Multiresolution Analysis

A function *f*(*x*) is projected at each step *j* onto the subset
*V*_{j}. This projection is defined by the scalar product *c*_{j}(*k*) of
*f*(*x*) with the scaling function
which is dilated and
translated:

(14.9) |

As is a scaling function which has the property:

or

(14.11) |

where is the Fourier transform of the function . We get:

(14.12) |

Equation 14.10 permits to compute directly the set

(14.13) |

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 *W*_{j+1} of
*V*_{j+1} in *V*_{j}.
This subspace can be generated by a suitable wavelet function
with translation and dilation.

(14.14) |

or

(14.15) |

We compute the scalar products
with:

(14.16) |

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:

(14.17) |

In order to get an exact restoration, two conditions are required for the conjugate filters:

*Dealiasing condition*:

*Exact restoration:*

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 14.3).
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 14.3).

Orthogonal wavelets correspond to the restricted case where:

= | (14.20) | ||

= | (14.21) | ||

= | (14.22) |

and

(14.23) |

We can easily see that this set satisfies the two basic relations 14.18 and 14.19. Daubechies wavelets are the only compact solutions. For biorthogonal wavelets [8] we have the relations:

= | (14.24) | ||

= | (14.25) |

and

(14.26) |

We also satisfy relations 14.18 and 14.19. 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

The 2D algorithm is based on separate variables leading to
prioritizing of x and y directions. The scaling function is defined by:

(14.27) |

The passage from a resolution to the next one is done by:

(14.28) |

The detail signal is obtained from three wavelets:

- a vertical wavelet :

- a horizontal wavelet:

- a diagonal wavelet:

The wavelet transform can be interpreted as the decomposition on frequency sets with a spatial orientation.