tween no-tiling and tiling at 64 × 64, while at 0.5 b/p this
difference is reduced to approximately 1.5 dB.
DC Level Shifting
Prior to computation of the forward discrete wavelet
transform (DWT) on each image tile, all samples of the
image tile component are dc level shifted by subtracting
the same quantity 2
P-1
, where P is the component’s preci
-
sion. DC level shifting is performed on samples of com
-
ponents that are unsigned only. Level shifting does not
affect variances. It actually converts an unsigned repre
-
sentation to a two’s complement representation, or vice
versa [55], [56]. If color transformation is used, dc level
shifting is performed prior to the computation of the for
-
ward component transform (Figs.
3 and 5). At the decoder side, in
-
verse dc level shifting is performed
on reconstructed samples by add
-
ing to them the bias 2
P-1
after the
computation of the inverse com
-
ponent transform.
Component Transformations
JPEG 2000 supports multi
-
ple-component images. Different
components need not have the
same bit depths nor need to all be
signed or unsigned [38], [39]. For
reversible (i.e., lossless) systems,
the only requirement is that the bit
depth of each output image com
-
ponent must be identical to the bit
depth of the corresponding input
image component.
Component transformations
improve compression and allow
for visually relevant quantization.
The standard supports two differ-
ent component transformations,
one irreversible component trans-
formation (ICT) that can be used
for lossy coding and one reversible
component transformation (RCT) that may be used for
lossless or lossy coding, and all this in addition to encod-
ing without color transformation. The block diagram of
the JPEG 2000 multicomponent encoder is depicted in
Fig. 5. (Without restricting the generality, only three
components are shown in the figure. These components
could correspond to the RGB of a color image.)
Since the ICT may only be used for lossy coding, it
may only be used with the 9/7 irreversible wavelet trans
-
form. (See also next section.) The forward and the inverse
ICT transformations are achieved by means of (1a) and
(1b), respectively [7], [38], [56]
Y
C
C
b
r
=− −
0299 0587 0114
016875 033126 05
...
.. .
05 0 41869 008131.. .−−
⋅
R
G
B
(1a)
R
G
B
=− −
10 0 1 402
10 034413 071414
10 1772
..
.. .
.. 0
⋅
Y
C
C
b
r
.
(1b)
Since the RCT may be used for lossless or lossy coding,
it may only be used with the 5/3 reversible wavelet trans
-
form. (See also next section). The RCT is a decorrelating
transformation, which is applied to the three first compo
-
nents of an image. Three goals are achieved by this trans
-
formation, namely, color decorrelation for efficient
40 IEEE SIGNAL PROCESSING MAGAZINE SEPTEMBER 2001
(a) (b)
c
d
▲ 4. Image :”ski” of size 720 × 576 (courtesy of Phillips Research, UK): (a) original image,
(b)-(d) reconstructed images after JPEG 2000 compression at 0.25 bpp: (b) without tiling,
(c) with 128 × 128 tiling, and (d) with 64 × 64 tiling.
Table 1. The Effect of Tiling on Image Quality.
Tiling
No Tiling
Tiles of
Size
128 × 128
Tiles of
Size
64 × 64
Bit Rate
(b/p)
0.125 24.75 23.42 20.07
0.25 26.49 25.69 23.95
0.5 28.27 27.79 26.80
PSNR (in dB) for the color image “ski” (of size 720 × 576
pixels per component)