Digital Signal Processing Reference
In-Depth Information
all recursive dyadic partitions of the frequency axis. Doing this by brute force is,
however, computationally prohibitive, requiring us to explore more than 2
2
J
−
1
trees
T
. The construction of the best basis can be accomplished much more efficiently in
O
(
N
log
N
) computation using the fast bottom-up algorithm introduced by Coif-
man and Wickerhauser (1992). This fast algorithm exploits the tree structure of the
wavelet packet dictionary and requires the recursive comparison of the cost function
of the child nodes to their parent nodes. See Coifman and Wickerhauser (1992) for
details.
2.8.5 Two-Dimensional Wavelet Packets
The wavelet packets can be extended to any dimension by separable (tensor) prod-
ucts. In 2-D, this leads to wavelet packets that are associated with quadtrees, and
with octrees in 3-D. For instance, in 2-D, wavelet packets are obtained by tensor
products of the functions
p
j
q
j
ψ
k
and
ψ
l
. The recursive relations between the wavelet
,
,
packet coefficients are given by
2
p
2
q
j
+
1
,
[
h h
p
q
j
+
1
]
↓
2
,
2
[
k
,
w
[
k
,
l
]
=
w
,
l
]
,
2
p
+
1
,
2
q
[
g h
p
q
j
+
,
w
[
k
,
l
]
=
w
1
]
2
[
k
,
l
]
,
↓
2
,
j
+
1
2
p
,
2
q
+
1
p
,
q
[
h g
w
[
k
,
l
]
=
w
1
]
↓
2
,
2
[
k
,
l
]
,
j
+
1
j
+
2
p
+
1
,
2
q
+
1
p
,
q
w
[
k
,
l
]
=
[
g g
w
1
]
↓
2
,
2
[
k
,
l
]
,
j
+
1
j
+
and the reconstruction is obtained from
p
q
j
+
1
=
,
2
p
2
q
j
+
1
+
,
2
p
+
1
,
2
q
2
p
,
2
q
+
1
2
p
+
1
,
2
q
+
1
w
4(
hh
w
˘
gh
w
˘
+
hg
w
˘
+
hg
w
˘
)[
k
,
l
]
.
j
+
1
j
+
1
j
+
1
2.9 GUIDED NUMERICAL EXPERIMENTS
2.9.1 Software
In the pedagogical numerical experiments of several chapters in this topic, we have
chosen the WaveLab toolbox (Buckheit and Donoho 1995). WaveLab contains a
fairly complete offering and a unified set of wavelet and time-frequency tools. It is
based on the MATLAB quantitative computing and interactive visualization envi-
ronment. WaveLab is installed as a MATLAB toolbox and has been available on-
line in some version since 1993. The latest version is WaveLab850, consisting of over
1,200 files, including programs, data, documentation, and scripts, which can be freely
retrieved from a link at this topic's Web site (http://www.SparseSignalRecipes.info;
http://www-stat.stanford.edu/
wavelab). Versions are provided for Unix, Linux,
Macintosh, and Windows platforms. WaveLab has become one of the most widely
∼
