Digital Signal Processing Reference
In-Depth Information
Fig. 13.6
The Mallat algorithm: decomposition of an image
13.2.1 Undecimated Discrete Wavelet Transform
The undecimated discrete wavelet transform (UDWT), also known as the stationary
wavelet transform, consists of keeping the filter bank construction which provides
a fast and dyadic algorithm, e.g., Mallat algorithm, but eliminating the decimation
step.
Due to the absence of downsamplers (decimation step) in the UDWT's imple-
mentation, each coefficient sequence from any level of decomposition has the same
length as the original: if the original signal has
N
samples, the UDWT
J
-level repre-
sentation
{
a
J
(k),d
j
(k)
}
0
<j<J
is of size
N(J
+
1
)
, making from the UDWT
J
-level
a highly redundant representation.
The implementation of the UDWT was initially performed by an algorithm called
the
à trous
algorithm (
à trous
, a French term, meaning
with holes
).
13.2.2 Wavelet Packets
Wavelet packets provide a finer analysis by decomposing, at each level, not only
the approximation spaces but also the detail spaces (see Fig.
13.7
). Wavelet packets,
defined by Coifman, Meyer, and Wickerhauser [
11
], therefore represent a general-
ization of multiresolution decomposition.
The wavelet packets transform is redundant and should only be used in cases
where an extremely fine analysis is required. The choice of the best decomposition
basis depends on the principle of minimal entropy.
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