Digital Signal Processing Reference
In-Depth Information
c
3
w
3
Horiz. Det.
w
2
j = 3
Horizontal Details
j = 2
w
3
w
3
w
1
Vert. Det.
Diag.Det.
j = 3
j = 3
Horizontal Details
j =1
w
2
w
2
Vertical Details
Diagonal Details
j = 2
j = 2
w
1
w
1
Vertical Details
Diagonal Details
j =1
j =1
Figure 2.6. Discrete wavelet transform represen-
tation of an image with corresponding horizontal,
vertical, diagonal, and approximation subbands.
2.5.3 Two-Dimensional Decimated Wavelet Transform
The preceding DWT algorithm can be extended to any dimension by
separable
(tensor) products of a scaling function
. For instance, the two-
dimensional algorithm is based on separate variables leading to prioritizing of hori-
zontal, vertical, and diagonal directions. The scaling function is defined by
φ
and a wavelet
ψ
φ
(
t
1
,
t
2
)
=
φ
(
t
1
)
φ
(
t
2
), and the passage from one resolution to the next is achieved by
c
j
+
1
[
k
,
l
]
=
h
[
m
−
2
k
]
h
[
n
−
2
l
]
c
j
[
m
,
n
]
m
,
n
[
h h
=
c
j
]
↓
2
,
2
[
k
,
l
]
,
(2.27)
where [
]
↓
2
,
2
stands for the decimation by factor 2 along both
x
- and
y
-axes (i.e.,
only even pixels are kept) and
c
j
.
h
1
h
2
is the 2-D discrete convolution of
c
j
by
the separable filter
h
1
h
2
(i.e., convolution first along the columns by
h
1
and then
convolution along the rows by
h
2
).
The detail coefficient images are obtained from three wavelets:
vertical wavelet:
1
(
t
1
,
ψ
t
2
)
=
φ
(
t
1
)
ψ
(
t
2
)
2
(
t
1
,
horizontal wavelet:
ψ
t
2
)
=
ψ
(
t
1
)
φ
(
t
2
)
3
(
t
1
,
(
t
2
)
which leads to three wavelet subimages (subbands) at each resolution level (see
Fig. 2.6):
diagonal wavelet:
ψ
t
2
)
=
ψ
(
t
1
)
ψ
1
j
[
g h
w
1
[
k
,
l
]
=
g
[
m
−
2
k
]
h
[
n
−
2
l
]
c
j
[
m
,
n
]
=
c
j
]
↓
2
,
2
[
k
,
l
]
,
+
m
,
n
[
h g
2
j
w
1
[
k
,
l
]
=
h
[
m
−
2
k
]
g
[
n
−
2
l
]
c
j
[
m
,
n
]
=
c
j
]
2
[
k
,
l
]
,
↓
2
,
+
m
,
n
j
w
1
[
k
,
l
]
=
g
[
m
−
2
k
]
g
[
n
−
2
l
]
c
j
[
m
,
n
]
=
[
g g
c
j
]
↓
2
,
2
[
k
,
l
]
.
+
m
,
n
