Databases Reference
In-Depth Information
F I GU R E 16 . 5
Reconstruction of Sinan image encoded using 0.5 bits per pixel and
the four-tap Daubechies filter.
16.3 Embedded Zerotree Coder
The embedded zerotree wavelet (EZW) coder was introduced by Shapiro [
215
]. It is a quan-
tization and coding strategy that incorporates some characteristics of the wavelet decomposi-
tion. Just as the quantization and coding approaches used in the JPEG standard, which were
motivated by the characteristics of the coefficients, were superior to the generic zonal cod-
ing algorithms, the EZW approach and its descendants significantly outperform some of the
generic approaches. The particular characteristic used by the EZW algorithm is that there are
wavelet coefficients in different subbands that represent the same spatial location in the image.
If the decomposition is such that the size of the different subbands is different (the first two
decompositions in Figure
16.3
), then a single coefficient in the smaller subband may represent
the same spatial location as multiple coefficients in the other subbands.
In order to put our discussion on more solid ground, consider the ten-band decomposition
shown in Figure
16.6
. The coefficient
a
in the upper-left corner of Band I represents the
same spatial location as coefficients
a
1
in Band II,
a
2
in Band III, and
a
3
in Band IV. In turn,
the coefficient
a
1
represents the same spatial location as coefficients
a
11
,
a
13
, and
a
14
in Band V. Each of these pixels represents the same spatial location as four pixels in Band
VIII, and so on. In fact, we can visualize the relationships of these coefficients in the form
of a tree. The coefficient
a
forms the root of the tree with three descendants
a
1
,
a
12
,
a
2
, and
a
3
.
The coefficient
a
1
has descendants
a
11
,
a
12
,
a
13
, and
a
14
. The coefficient
a
2
has descendants
a
21
,
a
33
, and
a
34
. Each of
these coefficients in turn has four descendants, making a total of 64 coefficients in this tree. A
pictorial representation of the tree is shown in Figure
16.7
.
a
22
,
a
23
, and
a
24
, and the coefficient
a
3
has descendants
a
31
,
a
32
,
