Digital Signal Processing Reference
In-Depth Information
Algorithm 6
2-D Dual-Tree Complex Wavelet Transform Algorithm
Task:
Compute dual-tree complex wavelet transform of
N
×
N
image
X
.
Parameters:
Filters
h
o
h
e
g
o
g
e
.
,
,
,
=
=
Initialization:
c
0
log
2
N
.
1. Compute the one-scale undecimated wavelet
X
,
J
1
2
3
W ={
w
1
,w
1
,w
1
,
c
1
}
:
h
(0)
h
(0)
c
1
=
c
0
,
1
1
g
(0)
h
(0)
w
=
c
0
,
h
(0)
g
(0)
1
w
=
c
0
,
3
1
g
(0)
g
(0)
w
=
c
0
.
2. Down-sample
c
1
along rows and columns to initialize the trees:
c
1
=
[
c
1
]
↓
2
e
,
2
e
↓
c
1
=
[
c
1
]
↓
2
e
,
↓
2
o
c
1
=
[
c
1
]
↓
2
o
↓
2
e
,
c
1
=
.
[
c
1
]
↓
2
o
↓
2
o
3. Compute the 2-D DWT of
c
1
with the filter pair (
h
e
g
e
). Get the coefficients
,
q
j
c
J
,w
}
2
≤
j
≤
J
,
q
∈{
1
,
2
,
3
}
.
4. Compute the 2-D DWT of
c
1
with the filter pair (
h
e
{
,
A
g
o
). Get the coefficients
,
q
j
c
J
,w
}
2
≤
j
≤
J
,
q
∈{
1
,
2
,
3
}
.
5. Compute the 2-D DWT of
c
1
with the filter pair (
h
o
{
,
B
,
g
e
). Get the coefficients
q
j
c
J
,w
.
6. Compute the 2-D DWT of
c
1
with the filter pair (
h
o
{
}
2
≤
j
≤
J
,
q
∈{
1
,
2
,
3
}
,
C
,
g
o
). Get the coefficients
q
j
c
J
,w
.
7. Form the wavelet complex coefficients
{
}
2
≤
j
≤
J
,
q
∈{
1
,
2
,
3
}
,
D
q
j
,
+
q
j
,
−
w
and
w
, according to equation
(3.8).
q
j
q
j
c
J
,w
Output:
W ={
,
+
,w
,
−
}
, the 2-D dual-tree complex wavelet
1
≤
j
≤
J
,
q
∈{
1
,
2
,
3
}
transform of
X
.
3.5 ISOTROPIC UNDECIMATED WAVELET TRANSFORM:
STARLET TRANSFORM
The isotropic undecimated wavelet transform (IUWT) algorithm is well known in
the astronomical domain because it is well adapted to astronomical data, where ob-
jects are more or less isotropic in most cases (Starck and Murtagh 1994; Starck and
Murtagh 2006). For this reason, and also because there is often confusion between
the UWT and IUWT, we call it the
starlet wavelet transform
.
Requirements for a good analysis of such data are as follows:
Filters must be symmetric (
h
=
h
, and
g
=
g
).
In 2-D or higher dimensions,
h
,
g
,ψ,φ
must be nearly isotropic.
