# From filters to wavelets

## Wavelets and scaling functions

Biorthogonal wavelets
and scaling functions are caracterized by a perfect reconstruction
filter bank; orthogonal wavelets and scaling functions are
caracterized by a pair of conjugate mirror filters. Nonetheless, a
perfect reconstruction filter bank (or any pair of conjugate mirror
filters) does not necessarily generate a wavelet system. Indeed, some
attention has to be paid to the stability of the decomposition and
reconstruction schemes as the number of scales increases, that is
when the number of filter bank cascades goes to the infinity. This is
expressed by an additional
condition (7.37) on the conjugate mirror
filter h for it to define a scaling function.

## Perfect reconstruction filter banks and algorithme à
trous

The decomposition can be performed on the signal
a_{1} to generate a signal a_{2} and a signal
d_{2}; repeating this construction produces a low resolution
signal a_{j} and a sequence of detail signals d_{1}
.... d_{j}.

A recursive decomposition which similar to the
previous one can be performed by the algorithme
à trous to generate low resolution
signal A_{j} and a sequence of detail signals D_{1}
.... D_{j}. The two decompositions are related by the
following equations:

a_{j}[n] =
A_{j}[2^{j}n]

d_{j}[n] =
D_{j}[2^{j}n]
## From algorithme à trous to scaling functions

In the Fourier domain, the transfer between
a_{0} and A_{j} is

Let us operate a time rescaling T =
2^{-j}t so that the interval between the non zero
coefficients of the slower filter is always one. Then the interval
between the non zero coefficients of the tightest filter is
2^{-j}. The transfer becomes

Let j go to the infinity. If the previous transfer
converges in L^{2}, the limit can be interpreted as the Fourier transform of the mirror of a function , and the processed signal can be written as

which can be interpreted as the scalar product between the original signal and a translate of the scaling function whose Fourier transform is

The scaling function necessarily satisfies a scaling
equation:

Such functions are at the core of multiresolution
analysis, which is itself the sarting
point for the definition of dyadic wavelets.

Multiresolution
analysis

Filter
Synthesis