter bank associated with the multiresolution
analysis
In the decomposition, the function is successively convolved with
the two filters H (low frequencies) and G (high frequencies). Each
resulting function is decimated by suppression of one sample out of two. The
high frequency signal is left, and we iterate with the low frequency signal
(upper part of figure ).
In the reconstruction, we restore the sampling by inserting a 0 between
each sample, then we convolve with the conjugate filters 
and
, we add the resulting functions and we multiply the result by 2.
We iterate up to the smallest scale
(lower part of figure ).
Orthogonal wavelets correspond to the restricted case where:
and
We can easily see that this set satisfies the two basic
relations and .
Daubechies wavelets are the only compact solutions.
For biorthogonal wavelets [8]
we have the relations:
and
We also satisfy relations and . A large class of
compact wavelet functions can be derived.
Many sets of filters were proposed, especially for coding. It was shown
[9] that the choice of these filters must be guided by the
regularity of the scaling and the wavelet functions.
The complexity is proportional to N. The algorithm provides a pyramid of
N elements.
The 2D algorithm is based on separate variables leading to prioritizing of x and y directions. The scaling function is defined by:
The passage from a resolution to the next one is done by:
The detail signal is obtained from three wavelets:
Figure: Wavelet transform representation of an image
The wavelet transform can be interpreted as the decomposition on frequency sets with a spatial orientation.
