Digital Signal Processing Reference
In-Depth Information
For 3-D data, seven wavelet subcubes are created at each resolution level, corre-
sponding to an analysis in seven directions.
For a discrete
N
×
N
image
X
, the algorithm is the following.
Algorithm 3
2-D DWT Algorithm
Task:
Compute DWT of a discrete image
X
.
Parameters:
Filters
h
h
.
,
Initialization:
c
0
=
X
,
J
=
log
2
N
.
for
j
=
0
to
J
−
1
do
h h
Compute
c
j
+
1
=
c
j
, down-sample by a factor 2 in each dimension.
g h
1
Compute
w
j
+
1
=
c
j
, down-sample by a factor 2 in each dimension.
h g
2
j
Compute
w
1
=
c
j
, down-sample by a factor 2 in each dimension.
+
3
j
Compute
w
=
g g
c
j
, down-sample by a factor 2 in each dimension.
+
1
1
1
2
1
3
1
1
J
2
J
3
J
Output:
W ={
w
,w
,w
,...,w
,w
,w
,
c
J
}
, the 2D DWT of
X
.
The reconstruction is obtained by
4(
h h
g h
1
j
+
1
h g
2
j
+
1
3
j
+
1
)
c
j
=
+
w
+
w
+
w
c
j
+
1
˘
˘
g g
˘
(2.28)
in a similar way to the 1-D case and with the proper generalization to 2-D.
Figure 2.7 shows the image “Einstein” (top right), the schematic separation of
the wavelet decomposition bands (top left), and the actual DWT coefficients (bot-
tom left) using the 7
9 filters (Antonini et al. 1992). The application of the DWT
to image compression, using the 7
/
9 filters (Antonini et al. 1992), leads to impres-
sive results compared to previous methods like JPEG. The inclusion of the wavelet
transform in JPEG 2000, the recent still-picture compression standard, testifies to
this lasting and significant impact. Figure 2.7 (bottom right) shows the decompressed
image for a compression ratio of 40, and as can be seen, the result is near-perfect.
/
2.6 NONDYADIC RESOLUTION FACTOR
Feauveau (1990) introduced
quincunx
analysis, based on Adelson's work (Adelson
et al. 1987). This analysis is
n
ot dyadic and allows an image decomposition with a
resolution factor equal to
√
2.
The advantage is that only one nonseparable wavelet is needed. At each step,
the image is undersampled by 2 in one direction (first and second direction, alter-
natively). This subsampling is made by keeping one pixel out of two, alternatively
even and odd. The following conditions must be satisfied by the filters:
ˆ
h
h
(
g
g
(
1
2
,ν
2
+
1
2
1
2
,ν
2
+
1
2
ν
1
+
ν
1
,ν
2
)
+
ν
1
+
ν
1
,ν
2
)
=
0
ν
1
,ν
2
)
h
(
ˆ
h
(
ν
1
,ν
2
)
g
(
ν
1
,ν
2
)
+
g
(
ν
1
,ν
2
)
=
1
.
