Digital Signal Processing Reference
In-Depth Information
Algorithm 5
2-D UWT Algorithm
Task:
Compute UWT of
N
×
N
-pixel image
X
.
h
.
Parameters:
Filters
h
,
=
=
Initialization:
c
0
X
,
J
log
2
N
.
for
j
=
0
to
J
−
1
do
(
h
(
j
)
h
(
j
)
,
=
,
1. Compute
c
j
+
1
[
k
l
]
c
j
)[
k
l
].
(
g
(
j
)
h
(
j
)
1
2. Compute
w
j
+
1
[
k
,
l
]
=
c
j
)[
k
,
l
].
(
h
(
j
)
g
(
j
)
2
j
3. Compute
w
1
[
k
,
l
]
=
c
j
)[
k
,
l
].
+
3
j
(
g
(
j
)
g
(
j
)
4. Compute
w
1
[
k
,
l
]
=
c
j
)[
k
,
l
].
+
1
2
3
J
J
J
Output:
W ={
w
1
,w
1
,w
1
,...,w
,w
,w
,
c
J
}
, the 2-D UWT of
X
.
Figure 3.3 shows the UWT of the “Einstein” image using five resolution levels.
Figures 3.3(1v), 3.3(1h), and 3.3(1d) correspond to the vertical, horizontal, and diag-
onal coefficients of the first resolution level, respectively. This transformation con-
tains 16 bands, each being of the same size as the original image. The redundancy
factor is therefore equal to 16.
3.3 PARTIALLY DECIMATED WAVELET TRANSFORM
The UWT is highly redundant with a redundancy factor for images of 3
J
+
1, where
N
image and using
six resolution levels, we need to store 19
N
2
real values in memory. When dealing
with very large images, this may not be acceptable in some applications for practical
reasons arising from a computation time constraint or available memory space. In
such cases, a compromise can be found by not decimating one or two coarse scales,
while decimating the others.
We will denote PWT
(
j
u
)
the wavelet transform, where only the first
j
u
resolution
levels are undecimated. For
j
u
equal to 0, PWT
(
j
u
)
corresponds to the biorthogonal
or orthogonal wavelet transform (OWT). Similarly, for
j
u
equal to
J
,PWT
(
J
)
corre-
sponds to the UWT. In general, it is easy to see that PWT
(
j
u
)
J
is the number of resolution levels. This means that for an
N
×
leads to a redundancy
1 in 2-D. For example, PWT
(1)
yields a redundancy
factor
j
u
+
+
1 in 1-D and 3
j
u
factor of 4 in 2-D.
The passage from a resolution
j
to the next resolution will require the same
operations as for the UWT when
j
j
u
. Denoting
j
=
≤
min(
j
,
j
u
), (3.5) becomes
(
h
(
j
)
h
(
j
)
c
j
+
1
[
k
,
l
]
=
c
j
)[
k
,
l
]
,
(
g
(
j
)
h
(
j
)
1
j
+
1
[
k
w
,
l
]
=
c
j
)[
k
,
l
]
,
(3.6)
(
h
(
j
)
g
(
j
)
2
j
w
1
[
k
,
l
]
=
c
j
)[
k
,
l
]
,
+
(
g
(
j
)
g
(
j
)
3
j
w
1
[
k
,
l
]
=
c
j
)[
k
,
l
]
.
+
After the
j
u
scale, the number of holes in the filters
h
and
g
remains unchanged.
