Digital Signal Processing Reference
In-Depth Information
Fig. 13.4
Decomposition of a signal on multiple levels
Fig. 13.5
Reconstruction or synthesis
2 symbol represents a decimation of factor 2, that is, accounting for the
use of every other (rather than every) sample.
Fast wavelet transforms are therefore calculated by the cascade of filterings by
h
and
The
↓
g
followed by subsampling (or decimation) by a factor of 2. Initialization of
the algorithm can present certain difficulties; however, it is possible to assimilate
the sampled values of the signal
X
with the coefficients
a
0
. The complexity of this
algorithm is of the order of
N
when the signal
X
is of size
N
.
Reconstruction or synthesis consists of an interpolation which inserts zeros in the
sequences
a
j
+
1
and
d
j
+
1
to double their length, followed by a filter, as shown in
Fig.
13.5
.
The
2 symbol represents an interpolation which inserts zeros between the sam-
ples of
a
j
+
1
and
d
j
+
1
.
Strictly speaking, the sequences
a
j
and
d
j
are respectively
a
j,k
=
↑
a
j
(k)
and
d
j,k
=
d
j
(k)
; where
k
is the time. There is a relationship between
h(k)
and
g(k)
:
1
)
k
h(
1
=
−
−
g(k)
(
k),
(13.9)
and the following notation will be adopted:
h(k)
k)
.Intwo
dimensions, decomposition into a separable wavelet basis is realized by an extension
of the Mallat algorithm. Therefore, for an image, the 1D algorithm is applied first,
to each row and then to each column, as shown in Fig.
13.6
.
=
h(
−
k)
and
g(k)
¯
=
g(
−
13.2 Redundant Multiresolution Analysis
When high accuracy is required in image analysis, redundant multiresolution anal-
ysis is applied, such as the undecimated discrete wavelet transform, the wavelet
packet transform, and the contourlet transform. This section introduces these trans-
forms and shows their benefits.
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