Digital Signal Processing Reference
In-Depth Information
Fig. 13.3
Discretization
process: (
a
) continuous
wavelet transform,
(
b
) discretization of the
(u,s)
-plane, (
c
) discrete
wavelet transform
To profit from a non-redundant signal representation while ensuring a perfect
reconstruction from its decomposition, an extremely effective tool, i.e., MRA, was
defined by Stéphane Mallat [
2
] and Yves Meyer [
9
]. This powerful concept allows
numerical implementation of wavelet decomposition; the definition of the discrete
wavelet transform thus necessarily underpins that of the MRA.
13.1.4 The Concept of MRA
The idea of multiresolution analysis of a signal consists in its representation as a
limit of its successive approximations, where each approximation is a smoothed
version of the preceding approximation. Successive approximations are presented
at different resolutions, hence the term multiresolution analysis.
When resolution increases, successive images approximate the signal increas-
ingly better, and in contrast, when resolution decreases, the amount of information
contained in an image also decreases, eventually to zero. The wavelet coefficients
encode the difference in information between two successive images, that is, the
details acquired by an image when its resolution doubles.
Readers interested in the theoretical aspects will find the axiomatic formulation
of MRA in references [
2
] and [
10
].
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