Digital Signal Processing Reference
In-Depth Information
The gain in PSNR when using the UWT (
j
u
=
4) instead of the biorthogonal
43 dB. Using a single undecimated scale, that is, PWT
(1)
,
leads to a PSNR increase of more than 1 dB, while requiring far less redundancy.
wavelet transform is 2
.
3.4 THE DUAL-TREE COMPLEX WAVELET TRANSFORM
In this section, we describe this transform only in the 2-D case. To obtain near-
translation invariance and directional selectivity with only one undecimated scale,
an additional refinement can be introduced to the PWT
(1)
by considering two pairs
of odd- and even-length biorthogonal linear-phase filters (
h
o
g
e
), in-
stead of one. This transform is called the
dual-tree complex wavelet transform
(Kingsbury 1998, 1999; Selesnick et al. 2005). The filters are real, but complex num-
bers are derived from the dual-tree wavelet coefficients. As depicted in Fig. 3.4, at
the first scale, a UWT is first computed on the image
c
0
using the filter pair (
h
o
,
g
o
) and (
h
e
,
g
o
),
which results in four subbands, each of the same size as
c
0
(i.e., the redundancy fac-
tor is equal to 4). Then the scaling (smooth) coefficients
c
1
are split into four parts
to extract four trees:
,
1.
c
1
=
[
c
1
]
↓
2
e
2
e
: coefficients at even row index and even column index
↓
2.
c
1
=
[
c
1
]
↓
2
e
↓
2
o
: coefficients at odd row index and even column index
3.
c
1
=
[
c
1
]
↓
2
o
2
e
: coefficients at even row index and odd column index
↓
4.
c
1
=
[
c
1
]
↓
2
o
↓
2
o
: coefficients at odd row index and odd column index
c
1
,
These four images
c
1
,
c
1
are decomposed, using the DWT, by separable
products of the odd- and even-length filter pairs.
c
1
,
Tree
T
ABCD
h
e
h
e
h
e
h
o
h
o
h
e
h
o
h
o
c
j
+
1
g
e
h
e
g
e
h
o
g
o
h
e
g
o
h
o
w
1
j
+
1
,
T
h
e
g
e
h
e
g
o
h
o
g
e
h
o
g
o
j
w
+
1
,
T
3
j
+
1
,
T
g
e
g
e
g
e
g
o
g
o
g
e
g
o
g
o
w
q
j
q
j
q
j
q
j
For each subband, the wavelet coefficients
w
,w
,w
,w
D
,
q
∈{
1
,
2
,
3
}
,are
,
A
,
B
,
C
,
combined to form real and imaginary parts of complex wavelet coefficients:
q
j
,
+
q
j
,
A
[
k
q
j
,
D
[
k
q
j
,
B
[
k
q
j
,
C
[
k
w
[
k
,
l
]
=
(
w
,
l
]
−
w
,
l
])
+
i
(
w
,
l
]
+
w
,
l
])
(3.8)
q
j
,
−
q
j
,
A
[
k
q
j
,
D
[
k
q
j
,
B
[
k
q
j
,
C
[
k
w
[
k
,
l
]
=
(
w
,
l
]
+
w
,
l
])
+
i
(
w
,
l
]
−
w
,
l
])
.
Therefore the three detail subbands lead to six complex subbands. Thus, in addi-
tion to near-translation invariance, the dual-tree complex transform has a better
directional selectivity compared to the three-orientation of the separable real
wavelet transform. This is achieved while maintaining a reasonable redundancy (4
in 2-D and 2
d
in
d
-D) and allowing fast finite impulse response filter bank-based
transform and reconstruction. The fast dual-tree complex wavelet transform and
reconstruction algorithms are pictorially detailed in Figs. 3.4 and 3.5.
