An overview of the JPEG 2000 still image compression standard

An overview of the JPEG 2000 still image compression standard
Signal Processing: Image Communication 17 (2002) 3–48
An overview of the JPEG 2000 still image
compression standard
Majid Rabbani*, Rajan Joshi
Eastman Kodak Company, Rochester, NY 14650, USA
Abstract
In 1996, the JPEG committee began to investigate possibilities for a new still image compression standard to serve
current and future applications. This initiative, which was named JPEG 2000, has resulted in a comprehensive standard
(ISO 154447ITU-T Recommendation T.800) that is being issued in six parts. Part 1, in the same vein as the JPEG
baseline system, is aimed at minimal complexity and maximal interchange and was issued as an International Standard
at the end of 2000. Parts 2–6 define extensions to both the compression technology and the file format and are currently
in various stages of development. In this paper, a technical description of Part 1 of the JPEG 2000 standard is provided,
and the rationale behind the selected technologies is explained. Although the JPEG 2000 standard only specifies the
decoder and the codesteam syntax, the discussion will span both encoder and decoder issues to provide a better
understanding of the standard in various applications. r 2002 Elsevier Science B.V. All rights reserved.
Keywords: JPEG 2000; Image compression; Image coding; Wavelet compression
1. Introduction and background
The Joint Photographic Experts Group (JPEG)
committee was formed in 1986 under the joint
auspices of ISO and ITU-T1 and was chartered
with the ‘‘digital compression and coding of
continuous-tone still images’’. The committee’s first
published standard [55,32], commonly known as
the JPEG standard,2 provides a toolkit of compression techniques from which applications can
*Corresponding author.
E-mail address: [email protected] (M. Rabbani).
1
Formerly known as the Consultative Committee for
International Telephone and Telegraph (CCITT).
2
Although JPEG Part 1 became an International Standard in
1993, the technical description of the algorithm was frozen as
early as 1988.
select various elements to satisfy particular requirements. This toolkit includes the following
components: (i) the JPEG baseline system, which
is a simple and efficient discrete cosine transform
(DCT)-based lossy compression algorithm that
uses Huffman coding, operates only in sequential
mode, and is restricted to 8 bits/pixel input; (ii) an
extended system, which introduces enhancements
to the baseline algorithm to satisfy a broader set of
applications; and (iii) a lossless mode, which is
based on a predictive coding approach using either
Huffman or arithmetic coding and is independent
of the DCT. The JPEG baseline algorithm has
since enjoyed widespread use in many digital
imaging applications. This is due, in part, to its
technical merits and status as a royalty-free
international standard, but perhaps more so, it is
0923-5965/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 3 - 5 9 6 5 ( 0 1 ) 0 0 0 2 4 - 8
4
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
due to the free and efficient software that is
available from the Independent JPEG Group
(IJG) [57].
Despite the phenomenal success of the JPEG
baseline system, it has several shortcomings that
become increasingly apparent as the need for
image compression is extended to such emerging
applications as medical imaging, digital libraries,
multimedia, internet and mobile. While the extended JPEG system addresses some of these
shortcomings, it does so only to a limited extent
and in some cases, the solutions are hindered by
intellectual property rights (IPR) issues. The desire
to provide a broad range of features for numerous
applications in a single compressed bit-stream
prompted the JPEG committee in 1996 to investigate possibilities for a new compression standard
that was subsequently named JPEG 2000. In
March 1997 a call for proposals was issued
[58,59], seeking to produce a standard to ‘‘address
areas where current standards failed to produce the
best quality or performance’’, ‘‘provide capabilities
to markets that currently do not use compression’’,
and ‘‘provide an open system approach to imaging
applications’’.
In November 1997, more than 20 algorithms
were evaluated, and a wavelet decomposition
approach was adopted as the backbone of the
new standard. A comprehensive requirements
document was developed that defined all the
various application areas of the standard, along
with a set of mandatory and optional requirements
for each application. In the course of the ensuing
three years, and after performing hundreds of
technical studies known as ‘‘core experiments’’, the
standard evolved into a state-of-the-art compression system with a diverse set of features, all of
which are supported in a single compressed bitstream.
The JPEG 2000 standard is scheduled to be
issued in six parts. Part 1, in the same vein as the
JPEG baseline system, defines a core coding
system that is aimed at minimal complexity while
satisfying 80% of the applications [60]. In addition, it defines an optional file format that includes
essential information for the proper rendering of
the image. It is intended to be available on a
royalty and fee-free basis and was issued as an
International Standard (IS) in December 2000.
Parts 2–6 define extensions to both the compression technology and the file format and are in
various stages of development. The history and the
timeline of the various parts of the standard are
shown in Table 1.
Part 2 is aimed at enhancing the performance of
Part 1 with more advanced technology, possibly at
the expense of higher complexity [61]. It is
intended to serve those applications where maximal interchange is less important than meeting
specific requirements. The codestream generated
by part 2 encoders is usually not decodable by Part
1 decoders, and some of the technology in Part 2
might be protected by IPR. Part 3 defines motion
JPEG 2000 (MJP2) and is primarily based on the
technology in Part 1 with the addition of a file
format [62]. It results in an encoder that is
significantly less complex than the popular MPEG
family of standards (due to lack of motion
estimation) and provides full random access to
the individually coded frames (albeit at the
Table 1
Timeline of the various parts of the JPEG 2000 standarda
Part
Title
CFP
WD
CD
FCD
FDIS
IS
1
2
3
4
5
6
JPEG 2000 image coding system: core coding system
JPEG 2000 image coding system: extensions
Motion JPEG 2000
Conformance testing
Reference software
Compound image file format
97/03
97/03
99/12
99/12
99/12
97/03
99/03
00/03
00/07
00/07
00/03
00/12
99/12
00/08
00/12
00/12
00/07
01/03
00/03
00/12
01/03
01/07
00/12
01/11
00/10
01/07
01/07
01/11
01/08
02/03
00/12
01/10
01/10
02/03
01/11
02/05
a
CFP=Call for Proposals, WD=Working Draft, CD=Committee Draft, FCD=Final Committee Draft, FDIS=Final Draft
International Standard, IS=International Standard.
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
expense of compression efficiency). It is intended
for applications such as digital still cameras with
burst capture mode, video editing in post-production environments, and digital cinema archive and
distribution. Part 4 defines conformance testing
[63], similar to the role of JPEG Part 2, to ensure
a high-quality implementation of the standard.
As mentioned earlier, a key factor in the JPEG
baseline system’s success as a widely used
standard was the availability of efficient and free
software. Part 5 defines a reference software
implementation for Part 1 of the JPEG 2000
standard [64]. Currently, two implementations are
available. One is a Java implementation by the
JJ2000 group [65] consisting of Canon Research
France, Ericsson and EPFL. The other is a C
implementation by Image Power and University of
British Columbia [2]. Finally, Part 6 defines a
compound image file format for document scanning and fax applications [66].
It is noteworthy that the real incentive behind
the development of the JPEG 2000 system was not
just to provide higher compression efficiency
compared to the baseline JPEG system. Rather,
it was to provide a new image representation with
a rich set of features, all supported within the same
compressed bit-stream, that can address a variety
of existing and emerging compression applications. In particular, the Part 1 of the standard
addresses some of the shortcomings of baseline
JPEG by supporting the following set of features:
* Improved compression efficiency.
* Lossy to lossless compression.
* Multiple resolution representation.
* Embedded bit-stream (progressive decoding
and SNR scalability).
* Tiling.
* Region-of-interest (ROI) coding.
* Error resilience.
* Random codestream access and processing.
* Improved performance to multiple compression/decompression cycles.
* A more flexible file format.
The JPEG 2000 standard makes use of several
recent advances in compression technology in
order to achieve these features. For example, the
low-complexity and memory efficient block DCT
of JPEG has been replaced by the full-frame
5
discrete wavelet transform (DWT). The DWT
inherently provides a multi-resolution image representation while also improving compression
efficiency due to good energy compaction and the
ability to decorrelate the image across a larger
scale. Furthermore, integer DWT filters can be
used to provide both lossless and lossy compression within a single compressed bit-stream.
Embedded coding is achieved by using a uniform
quantizer with a central deadzone (with twice the
step-size). When the output index of this quantizer
is represented as a series of binary symbols, a
partial decoding of the index is equivalent to using
a quantizer with a scaled version of the original
step-size, where the scaling factor is a power of
two. To encode the binary bitplanes of the
quantizer index, JPEG 2000 has replaced the
Huffman coder of baseline JPEG with a contextbased adaptive binary arithmetic coder with
renormalization-driven probability estimation,
known as the MQ coder. The embedded bitstream that results from bitplane coding
provides SNR scalability in addition to the
capability of compressing to a target file size.
Furthermore, the bitplanes in each subband are
coded in independent rectangular blocks and in
three fractional bitplane passes to provide an
optimal embedded bit-stream, improved error
resilience, partial spatial random access, ease of
certain geometric manipulations, and an extremely flexible codestream syntax. Finally, the
introduction of a canvas coordinate system
facilitates certain operations in the compressed
domain such as cropping, rotations by multiples
of 901, flipping, etc.
Several excellent review papers about JPEG
2000 Part 1 have recently appeared in the literature
[3,13,16,18,27,37], and a comprehensive book
describing all of the technical aspects of the
standard has been published [45]. In this paper, a
technical description of the fundamental building
blocks of JPEG 2000 Part 1 is provided, and the
rationale behind the selected technologies is
explained. Although the JPEG 2000 standard only
specifies the decoder and the codestream syntax,
many specific decoder implementation issues have
been omitted in our presentation in the interest of
brevity. Instead, the emphasis has been placed on
6
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
general encoder and decoder technology issues to
provide a better understanding of the standard in
various applications. Therefore, readers who plan
on implementing the standard should ultimately
refer to the actual standard [60].
This paper is organized as follows. In Section 2,
the fundamental building blocks of the JPEG 2000
Part 1 standard, such as pre-processing, DWT,
quantization, and entropy coding are described. In
Section 3, the syntax and organization of the
compressed bit-stream is explained. In Section 4,
various rate control strategies that can be used by
the JPEG 2000 encoder for achieving an optimal
SNR or visual quality for a given bit-rate are
discussed. In Section 5, the tradeoffs between the
various choices of encoder parameters are illustrated through an extensive set of examples.
Finally, Section 6 contains a brief description
of some additional JPEG 2000 features such as
ROI, error resilience and file format, as well as a
summary of the technologies used in Part 2.
2. JPEG 2000 fundamental building blocks
The fundamental building blocks of a typical
JPEG 2000 encoder are shown in Fig. 1. These
components include pre-processing, DWT, quantization, arithmetic coding (tier-1 coding), and bitstream organization (tier-2 coding). In the following, each of these components is discussed in more
detail.
The input image to JPEG 2000 may contain one
or more components. Although a typical color
image would have three components (e.g., RGB or
YCb Cr ), up to 16 384 (214) components can be
specified for an input image to accommodate
multi-spectral or other types of imagery. The
sample values for each component can be either
signed or unsigned integers with a bit-depth in the
range of 1–38 bits. Given a sample with a bit-depth
Original
Image Data
PreProcessing
Discrete Wavelet
Transform (DWT)
of B bits, the unsigned representation would
correspond to the range (0, 2B1), while the signed
representation would correspond to the range
(2B1, 2B11). The bit-depth, resolution, and
signed versus unsigned specification can vary for
each component. If the components have different
bit-depths, the most significant bits of the components should be aligned to facilitate distortion
estimation at the encoder.
2.1. Pre-processing
The first step in pre-processing is to partition the
input image into rectangular and non-overlapping
tiles of equal size (except possibly for those tiles at
the image borders). The tile size is arbitrary and
can be as large as the original image itself (i.e.,
only one tile) or as small as a single pixel. Each tile
is compressed independently using its own set of
specified compression parameters. Tiling is particularly useful for applications where the amount
of available memory is limited compared to the
image size.
Next, unsigned sample values in each component are level shifted (DC offset) by subtracting a
fixed value of 2B1 from each sample to make its
value symmetric around zero. Signed sample
values are not level shifted. Similar to the level
shifting performed in the JPEG standard, this
operation simplifies certain implementation issues
(e.g., numerical overflow, arithmetic coding context specification, etc.), but has no effect on the
coding efficiency. Part 2 of the JPEG 2000
standard allows for a generalized DC offset, where
a user-defined offset value can be signaled in a
marker segment.
Finally, the level-shifted values can be subjected
to a forward point-wise intercomponent transformation to decorrelate the color data. One restriction on applying the intercomponent transformation is that the components must have identical
Uniform Quantizer
with Deadzone
Adaptive Binary
Arithmetic Coder
(Tier-1 Coding)
Fig. 1. JPEG 2000 fundamental building blocks.
Compressed
Bit-stream
Image Data
Organization
(Tier-2 Coding)
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
bit-depths and dimensions. Two transform choices
are allowed in Part 1, where both transforms
operate on the first three components of an image
tile with the implicit assumption that these
components correspond to red–green–blue
(RGB). One transform is the irreversible color
transform (ICT), which is identical to the traditional RGB to YCb Cr color transformation and
can only be used for lossy coding. The forward
ICT is defined as
0
1 0
Y
0:299
B C B
@ Cb A ¼ @ 0:16875
0:500
Cr
0:587
0:33126
0:114
0:500
0:41869
0:08131
This can alternatively be written as
Cr ¼ 0:713ðR YÞ;
R
1
0
1:0
0
B C B
@ G A ¼ @ 1:0 0:34413
B
1:0
1:772
1:402
1 0
Y
1
C B C
0:71414 [email protected] Cb A:
0
Cr
ð2Þ
The other transform is the reversible color transform (RCT), which is a reversible integer-tointeger transform that approximates the ICT for
color decorrelation and can be used for both
lossless and lossy coding. The forward RCT is
defined as
Y¼
R þ 2G þ B
;
4
U ¼ R G;
At the decoder, the decompressed image is
subjected to the corresponding inverse color
transform if necessary, followed by the removal
of the DC level shift. Since each component of
each tile is treated independently, the basic
1 0
1
R
C B C
A @ G A:
ð1Þ
B
2.2. The discrete wavelet transform (DWT)
while the inverse ICT is given by
0
recovering the original RGB data, is given by
U þV
; R ¼ U þ G; B ¼ V þ G:
G¼Y
4
ð4Þ
compression engine for JPEG 2000 will only be
discussed with reference to a single tile of a
monochrome image.
Y ¼ 0:299ðR GÞ þ G þ 0:114ðB GÞ;
Cb ¼ 0:564ðB YÞ;
7
ð3Þ
V ¼ B G;
where bwc denotes the largest integer that is smaller
than or equal to w: The Y component has the same
bit-depth as the RGB components while the U
and V components have one extra bit of precision.
The inverse RCT, which is capable of exactly
The block DCT transformation in baseline
JPEG has been replaced with the full frame
DWT in JPEG 2000. The DWT has several
characteristics that make it suitable for fulfilling
some of the requirements set forth by the JPEG
2000 committee. For example, a multi-resolution
image representation is inherent to DWT. Furthermore, the full-frame nature of the transform
decorrelates the image across a larger scale and
eliminates blocking artifacts at high compression
ratios. Finally, the use of integer DWT filters
allows for both lossless and lossy compression
within a single compressed bit-stream. In the
following, we first consider a one-dimensional (1D) DWT for simplicity, and then extend the
concepts to two dimensions.
2.2.1. The 1-D DWT
The forward 1-D DWT at the encoder is best
understood as successive applications of a pair of
low-pass and high-pass filters, followed by downsampling by a factor of two (i.e., discarding odd
indexed samples) after each filtering operation as
shown in Fig. 2. The low-pass and high-pass filter
pair is known as analysis filter-bank. The low-pass
8
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
h0
2
2
g0
xˆ ( n )
x(n)
h1
2
Analysis filter bank
2
g1
Synthesis filter bank
Fig. 2. 1-D, 2-band wavelet analysis and synthesis filter-bank.
filter preserves the low frequencies of a signal while
attenuating or eliminating the high frequencies,
thus resulting in a blurred version of the original
signal. Conversely, the high-pass filter preserves
the high frequencies in a signal such as edges,
texture and detail, while removing or attenuating
the low frequencies.
Consider a 1-D signal xðnÞ (such as the pixel
values in a row of an image) and a pair of low-pass
and high-pass filters designated by h0 ðnÞ and h1 ðnÞ;
respectively. An example of a low-pass filter is
h0(n)=(1 2 6 2 1)/8, which is symmetric and
has five integer coefficients (or taps). An example
of a high-pass filter is h1(n)=(1 2 1)/2, which is
symmetric and has three integer taps. The analysis
filter-bank used in this example was first proposed
in [17] and is often referred to as the (5, 3) filterbank, indicating a low-pass filter of length five and
a high-pass filter of length three. To ensure that
the filtering operation is defined at the signal
boundaries, the 1-D signal must be extended in
both directions. When using odd-tap filters, the
signal is symmetrically and periodically extended
as shown in Fig. 3. The extension for even-tap
filters (allowed in Part 2 of the standard) is more
complicated and is explained in Part 2 of the
standard document [61].
The filtered samples that are output from the
forward DWT are referred to as wavelet coefficients. Because of the downsampling process, the
total number of wavelet coefficients is the same as
the number of original signal samples. When the
DWT decomposition is applied to sequences with
an odd number of samples, either the low pass or
Original signal samples
Fig. 3. Symmetric and periodic extension of the input signal at
boundaries.
the high-pass sequence will have one additional
sample to maintain the same number of coefficients as original samples. In JPEG 2000, this
choice is dictated by the positioning of the input
signal with respect to the canvas coordinate
system, which will be discussed in Section 3. The
(h0 ; h1 ) filter pair is designed in such a manner that
after downsampling the output of each filter by a
factor of two, the original signal can still be
completely recovered from the remaining samples
in the absence of any quantization errors. This is
referred to as the perfect reconstruction (PR)
property.
Reconstruction from the wavelet coefficients at
the decoder is performed with another pair of lowpass and high-pass filters (g0 ; g1 ), known as the
synthesis filter-bank. Referring to Fig. 2, the downsampled output of the low-pass filter h0 ðnÞ is first
upsampled by a factor of two by inserting zeroes in
between every two samples. The result is then
filtered with the synthesis low-pass filter, g0 ðnÞ: The
downsampled output of the high-pass filter h1 ðnÞ is
also upsampled and filtered with the synthesis
high-pass filter, g1 ðnÞ: The results are added
#
together to produce a reconstructed signal xðnÞ;
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
which assuming sufficient precision, will be identical to xðnÞ because of the PR property.
For perfect reconstruction, the analysis and
synthesis filters have to satisfy the following two
conditions:
H0 ðzÞG0 ðzÞ þ H1 ðzÞG1 ðzÞ ¼ 2;
ð5Þ
H0 ðzÞG0 ðzÞ þ H1 ðzÞG1 ðzÞ ¼ 0;
ð6Þ
where H0 ðzÞ is the Z-transform of h0 ðnÞ; G0 ðzÞ is
the Z-transform of g0 ðnÞ; etc. The condition in
Eq. (6) can be satisfied by choosing
G0 ðzÞ ¼ czl H1 ðzÞ and
G1 ðzÞ ¼ czl H0 ðzÞ;
ð7Þ
where l is an integer constant and c is a scaling
factor. Combining this result with Eq. (5) indicates
that the analysis filter pair (h0 ; h1 ) has to be chosen
to satisfy
czl H0 ðzÞH1 ðzÞ þ czl H1 ðzÞH0 ðzÞ ¼ 2:
ð8Þ
The constant l represents a delay term that
imposes a restriction on the spatial alignment of
the analysis and synthesis filters, while the
constant c affects the filter normalization. The
filter-bank that satisfies these conditions is known
as the bi-orthogonal filter-bank. This name stems
from the fact that h0 and g1 are orthogonal to each
other and h1 and g0 are orthogonal to each other.
A particular class of bi-orthogonal filters is one
where the analysis and synthesis filters are FIR
9
and linear phase (i.e., they satisfy certain symmetry conditions) [47]. Then, it can be shown that in
order to satisfy Eq. (8), the analysis filters h0 and
h1 have to be of unequal lengths. If the filters have
an odd number of taps, their length can differ only
by an odd multiple of two.
While the (5, 3) filter-bank is a prime example of
a bi-orthogonal filter-bank with integer taps, the
filter-banks that result in the highest compression
efficiency often have floating point taps [48]. The
most well-known filter-bank in this category is the
Daubechies (9, 7) filter-bank, introduced in [5] and
characterized by the filter taps given in Table 2.
For comparison, the analysis and synthesis filter
taps for the integer (5, 3) filter-bank are specified in
Table 3. It can be easily verified that these filters
satisfy Eqs. (7) and (8) with l ¼ 1 and c ¼ 1:0: As
is evident from Tables 2 and 3, the filter h0 is
centered at zero while h1 is centered at 1. As a
result, the downsampling operation effectively
retains the even indexed samples from the lowpass output and the odd indexed samples from the
high-pass output sequence.
After the 1-D signal has been decomposed into
two bands, the low-pass output is still highly
correlated and can be subjected to another stage of
two-band decomposition to achieve additional
decorrelation. In comparison, there is generally
little to be gained by further decomposing the
high-pass output. In most DWT decompositions,
only the low-pass output is further decomposed to
Table 2
Analysis and synthesis high-pass filter taps for floating point Daubechies (9, 7) filter-bank
n
Low-pass, h0 ðnÞ
Low-pass, g0 ðnÞ
0
71
72
73
74
+0.602949018236360
+0.266864118442875
0.078223266528990
0.016864118442875
+0.026748757410810
n
High-pass, h1 ðnÞ
n
High-pass, g1 ðnÞ
1
2, 0
3, 1
4, 2
+1.115087052457000
0.591271763114250
0.057543526228500
+0.091271763114250
1
0, 2
1, 3
2, 4
3, 5
+0.602949018236360
0.266864118442875
0.078223266528990
+0.016864118442875
+0.026748757410810
+1.115087052457000
+0.591271763114250
0.057543526228500
0.091271763114250
10
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
Table 3
Analysis and synthesis filter taps for the integer (5, 3) filter-bank
n
h0 ðnÞ
g0 ðnÞ
n
h1 ðnÞ
n
g1 ðnÞ
0
71
72
3/4
1/4
1/8
+1
+1/2
1
2, 0
+1
1/2
1
0, 2
1, 3
+3/4
1/4
1/8
produce what is known as a dyadic or octave
decomposition. Part 1 of the JPEG 2000 standard
supports only dyadic decompositions, while Part 2
also allows for the further splitting of the highfrequency bands.
2.2.2. The 2-D DWT
The 1-D DWT can be easily extended to two
dimensions (2-D) by applying the filter-bank in a
separable manner. At each level of the wavelet
decomposition, each row of a 2-D image is first
transformed using a 1-D horizontal analysis filterbank (h0 ; h1 ). The same filter-bank is then applied
vertically to each column of the filtered and
subsampled data. The result of a one-level wavelet
decomposition is four filtered and subsampled
images, referred to as subbands. Given the linear
nature of the filtering process, the order in which
the horizontal and the vertical filters are applied
does not affect the final values of the 2-D
subbands. In a 2-D dyadic decomposition, the
lowest frequency subband (denoted as the LL
band to indicate low-pass filtering in both directions) is further decomposed into four smaller
subbands, and this process may be repeated until
no tangible gains in compression efficiency can be
achieved. Fig. 4 shows a 3-level, 2-D dyadic
decomposition and the corresponding labeling
for each subband. For example, the subband label
kHL indicates that a horizontal high-pass (H) filter
has been applied to the rows, followed by a vertical
low-pass (L) filter applied to the columns during
the kth level of the DWT decomposition. As a
convention, the subband 0LL refers to the original
image (or image tile). Fig. 5 shows a 3-level, 2-D
DWT decomposition of the Lena image using the
(9, 7) filter-bank as specified in Table 2, and it
clearly demonstrates the energy compaction property of the DWT (i.e., most of the image energy is
3LL 3HL
2HL
3LH 3HH
1HL
2LH
2HH
1LH
1HH
Fig. 4. 2-D, 3-level wavelet decomposition.
Fig. 5. 2-D, 3-level wavelet decomposition of Lena using the
(9, 7) filter-bank.
found in the lower frequency subbands). To better
visualize the subband energies, the AC subbands
(i.e., all the subbands except for LL) have been
scaled up by a factor of four. However, as will be
explained in Section 2.2.3, in order to show the
actual contribution of each subband to the overall
image energy, the wavelet coefficients in each
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
Table 4
L2 -norms of the DWT subbands after a 2-D, 3-level wavelet
decomposition
pffiffiffi pffiffiffi
( 2; 2) normalization
(1, 2) normalization
Subband (5, 3)
filter-bank
(9, 7)
filter-bank
(5, 3)
(9, 7)
filter-bank filter-bank
3LL
3HL
3LH
3HH
2HL
2LH
2HH
1HL
1LH
1HH
1.05209
1.04584
1.04584
1.03963
0.99841
0.99841
0.96722
1.01129
1.01129
1.04044
5.37500
2.91966
2.91966
1.58594
1.59222
1.59222
0.92188
1.03833
1.03833
0.71875
0.67188
0.72992
0.72992
0.79297
0.79611
0.79611
0.92188
1.03833
1.03833
1.43750
8.41675
4.18337
4.18337
2.07926
1.99681
1.99681
0.96722
1.01129
1.01129
0.52022
subband should be scaled by the weights given in
the last column of Table 4.
The DWT decomposition provides a natural
solution for the multiresolution requirement of the
JPEG 2000 standard. The lowest resolution at
which the image can be reconstructed is referred to
as resolution zero. For example, referring to
Fig. 4, the 3LL subband would correspond to
resolution zero for a 3-level decomposition. For an
NL-level3 DWT decomposition, the image can be
reconstructed at NL þ 1 resolutions. In general, to
reconstruct an image at resolution r (r > 0),
subbands (NL r þ 1)HL, (NL r þ 1)LH and
(NL r þ 1)HH need to be combined with the
image at resolution (r 1). These subbands are
referred to as belonging to resolution r: Resolution
zero consists of only the NLLL band. If the
subbands are encoded independently, the image
can be reconstructed at any resolution level by
simply decoding those portions of the codestream
that contain the subbands corresponding to that
resolution and all the previous resolutions. For
example, referring to Fig. 4, the image can be
reconstructed at resolution two by combining the
3
NL is the notation that is used in the JPEG 2000 document
to indicate the number of resolution levels, although the
subscript L might be somewhat confusing as it would seem to
indicate a variable.
11
resolution one image and the three subbands
labeled 2HL, 2LH and 2HH.
2.2.3. Filter normalization
The output of an invertible forward transform
can generally have any arbitrary normalization
(scaling) as long as it is undone by the inverse
transform. In case of DWT filters, the analysis
filters h0 and h1 can be normalized arbitrarily.
Referring to Eq. (8), the normalization chosen for
the analysis filters will influence the value of c;
which in turn determines the normalization of the
synthesis filters, g0 and g1 : The normalization of
the DWT filters is often expressed in terms of the
DC gain of the low-pass analysis filter h0 ; and
the Nyquist gain of the high-pass analysis filter h1 :
The DC gain and the Nyquist gain of a filter hðnÞ;
denoted by GDC and GNyquist, respectively, are
defined as
X
X
n
GDC ¼ hðnÞ
; GNyquist ¼ ð1Þ hðnÞ
: ð9Þ
n
n
The (9, 7) and the (5, 3) analysis filter-banks as
defined in Tables 2 and 3 have been normalized so
that the low-pass filter has a DC gain of 1 and the
high-pass filter has a Nyquist gain of 2. This is
referred to as the (1, 2) normalization and it is the
one adopted by Part 1 of the JPEG 2000 standard.
Other common normalizations
pffiffiffi pffiffiffi that have appeared
in the literature are ( 2; 2) and (1, 1). Once the
normalization of the analysis filter-bank has been
specified, the normalization of the synthesis filterbank is automatically determined by reversing the
order and multiplying by the scalar constant c of
Eq. (8).
In the existing JPEG standard, the scaling of the
forward DCT is defined to create an orthonormal
transform, which has the property that the sum of
the squares of the image samples is equal to the
sum of the squares of the transform coefficients
(Parseval’s theorem). Furthermore, the orthonormal normalization of the DCT has the useful
property that the mean-squared error (MSE) of
the quantized DCT coefficients is the same as the
MSE of the reconstructed image. This provides
a simple means for quantifying the impact of
coefficient quantization on the reconstructed
12
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
image MSE. Unfortunately, this property does not
hold for a DWT decomposition.
Each wavelet coefficient in a 1-D DWT decomposition can be associated with a basis function.
# can be expressed as a
The reconstructed signal xðnÞ
weighted sum of these basis functions, where the
weights are the wavelet coefficients (either quantized or unquantized). Let cbm ðnÞ denote the basis
function corresponding to a coefficient yb ðmÞ; the
mth wavelet coefficient from subband b: Then,
XX
# ¼
yb ðmÞcbm ðnÞ:
ð10Þ
xðnÞ
b
m
For a simple 1-level DWT, the basis functions for
the wavelet coefficients in the low-pass or the highpass subbands are shifted versions of the corresponding low-pass or high-pass synthesis filters,
except near the subband boundaries. In general,
the basis functions of a DWT decomposition are
not orthogonal; hence, Parseval’s theorem does
not apply. Woods and Naveen [50] have shown
that for quantized wavelet coefficients under
certain assumptions on the quantization noise,
the MSE of the reconstructed image can be 10
approximately expressed as a weighted sum of the
MSE of the wavelet coefficients, where the weight
for subband b is
X
cb ðnÞ
2 :
a2b ¼
ð11Þ
n
The coefficient ab is referred to as the L2-norm4 for
subband b: For an orthonormal transform, all the
ab values would be unity. The knowledge of the
L2-norms is essential for the encoder, because they
represent the contribution of the quantization
noise of each subband to the overall MSE and
are a key factor in designing quantizers or
prioritizing the quantized data for coding.
The DWT filter normalization impacts both the
L2-norm and the dynamic range of each subband.
Given the normalization of the 1-D analysis filterbank, the nominal dynamic range of the 2-D
subbands can be easily determined in terms of the
bit-depth of the source image sample, RI : In
particular, for the (1, 2) normalization, the kLL
4
We have ignored the fact that, in general, the L2-norm for
the coefficients near the subband boundaries are slightly
different than the rest of the coefficients in the subband.
subband will have a nominal dynamic range of RI
bits. However, the actual dynamic range might be
slightly larger. In JPEG 2000, this situation is
handled by using guard bits to avoid the overflow
of the subband value. For the (1, 2) normalization,
the nominal dynamic ranges of the kLH and kHL
subbands are RI þ 1; while that of the kHH
subband is RI þ 2:
Table 4 shows the L2-norms of the DWT
subbands after a 3-level decomposition with
either the (9, 7) or
pffiffithe
ffi pffiffiffi(5, 3) filter-bank and
using either the ( 2; 2) p
orffiffiffi p
theffiffiffi (1, 2) filter
normalization. Clearly, the ( 2; 2) normalization results in a DWT that is closer to an
orthonormal transform (especially for the (9, 7)
filter-bank), while the (1, 2) normalization avoids
the dynamic range expansion at each level of the
decomposition.
2.2.4. DWT implementation issues and the lifting
scheme
In the development of the existing DCT-based
JPEG standard, great emphasis was placed on the
implementation complexity of the encoder and
decoder. This included issues such as memory
requirements, number of operations per sample,
and amenability to hardware or software implementation, e.g., transform precision, parallel
processing, etc. The choice of the 8 8 block size
for the DCT was greatly influenced by these
considerations.
In contrast to the limited buffering required for
the 8 8 DCT, a straightforward implementation
of the 2-D DWT decomposition requires the
storage of the entire image in memory. The use
of small tiles reduces the memory requirements
without significantly affecting the compression
efficiency (see Section 5.1.1). In addition, some
clever designs for line-based processing of the
DWT have been published that substantially
reduce the memory requirements depending on
the size of the filter kernels [14]. Recently, an
alternative implementation of the DWT has been
proposed, known as the lifting scheme [15,41–43].
In addition to providing a significant reduction in
the memory and the computational complexity of
the DWT, lifting provides in-place computation of
the wavelet coefficients by overwriting the memory
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
locations that contain the input sample values. The
wavelet coefficients computed with lifting are
identical to those computed by a direct filter-bank
convolution, in much the same manner as a fast
Fourier transform results in the same DFT
coefficients as a brute force approach. Because of
these advantages, the specification of the DWT
kernels in JPEG 2000 is only provided in terms of
the lifting coefficients and not the convolutional
filters.
The lifting operation consists of several steps.
The basic idea is to first compute a trivial wavelet
transform, also referred to as the lazy wavelet
transform, by splitting the original 1-D signal into
odd and even indexed subsequences, and then
modifying these values using alternating prediction
and updating steps. Fig. 6 depicts an example of
the lifting steps corresponding to the integer (5, 3)
filter-bank. The sequences fs0i g and {di0 } denote
the even and odd sequences, respectively, resulting
from the application of the lazy wavelet transform
to the input sequence.
In JPEG 2000, a prediction step consists of
predicting each odd sample as a linear combination of the even samples and subtracting it from
the odd sample to form the prediction error {di1 }.
Referring to Fig. 6, for the (5, 3) filter-bank, the
prediction step consists of predicting {di0 } by
averaging the two neighboring even sequence
pixels and subtracting the average from the odd
sample value, i.e.,
di1 ¼ di0 12ðs0i þ s0iþ1 Þ:
ð12Þ
Due to the simple structure of the (5, 3) filter-bank,
the output of this stage, {di1 }, is actually the high-
d0 0
s00
s10
_1 1 _ 1
2
2
1
d1 0
_1 1 _ 1
2
2
d01 1
1
4
s20
1
4
d2 0
Input sequence
_1 1 _1
2
2
d1 1 1
1
4
s30
1
4
d21 1
1
4
High-pass outpu
1
4
Low-pass outpu
s01
s11
s21
s31
Fig. 6. Lifting prediction/update steps for the (5, 3) filter-bank.
13
pass output of the DWT filter. In general, the
number of even pixels employed in the prediction
and the actual weights applied to the samples
depend on the specific DWT filter-bank.
An update step consists of updating the even
samples by adding to them a linear combination of
the already modified odd samples, {di1 }, to form
the updated sequence {s1i }. Referring to Fig. 6, for
the (5, 3) filter-bank, the update step consists of the
following:
1
s1i ¼ s0i þ 14ðdi1
þ di1 Þ:
ð13Þ
For the (5, 3) filter-bank, the output of this stage,
{s1i }, is actually the low-pass output of the DWT
filter. Again, the number of odd pixels employed in
the update and the actual weights applied to each
sample depend on the specific DWT filter-bank.
The prediction and update steps are generally
iterated N times, with different weights used at
each iteration. This can be summarized as
X
din ¼ din1 þ
Pn ðkÞsn1
nA½1; 2; y; N; ð14Þ
k ;
k
sni ¼ sn1
þ
i
X
Un ðkÞdkn ;
nA½1; 2; y; N;
ð15Þ
k
where Pn ðkÞ and Un ðkÞ are, respectively, the
prediction and update weights at the nth iteration.
For the (5, 3) filter-bank N ¼ 1; while for the
Daubechies (9, 7) filter-bank, N ¼ 2: The output of
the final prediction step will be the high-pass
coefficients up to a scaling factor K1 ; while the
output of the final update step will be the low-pass
coefficients up to a scaling constant K0 : For
the (5, 3) filter-bank, K0 ¼ K1 ¼ 1: The lifting
steps corresponding to the (9, 7) filter-bank (as
specified in Table 2) are shown in Fig. 7. The
general block diagram of the lifting process is
shown in Fig. 8.
A nice feature of the lifting scheme is that it
makes the construction of the inverse transform
straightforward. Referring to Fig. 8 and working
from right to left, first the low-pass and high-pass
wavelet coefficients are scaled by 1=K0 and 1=K1
N
N
to produce {sN
i } and {di }. Next, {di } is taken
through the update stage UN ðzÞ and subtracted
N1
from {sN
}. This process
i } to produce {si
continues, where each stage of the prediction and
update is undone in the reverse order that it was
14
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
constructed at the encoder until the image samples
have been reconstructed.
2.2.5. Integer-to-integer transforms
Although the input image samples to JPEG
2000 are integers, the output wavelet coefficients
are floating point when using floating point DWT
Input sequence
p1
u1
p2
p1
u1
p1
u1
p2
p2
p1
p1
u1
u1
p2
p2
p1
u1
p2
p1
p1
u1
u1
p2
Intermediate
stages
p2
High-pass
u2
u2
u2
u2
u2
u2
u2
K1
u2
Low-pass
K0
p1
u1
- 1.586134342059924
- 0.052980118572961
p2
+0.882911075530934
u2
+0.443506852043971
K1 = 1/ K0
+1.230174104914001
Fig. 7. Lifting steps for the (9, 7) filter-bank.
si0
filters. Even when dealing with integer filters such
as the (5, 3) filter-bank, the precision required for
achieving mathematically lossless performance
increases significantly with every level of the
wavelet decomposition and can quickly become
unmanageable. An important advantage of the
lifting approach is that it can provide a convenient
framework for constructing integer-to-integer
DWT filters from any general filter specification
[1,10].
This can be best understood by referring to
Fig. 9, where quantizers are inserted immediately
after the calculation of the prediction and the
update terms but before modifying the odd or the
even sample value. The quantizer typically performs an operation such as truncation or rounding
to the nearest integer, thus creating an integervalued output. If the values of K0 and K1 are
approximated by rational numbers, it is easy to
verify that the resulting system is mathematically
invertible despite the inclusion of the quantizer. If
the underlying floating point filter uses the (1, 2)
normalization and K0 ¼ K1 ¼ 1 (as is the case for
the (5, 3) filter-bank), the final low-pass output will
have roughly the same bit precision as that of the
+
xi
Lazy
P1(z)
siN
siN-1+
+
K0
Low-pass
+
U1(z)
PN(z)
UN(z)
Transform
+
di 0
diN-1
+
+
diN
+
High-pass
K1
Fig. 8. General block diagram of the lifting process.
si 0
+
xi
Lazy
Transform
+
K0
Low-pass
+
P1(z)
QU1
PN(z)
QUN
QP1
U1(z)
QPN
UN(z)
+
d i0
si N
siN-1+
+
diN-1
+
+
diN
High-pass
K1
Fig. 9. General block diagram of a forward integer-to-integer transform using lifting.
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
input sample while the high-pass output will have
an extra bit of precision. This is because input
samples with a large enough dynamic range (e.g.,
8 bits or higher), rounding at each lifting step have
a negligible effect on the nominal dynamic range of
the output.
As described in the previous section, the inverse
transformation is simply performed by undoing all
the prediction and update steps in the reverse
order that they were performed at the encoder.
However, the resulting integer-to-integer transform is nonlinear and hence when extended to two
dimensions, the order in which the transformation
is applied to the rows or the columns will impact
the final output. To recover the original sample
values losslessly, the inverse transform must be
applied in exactly the reverse row–column order of
the forward transform. An extensive performance
evaluation and analysis of reversible integer-tointeger DWT for image compression has been
published in [1].
As an example, consider the conversion of
the (5, 3) filter-bank into an integer-to-integer
transform by adding the two quantizers
QP1 ðwÞ ¼ I wm and QU1 ðwÞ ¼ Iw þ 1=2m to
the prediction and update steps, respectively, in
the lifting diagram of Fig. 6. The resulting
forward transform is given by
xð2nÞ þ xð2n þ 2Þ
yð2n þ 1Þ ¼ xð2n þ 1Þ ;
2
ð16Þ
yð2n 1Þ þ yð2n þ 1Þ þ 2
:
yð2nÞ ¼ xð2nÞ þ
4
The required precision for the low-pass band stays
roughly the same as the original sample while the
precision of the high-pass band grows by one bit.
The inverse transform, which losslessly recovers
the original sample values, is given by
yð2n 1Þ þ yð2n þ 1Þ þ 2
xð2nÞ ¼ yð2nÞ ;
4
ð17Þ
xð2nÞ þ xð2n þ 2Þ
:
xð2n þ 1Þ ¼ yð2n þ 1Þ 2
2.2.6. DWT filter choices in JPEG 2000 Part 1
Part 1 of the JPEG 2000 standard has adopted
only two choices for the DWT filters. One is the
15
Daubechies (9, 7) floating point filter-bank (as
specified in Table 2), which has been chosen for its
superior lossy compression performance. The
other is the lifted integer-to-integer (5, 3) filterbank, also referred to as the reversible (5, 3)
filter-bank, as specified in Eqs. (16) and (17). This
choice was driven by requirements for low
implementation complexity and lossless capability.
The performance of these filters is compared in
Section 5.1.3. Part 2 of the standard allows for
arbitrary filter specifications in the codestream,
including filters with an even number of taps.
2.3. Quantization
The JPEG baseline system employs a uniform
quantizer and an inverse quantization process that
reconstructs the quantized coefficient to the midpoint of the quantization interval. A different stepsize is allowed for each DCT coefficient to take
advantage of the sensitivity of the human visual
system (HVS), and these step-sizes are conveyed
to the decoder via an 8 8 quantization table
(q-table) using one byte per element. The quantization strategy employed in JPEG 2000 Part 1 is
similar in principle to that of JPEG, but it has a
few important differences to satisfy some of the
JPEG 2000 requirements.
One difference is in the incorporation of a
central deadzone in the quantizer. It was shown
in [40] that the R–D optimal quantizer for a
continuous signal with Laplacian probability
density (such as DCT or wavelet coefficients) is a
uniform quantizer with a central deadzone. The
size of the optimal deadzone as a fraction of the
step-size increases as the variance of the Laplacian
distribution decreases; however, it always stays less
than two and is typically closer to one. In Part 1,
the deadzone has twice the quantizer step-size as
depicted in Fig. 10, while in Part 2, the size of the
deadzone can be parameterized to have a different
value for each subband.
Part 1 adopted the deadzone with twice the stepsize due to its optimal embedded structure. Briefly,
this means that if an Mb-bit quantizer index
resulting from a step-size of Db is transmitted
progressively starting with the most significant bit
(MSB) and proceeding to the least significant bit
16
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
Fig. 10. Uniform quantizer with deadzone with step-size Db :
(LSB), the resulting index after decoding only Nb
bits is identical to that obtained by using a similar
quantizer with a step-size of Db 2Mb Nb . This
property allows for SNR scalability, which in its
optimal sense means that the decoder can cease
decoding at any truncation point in the codestream and still produce exactly the same image
that would have been encoded at the bit-rate
corresponding to the truncated codestream. This
property also allows a target bit-rate or a target
distortion to be achieved exactly, while the current
JPEG standard generally requires multiple
encoding cycles to achieve the same goal. This
allows an original image to be compressed
with JPEG 2000 to the highest quality required
by a given set of clients (through the proper
choice of the quantization step-sizes) and
then disseminated to each client according to
the specific image quality (or target filesize)
requirement without the need to decompress and
recompress the existing codestream. Importantly,
the codestream can also be reorganized in other
ways to meet the various requirements of
the JPEG 2000 standard as will be described in
Section 3.
Another difference is that the inverse quantization of JPEG 2000 explicitly allows for a
reconstruction bias from the quantizer midpoint
for non-zero indices to accommodate the skewed
probability distribution of the wavelet coefficients.
In JPEG baseline, a simple biased reconstruction
strategy has been shown to improve the decoded
image PSNR by about 0.25 dB [34]. Similar gains
can be expected with the biased reconstruction of
wavelet coefficients in JPEG 2000. The exact
operation of the quantization and inverse quantization is explained in more detail in the following
sections.
2.3.1. Quantization at the encoder
For each subband b; a basic quantizer step-size
Db is selected by the user and is used to quantize all
the coefficients in that subband. The choice of Db
can be driven by the perceptual importance of each
subband based on HVS data [4,19,31,49], or it can
be driven by other considerations such as rate
control. The quantizer maps a wavelet coefficient
yb ðu; vÞ in subband b to a quantized index value
qb ðu; vÞ; as shown in Fig. 10. The quantization
operation is an encoder issue and can be implemented in any desired manner. However, it is most
efficiently performed according to
jyb ðu; vÞj
qb ðu; vÞ ¼ signð yb ðu; vÞÞ
:
ð18Þ
Db
The step-size Db is represented with a total of two
bytes; an 11-bit mantissa mb ; and a 5-bit exponent
eb ; according to the relationship
mb Db ¼ 2Rb eb 1 þ 11
;
ð19Þ
2
where Rb is the number of bits representing the
nominal dynamic range of the subband b; which is
explained in Section 2.2.3. This limits the largest
possible step-size to about twice the dynamic range
of the input sample (when mb has its maximum
value and eb ¼ 0), which is sufficient for all
practical cases of interest. When the reversible
(5, 3) filter-bank is used, Db is set to one by
choosing mb ¼ 0 and eb ¼ Rb : The quantizer index
qb ðu; vÞ will have Mb bits if fully decoded, where
Mb ¼ G þ eb 1: The parameter G is the number
of guard bits signaled to the decoder, and it is
typically one or two.
Two modes of signaling the value of Db to the
decoder are possible. In one mode, which is similar
to the q-table specification used in the current
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
JPEG, the (eb ; mb ) value for every subband is
explicitly transmitted. This is referred to as
expounded quantization. The values can be chosen
to take into account the HVS properties and/or the
L2-norm of each subband in order to align the
bitplanes of the quantizer indices according to
their true contribution to the MSE. In another
mode, referred to as derived quantization, a single
value (e0 ; m0 ) is sent for the LL subband and the
(eb ; mb ) values for each subband are derived by
scaling the D0 value by some power of two
depending on the level of decomposition associated with that subband. In particular,
ðeb ; mb Þ ¼ ðe0 NL þ nb ; m0 Þ;
ð20Þ
where NL is the total number of decomposition
levels and nb is the decomposition level corresponding to subband b: It is easy to show that
Eq. (20) scales the step-sizes for each subband
according to a power of two that best approximates the L2-norm of a subband relative to the LL
band (refer to Table 4). This procedure approximately aligns the quantized subband bitplanes
according to their proper MSE contribution.
2.3.2. Inverse quantization at the decoder
When the irreversible (9, 7) filter-bank is used,
the reconstructed transform coefficient, Rqb ðu; vÞ;
for a quantizer step-size of Db is given by
8
>
< ðqb ðu; vÞ þ gÞDb if qb ðu; vÞ > 0;
Rqb ðu; vÞ ¼
>
:
ðqb ðu; vÞ gÞDb
0
if qb ðu; vÞo0; ð21Þ
otherwise;
where 0pgo1 is a reconstruction parameter
arbitrarily chosen by the decoder. A value of g ¼
0:50 results in midpoint reconstruction as in the
existing JPEG standard. A value of go0:50 creates
a reconstruction bias towards zero, which can
result in improved reconstruction PSNR when the
probability distribution of the wavelet coefficients
falls off rapidly away from zero (e.g., a Laplacian
distribution). A popular choice for biased reconstruction is g ¼ 0:375: If all of the Mb bits for a
quantizer index are fully decoded, the step-size is
equal to Db : However, when only Nb bits are
decoded, the step-size in Eq. (21) is equivalent to
Db 2Mb Nb : The reversible (5, 3) filter-bank is
treated the same way (with Db ¼ 1), except when
17
the index is fully decoded to achieve lossless
reconstruction, in which case Rqb ðu; vÞ ¼ qb ðu; vÞ:
2.4. Entropy coding
The quantizer indices corresponding to the
quantized wavelet coefficients in each subband
are entropy encoded to create the compressed
bit-stream. The choice of the entropy coder in
JPEG 2000 is motivated by several factors. One is
the requirement to create an embedded bit-stream,
which is made possible by bitplane encoding of the
quantizer indices. Bitplane encoding of wavelet
coefficients has been used by several well-known
embedded wavelet coders such as EZW [38] and
SPIHT [36]. However, these coders use coding
models that exploit the correlation between subbands to improve coding efficiency.
Unfortunately, this adversely impacts error
resilience and severely limits the flexibility of a
coder to arrange the bit-stream in an arbitrary
progression order. In JPEG 2000, each subband is
encoded independently of the other subbands. In
addition, JPEG 2000 uses a block coding paradigm in the wavelet domain as in the embedded
block coding with optimized truncation (EBCOT)
algorithm [44], where each subband is partitioned
into small rectangular blocks, referred to as
codeblocks, and each codeblock is independently
encoded. The nominal dimensions of a codeblock
are free parameters specified by the encoder but
are subject to the following constraints: they must
be an integer power of two; the total number of
coefficients in a codeblock can not exceed 4096;
and the height of the codeblock cannot be less
than four.
The independent encoding of the codeblocks has
many advantages including localized random
access into the image, parallelization, improved
cropping and rotation functionality, improved
error resilience, efficient rate control, and maximum flexibility in arranging progression orders
(see Section 3). It may seem that failing to exploit
inter-subband redundancies would have a sizable
adverse effect on coding efficiency. However, this is
more than compensated by the finer scalability
that results from multiple-pass encoding of the
codeblock bitplanes. By using an efficient rate
18
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
control strategy that independently optimizes the
contribution of each codeblock to the final bitstream (see Section 4.2), the JPEG 2000 Part 1
encoder achieves a compression efficiency that is
superior to other existing approaches [46].
Fig. 11 shows a schematic of the multiple
bitplanes that are associated with the quantized
wavelet coefficients. The symbols that represent
the quantized coefficients are encoded one bit at a
time starting with the MSB and proceeding to the
LSB. During this progressive bitplane encoding, a
quantized wavelet coefficient is called insignificant
if the quantizer index is still zero (e.g., the example
coefficient in Fig. 11 is still insignificant after
encoding its first two MSBs). Once the first nonzero bit is encoded, the coefficient becomes
significant, and its sign is encoded. Once a
coefficient becomes significant, all subsequent
bits are referred to as refinement bits. Since the
DWT packs most of the energy in the lowfrequency subbands, the majority of the wavelet
coefficients will have low amplitudes. Consequently, many quantized indices will be insignificant in the earlier bitplanes, leading to a very
low information content for those bitplanes.
JPEG 2000 uses an efficient coding method
for exploiting the redundancy of the bitplanes
known as context-based adaptive binary arithmetic coding.
2.4.1. Arithmetic coding and the MQ-coder
Arithmetic coding uses a fundamentally different approach from Huffman coding in that the
2Mb-1
2Mb-2
2Mb-3
0
0
1
...
2Mb-Nb
...
0
...
Fig. 11. Bitplane coding of quantized wavelet coefficients.
entire sequence of source symbols is mapped into a
single codeword (albeit a very long codeword).
This codeword is developed by recursive interval
partitioning using the symbol probabilities, and
the final codeword represents a binary fraction
that points to the subinterval determined by the
sequence.
An adaptive binary arithmetic coder can be
viewed as an encoding device that accepts the
binary symbols in a source sequence, along with
their corresponding probability estimates, and
produces a codestream with a length at most two
bits greater than the combined ideal codelengths of
the input symbols [33]. Adaptivity is provided by
updating the probability estimate of a symbol
based upon its present value and history. In
essence, arithmetic coding provides the compression efficiency that comes with Huffman coding of
large blocks, but only a single symbol is encoded at
a time. This single-symbol encoding structure
greatly simplifies probability estimation, since
only individual symbol probabilities are needed
at each sub-interval iteration (not the joint
probability estimates that are necessary in block
coding). Furthermore, unlike Huffman coding,
arithmetic coding does not require the development of new codewords each time the symbol
probabilities change. This makes it easy to adapt
to the changing symbol probabilities within a
codeblock of quantized wavelet coefficient bitplanes.
Practical implementations of arithmetic coding
are always less efficient than an ideal one. Finitelength registers limit the smallest probability that
can be maintained, and computational speed
requires approximations, such as replacing
multiplies with adds and shifts. Moreover,
symbol probabilities are typically chosen from
a finite set of allowed values, so the true
symbol probabilities must often be approximated. Overall, these restrictions result in a
coding inefficiency of approximately 6% compared
to the ideal codelength of the symbols encoded
[32]. It should be noted that even the most
computationally efficient implementations of arithmetic coding are significantly more complex
than Huffman coding in both software and
hardware.
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
One of the early practical implementations of
adaptive binary arithmetic coding was the Q-coder
developed by IBM [33]. Later, a modified version
of the Q-coder, known as the QM-coder, was
chosen as the entropy coder for the JBIG standard
and the extended JPEG mode [32]. However, IPR
issues have hindered the use of the QM-coder in
the JPEG standard. Instead, the JPEG 2000
committee adopted another modification of the
Q-coder, named the MQ-coder. The MQ-coder
was also adopted for use in the JBIG2 standard
[67]. The companies that own IPR on the MQcoder have made it available on a license-free and
royalty-free basis for use in the JPEG 2000
standard. Differences between the MQ and the
QM coders include ‘‘bit stuffing’’ versus ‘‘byte
stuffing’’, decoder versus encoder carry resolution,
hardware versus software coding convention, and
the number of probability states. The specific
details of these coders are beyond the scope of this
paper, and the reader is referred to [39] and the
MQ-coder flowcharts in the standard document
[60]. We mention in passing that the specific
realization of the ‘‘bit stuffing’’ procedure in the
MQ-coder (which costs about 0.5% in coding
efficiency), creates a redundancy such that any two
consecutive bytes of coded data are always
forced to lie in the range of hexadecimal
‘‘0000’’ through ‘‘FF8F’’ [45]. This leaves the
range of ‘‘FF90’’ through ‘‘FFFF’’ unattainable
by coded data, and the JPEG 2000 syntax uses
this range to represent unique marker codes
that facilitate the organization and parsing
of the bit-stream as well as improve error
resilience.
In general, the probability distribution of each
binary symbol in a quantized wavelet coefficient is
influenced by all the previously coded bits
corresponding to that coefficient as well as the
value of its immediate neighbors. In JPEG 2000,
the probability of a binary symbol is estimated
from a context formed from its current significance
state as well as the significance states of its
immediate eight neighbors as determined from
the previous bitplane and the current bitplane,
based on coded information up to that point. In
context-based arithmetic coding, separate probability estimates are maintained for each context,
19
which is updated according to a finite-state
machine every time a symbol is encoded in that
context.5 For each context, the MQ-coder can
choose from a total of 46 probability states
(estimates), where states 0 through 13 correspond
to start-up states (also referred to as fast-attack)
and are used for rapid convergence to a stable
probability estimate. States 14 through 45 correspond to steady-state probability estimates and
once entered from a start-up state, can never be
left by the finite-state machine. There is also an
additional non-adaptive state (state 46), which is
used to encode symbols with equal probability
distribution, and can neither be entered nor exited
from any other probability state.
2.4.2. Bitplane coding passes
The quantized coefficients in a codeblock are
bitplane encoded independently from all other
codeblocks when creating an embedded bitstream. Instead of encoding the entire bitplane in
one coding pass, each bitplane is encoded in three
sub-bitplane passes with the provision of truncating the bit-stream at the end of each coding pass.
A main advantage of this approach is near-optimal
embedding, where the information that results in
the largest reduction in distortion for the smallest
increase in file size is encoded first. Moreover, a
large number of potential truncation points facilitates an optimal rate control strategy where a
target bit-rate is achieved by including those
coding passes that minimize the total distortion.
Referring to Fig. 12, consider the encoding of a
single bitplane from a codeblock in three coding
passes (labeled A, B and C), where a fraction of
the bits are encoded at each pass. Let the
distortion and bit-rate associated with the reconstructed image prior and subsequent to the
encoding of the entire bitplane be given by
(D1 ; R1 ) and (D2 ; R2 ), respectively. The two coding
paths ABC and CBA correspond to coding the
same data in a different order, and they both start
and end at the same rate-distortion points.
However, their embedded performances are
5
In the MQ-coder implementation, a symbol’s probability
estimate is actually updated only when at least one bit of coded
output is generated.
20
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
R2
A
C
Rate
B
B
A
C
Distortion
D1
R1
D2
Fig. 12. R–D path for optimal embedding.
significantly different. In particular, if the coded
bit-stream is truncated at any intermediate point
during the encoding of the bitplane, the path ABC
would have less distortion for the same rate, and
hence would possess a superior embedding property. In creating optimal embedding, the data with
the highest distortion reduction per average bit
of compressed representation should be coded
first [23].
For a coefficient that is still insignificant, it can
be shown that given reasonable assumptions about
its probability distribution, the distortion reduction per average bit of compressed representation
increases with increasing probability of becoming
significant, ps [23,30]. For a coefficient that is being
refined, the distortion reduction per average bit is
smaller than an insignificant coefficient, unless ps
for that coefficient is less than 1%. As a result,
optimal embedding can theoretically be achieved
by first encoding the insignificant coefficients
starting with the highest ps until that probability
reaches about 1%. At that point, all the refinement
bits should be encoded, followed by all the
remaining coefficients in the order of their
decreasing ps : However, the calculation of the ps
values for each coefficient is a tedious and
approximate task, so the JPEG 2000 coder instead
divides the bitplane data into three groups and
encodes each group during a fractional bitplane
pass. Each coefficient in a block is assigned a
binary state variable called its significance state
that is initialized to zero (insignificant) at the start
of the encoding. The significance state changes
from zero to one (significant) when the first non-
zero magnitude bit is found. The context vector for
a given coefficient is the binary vector consisting of
the significance states of its eight immediate
neighbor coefficients as shown in Fig. 13. During
the first pass, referred to as the significance
propagation pass, the insignificant coefficients that
have the highest probability of becoming significant, as determined by their immediate eight
neighbors, are encoded. In the second pass, known
as the refinement pass, the significant coefficients
are refined by their bit representation in the
current bitplane. Finally, during the cleanup pass,
all the remaining coefficients in the bitplane are
encoded as they have the lowest probability of
becoming significant. The order in which the data
in each pass are visited is data dependent and
follows a deterministic stripe-scan order with a
height of four pixels as shown in Fig. 14. This
stripe-based scan has been shown to facilitate
software and hardware implementations [26]. The
bit-stream can be truncated at the end of each
d
v
h
d
d
h
v
d
Fig. 13. Neighboring pixels used in context selection.
××
××
××
××
Fig. 14. Scan order within a codeblock.
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
coding pass. In the following, each coding pass is
described in more detail.
2.4.2.1. Significance propagation pass. During this
pass, the insignificant coefficients that have the
highest probability of becoming significant in the
current bitplane are encoded. The data is scanned
in the stripe order shown in Fig. 14, and every
sample that has at least one significant immediate
neighbor, based on coded information up to that
point, is encoded. As soon as a coefficient is coded,
its significance state is updated so that it can effect
the inclusion of subsequent coefficients in that
coding pass. The significance state of the coefficient is arithmetic coded using contexts that are
based on the significance states of its immediate
neighbors. In general, the significance states of the
eight neighbors can create 256 different contexts,6
however, many of these contexts have similar
probability estimates and can be merged together.
A context reduction mapping reduces the total
number of contexts to only nine to improve the
efficiency of the MQ-coder probability estimation
for each context. Since the codeblocks are encoded
independently, if a sample is located at the
codeblock boundary, only its immediate neighbors
that belong to the current codeblock are considered and the significance state of the missing
neighbors are assumed to be zero. Finally, if a
coefficient is found to be significant, its sign needs
to be encoded. The sign value is also arithmetic
encoded using five contexts that are determined
from the significance and the sign of the coefficient’s four horizontal and vertical neighbors.
2.4.2.2. Refinement pass. During this pass, the
magnitude bit of a coefficient that has already
become significant in a previous bitplane is
arithmetic encoded using three contexts. In general, the refinement bits have an even distribution
unless the coefficient has just become significant in
the previous bitplane (i.e., the magnitude bit to be
encoded is the first refinement bit). This condition
6
Technically, the combination where all the neighbors are
insignificant cannot happen in this pass. However, this
combination is given its own context (labeled zero) and is used
during the cleanup pass.
21
is first tested and if it is satisfied, the magnitude bit
is encoded using two coding contexts based on the
significance of the eight immediate neighbors.
Otherwise, it is coded with a single context
regardless of the neighboring values.
2.4.2.3. Cleanup pass. All the remaining coefficients in the codeblock are encoded during the
cleanup pass. Generally, the coefficients coded in
this pass have a very small ps value and are
expected to remain insignificant. As a result, a
special mode, referred to as the run mode, is used
to aggregate the coefficients that have the highest
probability of remaining insignificant. More specifically, a run mode is entered if all the four
samples in a vertical column of the stripe have
insignificant neighbors. In the run mode, a binary
symbol is arithmetic encoded in a single context to
specify whether all the four samples in the vertical
column remain insignificant. An encoded value of
zero implies insignificance for all four samples,
while an encoded value of one implies that at least
one of the four samples becomes significant in the
current bitplane. An encoded value of one is
followed by two additional arithmetic encoded bits
that specify the location of the first nonzero
coefficient in the vertical column. Since the
probability of these additional two bits is nearly
evenly distributed, they are encoded with a uniform
context, which uses state 46 of the MQ-coder as its
probability estimate. It should be noted that the
run mode has a negligible impact on the coding
efficiency, and it is primarily used to improve the
throughput of the arithmetic coder through
symbol aggregation.
After the position of the first nonzero coefficient
in the run is specified, the remaining samples in the
vertical column are encoded in the same manner as
in the significance propagation pass and use the
same nine coding contexts. Similarly, if at least one
of the four coefficients in the vertical column has a
significant neighbor, the run mode is disabled and
all the coefficients in that column are coded
according to the procedure employed for the
significance propagation pass.
For each codeblock, the number of MSB planes
that are entirely zero is signaled in the bit-stream.
Since the significance state of all the coefficients in
22
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
the first non-zero MSB is zero, only the cleanup
pass is applied to the first non-zero bitplane.
2.4.3. Entropy coding options
The coding models used by the JPEG 2000
entropy coder employ 18 coding contexts in
addition to a uniform context according to the
following assignment. Contexts 0–8 are used for
significance coding during the significance propagation and cleanup passes, context 9 is used for
run coding the cleanup pass, contexts 10–14 are
used for sign coding, contexts 15–17 are used
during the refinement pass. Each codeblock
employs its own MQ-coder to generate a single
arithmetic codeword for the entire codeblock. In
the default mode, the coding contexts for each
codeblock are initialized at the start of the coding
process and are not reset at any time during the
encoding process. Furthermore, the resulting
codeword can only be truncated at the coding
pass boundaries to include a different number of
coding passes from each codeblock in the final
codestream. All contexts are initialized to uniform
probabilities except for the zero context (all
insignificant neighbors) and the run context, where
the initial less probable symbol (LPS) probabilities
are set to 0.0283 and 0.0593, respectively.
In order to facilitate the parallel encoding or
decoding of the sub-bitplane passes of a single
codeblock, it is necessary to decouple the arithmetic encoding of the sub-bitplane passes from
one another. Hence, JPEG 2000 allows for the
termination of the arithmetic coded bit-stream as
well as the re-initialization of the context probabilities at each coding pass boundary. If any of
these two options is flagged in the codestream, it
must be executed at every coding pass boundary.
The JPEG 2000 also provides for another coding
option known as vertically stripe-causal contexts.
This option is aimed at enabling the parallel
decoding of the coding passes as well as reducing
the external memory utilization. In this mode,
during the encoding of a certain stripe of a
codeblock, the significances of the samples in
future stripes within that codeblock are ignored.
Since the height of the vertical columns is four
pixels, this mode only affects the pixels in the last
row of each stripe. The combination of these three
options, namely arithmetic coder termination, reinitialization at each coding pass boundary, and
the vertically stripe-causal context, is often referred to as the parallel mode.
Another entropy coding option, aimed at
reducing computational complexity, is the lazy
coding mode, where the arithmetic coder is entirely
bypassed in certain coding passes. More specifically, after the encoding of the fourth most
significant bitplane of a codeblock, the arithmetic
coder is bypassed during the encoding of the first
and second sub-bitplane coding passes of subsequent bitplanes. Instead, their content is included
in the codestream as raw data. In order to
implement this mode, it is necessary to terminate
the arithmetic coder at the end of the cleanup pass
preceding each raw coding pass and to pad the raw
coding pass data to align it with the byte
boundary. However, it is not necessary to reinitialize the MQ-coder context models. The lazy
mode can also be combined with the parallel
mode. The impact of the lazy and parallel modes
on the coding efficiency is studied in Section 5.1.5.
2.4.4. Tier-1 and tier-2 coding
The arithmetic coding of the bitplane data is
referred to as tier-1 (T1) coding. Fig. 15 illustrates
a simple example of the compressed data generated at the end of tier-1 encoding. The example
image (shown at the top right of Fig. 15) is of size
256 256 with two levels of decomposition, and
the codeblock size is 64 64. Each square box in
the figure represents the compressed data associated with a single coding pass of a 22 single
codeblock. Since the codeblocks are independently
encoded, the compressed data corresponding to
the various coding passes can be arranged in
different configurations to create a rich set of
progression orders to serve different applications.
The only restriction is that the sub-bitplane coding
passes for a given codeblock must appear in a
causal order starting from the most significant
bitplane. The compressed sub-bitplane coding
passes can be aggregated into larger units named
packets. This process of packetization along with
its supporting syntax, as will be explained in
Section 3, is often referred to as tier-2 (T2) coding.
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
23
Fig. 15. Example of compressed data associated with various sub-bitplane coding passes.
3. JPEG 2000 bit-stream organization
JPEG 2000 offers significant flexibility in the
organization of the compressed bit-stream to
enable such features as random access, region of
interest coding, and scalability. This flexibility is
achieved partly through the various structures of
components, tiles, subbands, resolution levels, and
codeblocks that are discussed in Section 2. These
structures partition the image data into: (1) color
channels (through components); (2) spatial regions
(through tiles); (3) frequency regions (through
subbands and resolution levels), and (4) space–
frequency regions (through codeblocks). Tiling
provides access to the image data over large
spatial regions, while the independent coding of
the codeblocks provides access to smaller units.
Codeblocks can be viewed as a tiling of the
coefficients in the wavelet domain. JPEG 2000
also provides an intermediate space-frequency
structure known as a precinct. A precinct is a
collection of spatially contiguous codeblocks from
all subbands at a particular resolution level.
In addition to these structures, JPEG 2000
organizes the compressed data from the codeblocks into units known as packets and layers
during the tier-2 coding step. For each precinct,
the compressed data for the codeblocks is first
organized into one or more packets. A packet is
simply a continuous segment in the compressed
codestream that consists of a number of bitplane
coding passes for each codeblock in the precinct.
The number of coding passes can vary from
codeblock to codeblock (including zero coding
passes). Packets from each precinct at all resolution levels in a tile are then combined to form
layers. In order to discuss packetization of the
compressed data, it is first necessary to introduce
the concepts of resolution grids and precinct
partitions. Throughout the following discussion,
it will be assumed that the image has a single tile
and a single component. The extension to multiple
tiles and components (which are possibly subsampled) is straightforward, but tedious, and it is
not necessary for understanding the basic concepts. Section B.4 of the JPEG 2000 standard [60]
provides a detailed description and examples for
the more general case.
3.1. Canvas coordinate system
During the application of the DWT to the input
image, successively lower resolution versions of the
input image are created. The input image can be
thought of as the highest resolution version. The
pixels of the input image are referenced with
respect to a high-resolution grid, known as the
reference grid. The reference grid is a rectangular
24
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
grid of points with indices from (0, 0) to (Xsiz1, Ysiz-1).7 If the image has only one component,
each image pixel corresponds to a high-resolution
grid. In case of multiple components with differing
sampling rates, the samples of each component are
at integer multiples of the sampling factor on the
high-resolution grid. An image area is defined by
the parameters (XOsiz, YOsiz) that specify the
upper left corner of the image, and extends to
(Xsiz-1, Ysiz-1) as shown in Fig. 16.
The spatial positioning of each resolution level,
as well as each subband, is specified with respect to
its own coordinate system. We will refer to each
coordinate system as a resolution grid. The
collection of these coordinate systems is known
as the canvas coordinate system. The relative
positioning of the different coordinate systems
corresponding to the resolution levels and subbands is defined in Section B.5 of the JPEG 2000
standard [60], and is also specified later in this
section. The advantage of the canvas coordinate
system is that it facilitates the compressed domain
implementation of certain spatial operations, such
as cropping and rotation by multiples of 901. As
will be described in Section 5.1.6, proper use of the
canvas coordinate system improves the performance of the JPEG 2000 encoder in case of
image at resolution level r (0prpNL ) is represented by the subband (NLr)LL. Recall from
Section 2.2.2 that the image at resolution r (r > 0)
is formed by combining the image at resolution
(r 1) with the subbands at resolution r; i.e.
subbands (NLr+1)HL, (NLr+1)LH and
(NLr+1)HH. The image area on the highresolution reference grid as specified by (Xsiz, Ysiz)
and (XOsiz, YOsiz) is propagated to lower resolution levels as follows. For the image area at
resolution level r (0prpNL ) the upper left-hand
corner is (xr0, yr0) and the lower right-hand corner
is (xr11, yr11), where
XOsiz
YOsiz
Xsiz
xr0 ¼ N r ; yr0 ¼ N r ; xr1 ¼ N r
2 L
2 L
2 L
Ysiz
ð22Þ
and yr1 ¼ N r ;
2 L
and Jwn denotes the smallest integer that is
greater than or equal to w:
The high-resolution reference grid is also
propagated to each subband as follows. The
positioning of the subband nbLL is the same as
that of the image at a resolution of (NLnb). The
positioning of subbands nbHL, nbLH, and nbHH is
specified as
8 XOsiz 2nb 1
YOsiz
>
>
>
;
>
>
2nb
2nb
>
>
>
<
XOsiz
YOsiz 2nb 1
ðxb0 ; yb0 Þ ¼
;
>
2nb
2nb
>
>
> >
>
XOsiz 2nb 1
YOsiz 2nb 1
>
>
;
:
2nb
2nb
multiple compression cycles when the image is
being cropped between compression cycles.
3.2. Resolution grids
Consider a single component image that is
wavelet transformed with NL decomposition levels, creating NL þ 1 distinct resolution levels. An
7
The coordinates are specified as (x, y), where x refers to the
column index and y refers to the row index.
for nb HL band;
for nb LH band;
ð23Þ
for nb HH band:
The coordinates (xb1, yb1) can be obtained from
Eq. (23) by substituting XOsiz with Xsiz and
YOsiz with Ysiz. The extent of subband b is from
(xb0, yb0) to (xb11, yb11). These concepts are
best illustrated by a simple example. Consider a
3-level wavelet decomposition of an original
image of size 768 (columns) 512 (rows). Let
the upper left reference grid point (XOsiz,
YOsiz) be (7, 9) for the image area. Then,
(Xsiz, Ysiz) is (775, 521). Resolution one extends
from (2, 3) to (193, 130) while subband 3HL,
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
(0,0)
(XOsiz,YOsiz)
Image
(Xsiz-1,Ysiz-1)
Fig. 16. The canvas coordinate system.
which belongs to resolution one, extends from
(1, 2) to (96, 65).
3.3. Precinct and codeblock partitioning
Each resolution level of a tile is further
partitioned into rectangular regions known as
precincts. Precinct partitioning makes it easier to
access the wavelet coefficients corresponding to a
particular spatial region of the image. The precinct
partition at resolution r induces a precinct
partitioning of the subbands at the same resolution
level, i.e. subbands (NLr+1)HL, (NLr+1)LH
and (NLr+1)HH. The precinct size can vary
from resolution to resolution, but is restricted to
be a power of two. Each subband is also divided
into rectangular codeblocks with dimensions that
25
are a power of two. The precinct and codeblock
partitions are both anchored at (0, 0). Each
precinct boundary coincides with a codeblock
boundary, but the reverse is not true, because a
precinct may consist of multiple codeblocks.
Codeblocks from all resolution levels are constrained to have the same size, except due to the
constraints imposed by the precinct size. For
codeblocks having the same size, those from lower
resolutions correspond to progressively larger
regions of the original image. For example, for
a three-level decomposition, a 64 64 codeblock
in subbands 1LL, 2LL and 3LL corresponds to
original image regions of size 128 128, 256 256
and 512 512, respectively. This diminishes the
ability of the codeblocks to provide spatial
localization. To alleviate this problem, the codeblock size at a given resolution is bounded by the
precinct size at that resolution. For example,
consider a 768 512 image that we wish to
partition into six 256 256 regions for efficient
spatial access. For a codeblock size of 64 64, the
precinct sizes for resolutions 0–3 can be chosen to
be 32 32, 32 32, 64 64 and 128 128, respectively. In this case, the actual codeblock size for
the 3LL, 3LH, 3HL and 3HH subbands would be
32 32. Fig. 17 shows the precinct partitions for a
three-level decomposition of a 768 512 image.
The highlighted precincts in resolutions 0–3
correspond roughly to the same 256 256 region
in the original image.
Fig. 17. Examples of precincts and codeblocks.
26
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
3.4. Layers and packets
The compressed bit-stream for each codeblock is
distributed across one or more layers in the
codestream. All of the codeblocks from all
subbands and components of a tile contribute
compressed data to each layer. For each codeblock, a number of consecutive coding passes
(including zero) is included in a layer. Each layer
represents a quality increment. The number of
coding passes included in a specific layer can vary
from one codeblock to another and is typically
determined by the encoder as a result of
post-compression rate-distortion optimization as
will be explained in Section 4.2. This feature offers
great flexibility in ordering the codestream. It also
enables spatially adaptive quantization. Recall
that all the codeblocks in a subband must use
the same quantizer step-size. However, the layers
can be formed in such a manner that certain
codeblocks, which are deemed perceptually more
significant, contribute a greater number of coding
passes to a given layer. As discussed in Section
2.3.1, this reduces the effective quantizer step-size
for those codeblocks by a power of two compared
to other codeblocks with less coding passes in
that layer.
The compressed data belonging to a specific tile,
component, resolution, layer and precinct is
aggregated into a packet. The compressed data
in a packet needs to be contiguous in the
codestream. If a precinct contains data from more
than one subband, it appears in the order HL, LH
and HH. Within each subband, the contributions
from codeblocks appear in the raster order. Fig. 17
shows an example of codeblocks belonging to a
precinct. The numbering of the codeblocks represents the order in which the coded data from the
codeblocks will appear in a packet.
3.5. Packet header
A packet is the fundamental building block in a
JPEG 2000 codestream. Each packet starts with a
packet header. The packet header contains information about the number of coding passes for
each codeblock in the packet. It also contains the
length of the compressed data for each codeblock.
The first bit of a packet header indicates whether
the packet contains data or is empty. If the packet
is non-empty, codeblock inclusion information is
signaled for each codeblock in the packet. This
information indicates whether any compressed
data from a codeblock is included in the packet.
If compressed codeblock data has already been
included in a previous packet, this information is
signaled using a single bit. Otherwise, it is signaled
with a separate tag-tree for the corresponding
precinct. The tag-tree is a hierarchical data
structure that is capable of exploiting spatial
redundancy. If codeblock data is being included
for the first time, the number of most significant
bitplanes that are entirely zero is also signaled with
another set of tag-trees for the precinct. After this,
the number of coding passes for the codeblock and
the length of the corresponding compressed data
are signaled.
The arithmetic encoding of the bitplanes is
referred to as tier-1 coding, whereas the packetization of the compressed data and encoding of the
packet header information is known as tier-2
coding. In order to change the sequence in which
the packets appear in the codestream, it is necessary to decode the packet header information, but
it is not necessary to perform arithmetic decoding.
This allows the codestream to be reorganized with
minimal computational complexity.
3.6. Progression order
The order in which packets appear in the
codestream is called the progression order and is
controlled by specific markers. Regardless of the
ordering, it is necessary that coding passes for each
codeblock appear in the codestream in causal
order from the most significant bit to the least
significant bit. For a given tile, four parameters are
needed to uniquely identify a packet. These are
component, resolution, layer and position (precinct). The packets for a particular component,
resolution and layer are generated by scanning the
precincts in a raster order. All the packets for a
tile can be ordered by using nested ‘‘for loops’’
where each ‘‘for loop’’ varies one parameter from
the above list. By changing the nesting order of
the ‘‘for loops’’, a number of different progression
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
orders can be generated. JPEG 2000 Part 1 allows
only five progression orders, which have been
chosen to address specific applications. They are
(i) layer–resolution–component–position progression; (ii) resolution–layer–component–position
progression; (iii) resolution–position–component–
layer progression; (iv) position–component–resolution–layer progression; and (v) component–
position–resolution–layer progression. These progression orders share some similarities with the
different modes of the extended DCT-based JPEG
standard as will be pointed out in the subsequent
subsections.
To illustrate these different orderings, consider a
three-component color image of size 768 512
with two layers and three decomposition levels
(corresponding to four resolution levels). The
precinct partition is as shown in Fig. 17. The
component, resolution, layer and position are
indexed by c; r; l and k; respectively. It is possible
that the components of an image have different
number of resolution levels. In that case, the LL
subbands of different components are aligned.
3.6.1. Layer–resolution–component–position
progression (LRCP)
This type of progression is obtained by arranging the packets in the following order:
for each l ¼ 0; 1
for each r ¼ 0; 1, 2, 3
for each c ¼ 0; 1, 2
for each k ¼ 0; 1, 2, 3, 4, 5
packet for component c; resolution r;
layer l; and position k:
This type of progression order is useful in an image
database browsing application, where progressively refining the quality of an image may be
desirable. This mode has no exact counterpart in
the existing JPEG. However, the ‘‘sequential
progressive’’ mode of extended JPEG (component
non-interleaved format) provides similar functionality for a single resolution image.
3.6.2. Resolution–layer–component–position
progression (RLCP)
This type of progression order is obtained
by interleaving the ‘‘for loops’’ in the order
27
r; l; c and k; starting with the outermost
‘‘for loop’’. It is useful in a client–server application, where different clients might demand
images at different resolutions. This progression
order is similar to ‘‘hierarchical progressive’’
mode of extended JPEG where each resolution is further encoded with the ‘‘sequential
progressive’’ mode (component non-interleaved
format).
3.6.3. Resolution–position–component–layer
progression (RPCL)
This type of progression order is obtained by
interleaving the ‘‘for loops’’ in the order r; k; c and
l; starting with the outermost ‘‘for loop’’. It can be
used when resolution scalability is needed, but
within each resolution, it is desirable that all
packets corresponding to a precinct appear contiguously in the compressed bit-stream. The
‘‘resolution–position–component’’ order for a
single layer can be obtained using the 27 ‘‘hierarchical progressive’’ mode of extended JPEG
with each resolution encoded with baseline
JPEG (component interleaved format).
3.6.4. Position–component–resolution–layer
progression (PCRL)
This type of progression order is obtained by
arranging the ‘‘for loops’’ in the order k; c; r and l;
starting with the outermost ‘‘for loop’’. It should
be used if it is desirable to refine the image quality
at a particular spatial location. The ‘‘position–
component’’ order is similar to JPEG baseline
where the image is sequentially compressed by
compressing component interleaved 8 8 blocks
in a raster order fashion.
3.6.5. Component–position–resolution–layer
progression (CPRL)
This type of progression order is obtained by
arranging the ‘‘for loops’’ in the order c; k; r and l;
starting with the outermost ‘‘for loop’’. It should
be used if it is desirable to obtain highest quality
image for a particular spatial location only for
a specific image component. The ‘‘component–
position’’ order is similar to the JPEG baseline
where the image is sequentially compressed by
28
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
compressing each color component separately in a
raster order fashion.
In the last three progression orders, the
‘‘for loop’’ corresponding to the variable k, which
determines the order in which the precincts appear
in the codestream, can become complicated if
different components have different precinct sizes
as is explained in the standard document [60]. The
JPEG 2000 syntax offers the flexibility of changing
from one progression order to another in the
middle of the codestream. For example, a digital
camera image might start out in the RLCP order
to provide a thumbnail. The order then may be
switched to LRCP to facilitate rate control and
truncation after the image has been captured.
Figs. 18–20 illustrate some of these progression
orders for the ‘‘boy’’ image (768 512, monochrome). In these examples, the DWT has three
decomposition levels, the (9, 7) filter-bank is used,
and the precinct sizes for the subbands at
resolutions 0, 1, 2 and 3 are 32 32, 32 32,
64 64 and 128 128, respectively. The codeblock
size is 64 64, except for resolutions 0 and 1 where
the codeblock size is constrained to the precinct
size of 32 32. Thus, there are four resolutions, six
precincts per resolution, and two layers, resulting
in 48 packets. Fig. 18 shows the LRCP progression
order (Section 3.6.1). The image has been reconstructed at the two quality levels of 0.125 bits/pixel
and 0.5 bits/pixel by decoding 24 and 48 packets,
respectively. Fig. 19 illustrates the RLCP ordering
(Section 3.6.2). The figure shows images reconstructed after decoding resolutions 0, 1, 2 and 3
(12, 24, 36 and 48 packets), respectively. Fig. 20
illustrates the PCRL ordering (Section 3.6.4). The
image has been reconstructed after decoding 32
packets corresponding to the first four precincts.
It should be noted that due to the prediction
step in the ‘‘hierarchical progressive’’ mode of
extended JPEG, before decoding any data at a
given resolution, it is necessary to fully decode all
the data corresponding to the lower resolution
versions of the image. This inter-resolution dependency makes it impossible to achieve 28 certain
progression orders, e.g. LRCP. Also, rearranging
the JPEG compressed data from one progression
Fig. 18. Example of layer progressive bit-stream ordering, (left) 0.125 bpp; (right) 0.50 bpp.
Fig. 19. Example of resolution progressive bit-stream ordering.
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
29
a codestream that conforms to the standardized
syntax. In the following, a brief discussion of
several possible approaches to JPEG 2000 rate
control is provided.
4.1. Rate control using explicit q-table
Fig. 20. Example of spatially progressive bit-stream ordering
(four precincts decoded).
mode to another generally requires an inverse
DCT, for example, when converting from the
‘‘hierarchical progressive’’ to the ‘‘sequential
progressive’’ mode. With JPEG 2000, a given
progression order can be converted into another
without the need for arithmetic decoding or
inverse wavelet transform by simply rearranging
the packets. This only requires decoding of the
packet headers to determine the length of each
packet.
4. Rate control
Rate control refers to the process of generating
an optimal image for a target file size (bit-rate) and
is strictly an encoder issue. The criterion for
optimality can be based on mean squared error
(MSE) between the original and reconstructed
image, visual distortion, or any other metric. In
the existing JPEG standard, the user only has
control over the selection of the quantization and
Huffman tables, which does not provide an easy
mechanism for compressing an image to a desired
bit-rate. A typical JPEG rate control algorithm
starts with a basic q-table and iteratively modifies
the q-table elements (e.g., by a scale factor) until
the desired bit-rate is achieved. In contrast, the
embedded block coding scheme of the JPEG 2000
and its flexible codestream syntax allow for the
generation of a rate-distortion (R–D) optimized
codestream for a given file size. Each JPEG 2000
encoder can perform its own optimization
(based on the distortion metric used) to generate
One approach is to use an explicit q-table similar
to JPEG, where a quantizer step-size is specified
for each subband and signaled explicitly as header
information. As mentioned before, this approach
suffers from the drawback that the q-table needs to
be modified (e.g., scaled) iteratively to achieve the
desired bit-rate. Although there is no need to
perform the wavelet transform at each iteration,
the quantization and coding processes still need to
be performed.
In most applications, humans are the ultimate
judges of perceived image quality, and it is
important to consider the properties of the HVS
when designing a q-table [4,19,31,49]. The general
approach to q-table design is to take advantage of
the sensitivity variations of the HVS to different
spatial frequencies. Although q-tables can be
designed through actual observer experiments, as
was done in developing the example q-tables
specified in the existing JPEG standard, such
experiments are laborious and must be repeated
each time that the viewing conditions are changed.
A more efficient approach is to use a contrast
sensitivity function (CSF) model as described in
[19]. The application of a CSF model requires
knowledge of the intended viewing conditions,
such as viewing distance, displayed pixel size,
display noise, and light adaptation level. In
general, the use of a CSF model implies that
the q-tables are designed for threshold-level
compression. That is, the q-tables will produce
errors that are barely perceptible in the compressed image under the viewing conditions for
which the q-tables were developed.
4.2. Rate control using the EBCOT algorithm
In [44], Taubman proposed an efficient rate
control method for the EBCOT compression
algorithm that achieves a desired rate in a single
iteration with minimum distortion. This method
30
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
can also be used by a JPEG 2000 encoder, with
several possible variations.
In the basic approach, each subband is first
quantized using a very fine step-size, and the
bitplanes of the resulting codeblocks are entropy
coded. This typically generates more coding passes
for each codeblock than will be eventually
included in the final codestream. If the quantizer
step-size is chosen to be small enough, the R–D
performance of the algorithm is independent of the
initial choice of the step-size. Next, a Lagrangian
R–D optimization is performed to determine the
number of coding passes from each codeblock that
should be included in the final compressed bitstream to achieve the desired bit-rate. If more than
a single layer is desired, this process can be
repeated at the end of each layer to determine
the additional number of coding passes from
each codeblock that need to be included in the
next layer.
The Lagrangian R–D optimization works in the
following manner. The compressed bit-stream
from each codeblock contains a large number of
potential truncation points that can occur at the
end of each sub-bitplane pass. The wavelet
coefficients y(u, v) contained in a codeblock of
subband b are initially quantized with a step-size
of Db ; resulting in an Mb -bit quantizer index for
each coefficient. If the codeblock bit-stream is
truncated so that only Nb bits are decoded, the
effective quantizer step-size for the coefficients is
Db 2Mb Nb : The inclusion of each additional bitplane in the compressed bit-stream will decrease
the effective quantizer step-size by a factor of two.
However, the effective quantizer step-size might
not be the same for every coefficient in a given
codeblock due to the inclusion of some coefficients
in the sub-bitplane at which the truncation occurs.
For each sub-bitplane, the increase in bit-rate and
the reduction in distortion resulting from the
inclusion of that sub-bitplane in the bit-stream
are calculated. The distortion measure selected is
usually MSE or visually weighted MSE, although
any general distortion measure that is additive
across codeblocks can be used. Let the total
number of codeblocks for the entire image be P;
and let the 30 codeblocks in the image be denoted
by Bi ; 1pipP: For a given truncation point t in
codeblock Bi ; the associated weighted MSE
distortion Dti is given by
X
Dti ¼ a2b
wi ðu; vÞ½yi ðu; vÞ yti ðu; vÞ2 ;
ð24Þ
u;v
where u and v represent the coefficient row and
column indices within the codeblock Bi ; yi(u, v) is
the original coefficient value; yti (u, v) is the quantized coefficient value for truncation point t;
wi(u, v) is a weighting factor for coefficient yi(u, v);
and ab is the L2-norm for subband b: Under
certain assumptions about the quantization noise,
this distortion is additive across codeblocks. At the
given truncation point t; the size of the associated
compressed bit-stream (i.e., the rate) for the
codeblock Bi is determined and denoted by Ri t.
Given a total bit budget of R bytes for the
compressed bit-stream, the EBCOT rate control
algorithm finds the truncation point for each
codeblock that minimizes the total distortion D:
This is equivalent to finding the optimal bit
allocation for all of the codeblocks, Ri , 1pipP;
such that
X
X
D¼
Dni is minimized subject to
Rni pR:
i
i
ð25Þ
In the JPEG 2000 literature, this rate control
algorithm is also referred to as post-compression
R–D optimization. If the weighting factor wi(u, v) is
set to unity for all subband coefficients, the
distortion metric reduces to the mean-squared
error. A visual weighting strategy can also be used
in conjunction with the EBCOT rate control
algorithm as will be discussed next.
4.2.1. Fixed visual weighting
The CSF model used to design the q-tables for
explicit quantization of the wavelet coefficients
can also be used to derive the weighting factors
wi (u, v). For example, once the CSF-based
quantization step-sizes have been computed for a
given viewing condition (Section 4.1), the weighting factor for all the coefficients in a subband can
be set equal to the square of the reciprocal of these
step-sizes. Table J-24 from Part 1 of the JPEG
2000 standard [60] lists recommended frequency
weightings for three different viewing conditions.
This approach is known as fixed visual weighting.
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
4.2.2. Bit-stream truncation and layer
construction
In the EBCOT rate control algorithm, an image
is compressed in such a way that the minimum
distortion is achieved at the desired bit-rate.
However, it is sometimes desirable to truncate an
existing JPEG 2000 codestream to achieve a
smaller bit-rate. For example, this scenario could
take place in a digital camera where the already
captured and compressed images have to be
truncated to enable storage of a newly captured
image. The question that arises is whether the
truncated bit-stream also achieves the minimum
distortion for the smaller bit-rate. In other words,
we want the visual quality of the image from
the truncated codestream to be as close as possible to the visual quality of the image that
would be produced by compressing directly to
that bit-rate.
This property can be achieved only if the image
was initially encoded with a number of layers using
the LRCP ordering of the packets as described in
Section 3.6.1. The layers can be designed using
R–D optimization so that the minimum distortion
for the resulting bit-rate is achieved at each layer
boundary. However, the quality of the resulting
truncated image might not be optimal if the
truncation point for the desired bit-rate does not
fall on a layer boundary. This is because the nonboundary truncation of a layer in LCRP ordering
will result in a number of packets being discarded.
If the desired bit-rates or quality levels are known
in advance for a given application, it is recommended that the layers be constructed accordingly.
If the exact target bit-rates are not known a priori,
it is recommended that a large number of layers
(e.g., 50) be formed. This provides the ability to
approximate a desired bit-rate while still truncating at a layer boundary. As demonstrated in
Section 5.1.7, the impact of the resulting overhead
on PSNR is quite small.
4.2.3. Progressive visual weighting
In fixed visual weighting, the visual weights are
chosen according to a single viewing condition.
However, if the bit-stream is truncated, this
viewing condition may be inappropriate for the
reduced quality image. Consider a case where the
31
compressed bit-stream has a number of layers,
each corresponding to a potential truncation
point. If the bit-stream is truncated at a layer
boundary with a very low bit-rate, the resulting
image quality would be poor and the image might
be viewed at a larger viewing distance than the one
intended for the original compressed bit-stream.
As a result, in some applications it might be
desirable to have each layer correspond to a
different viewing condition. In an embedded coder
such as JPEG 2000, it is not possible to change the
subband quantization step-sizes for each layer.
However, if a nominal viewing condition can be
associated with each layer, a corresponding set of
visual weighting factors wi (u, v) can be used during
the R–D optimization process for that layer [22].
This is known as progressive visual weighting.
5. Performance comparison of JPEG 2000 encoder
options
The JPEG 2000 standard offers a number of
encoder options that directly affect the coding
efficiency, speed and implementation complexity.
In this section, we primarily compare the effects of
various coding options on the coding efficiency for
lossless compression and on the rate-distortion
performance for lossy compression. It is more
difficult to accurately compare the speed and
implementation complexity of different coding
options, so we only point out the relative speed/
complexity advantages of certain options.
To obtain the reported results, the three test
images shown in Fig. 21, ‘‘Bike’’, ‘‘Cafe! ’’ and
‘‘Woman’’ of size 2048 (columns) 2560 (rows)
were chosen from the JPEG 2000 test set. All three
images are grayscale and have a bit-depth of 8 bits/
sample. For lossy compression, distortion was
characterized by the peak signal to noise ratio
(PSNR), which for an 8-bit decompressed image is
defined as
2552
PSNR ¼ 10 log10
;
ð26Þ
MSE
where MSE refers to the mean squared error
between the original image and the reconstructed
image. In most cases, the results are presented as
32
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
Fig. 21. Test images (from left to right) Bike, Caf!e, Woman.
the average PSNR of the three images. We use the
average PSNR instead of the PSNR corresponding
to the average MSE in accordance to the practice
of JPEG 2000 core experiments.
During the development of the JPEG 2000
standard, the committee maintained a software
implementation of an encoder and decoder that
contained all the technologies considered for the
inclusion in the standard as of that time. This was
also accompanied by a textual description of the
technologies. Both the software and the textual
description were referred to as the Verification
Model (VM). After each meeting of the JPEG
committee, the VM was updated to reflect any
approved modifications. For the results in this
section, we used the JPEG 2000 Verification
Model Version 8.6 (VM8.6) [68]. During each
simulation, only one 32 parameter was varied
while the others were kept constant in order to
study the effect of a single parameter on compression performance.
As a reference implementation, we used the
following set of compression parameters: single
tile; 5 levels of wavelet decomposition; 64 64
codeblocks; and a single layer. The subbands at
each resolution were treated as a single precinct. In
case of irreversible (9, 7) lossy compression, the
reciprocal of the L2-norm was used as the
fundamental quantization step-size for each subband. In the case of reversible (5, 3) lossless and
lossy compression, the quantizer step-size was set
to unity for all subbands as required by the JPEG
2000 standard. Hence, when using the reversible
(5, 3) filter-bank for lossy compression, rate control is possible only by discarding bits from the
integer representation of the index for the quantized wavelet coefficient.
The results for lossy coding are reported for bitrates of 0.0625, 0.125, 0.5, 1.0 and 2.0 bits/pixel
(bpp). To achieve a target bit-rate, the compressed
codeblock bit-streams were truncated to form a
single layer. The truncation points are determined
using the EBCOT post-compression R–D optimization procedure as described in Section 4.2. We
chose this alternative instead of varying the
quantizer step-size for the following reason.
Suppose that a particular step-size is used to
achieve a target bit-rate without any truncation of
the compressed bit-stream. Then, varying a single
coding parameter while keeping the step-size the
same results in a different distortion as well as a
different rate. In that case, the only meaningful
way to compare results is by plotting the ratedistortion curves (as opposed to single R–D
points). Hence, it is more effective to present the
comparisons in a tabular form by comparing the
PSNR’s of different coding options for fixed bitrates by using the EBCOT rate control algorithm.
5.1. Lossy results
5.1.1. Tile size
JPEG 2000 allows spatial partitioning of the
image into tiles. Each tile is wavelet transformed
and coded independently. In fact, a different
number of decomposition levels can be specified
for each component of each tile. Smaller tile sizes
are particularly desirable in memory-constrained
applications or when access to only a small portion
of the image is desired (e.g., remotely roaming
33
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
the precinct size in a particular subband is less
than the codeblock size, the codeblock size is set
equal to the precinct size.
Table 6 compares the effect of varying the
codeblock size on R–D performance with the (9, 7)
filter. There is very little loss of PSNR (maximum
of 0.14 dB) in going from a codeblock size of
64 64 to a codeblock size of 32 32. However,
codeblock sizes smaller than 32 32 result in a
significant drop in PSNR. There are several factors
that contribute to this phenomenon. One factor is
the overhead information contained in the packet
header. The packet header contains information
about the number of coding passes and the length
of compressed data for each codeblock, so the
total size of the header information increases with
an increasing number of codeblocks. Another
factor is the independent encoding of each codeblock that requires the re-initialization of the
arithmetic coding models. As the codeblock size
becomes smaller, the number of samples required
to adapt to the underlying probability models
constitutes a greater portion of the total number of
over a large image). Table 5 compares the R–D
performance of the JPEG 2000 encoder for various
tile sizes with the (9, 7) filter. It is evident that the
compression performance decreases with decreasing tile size, particularly at low bit-rates. Furthermore, at low bit-rates where the tile boundaries are
visible in the reconstructed image, the perceived
quality of the image might be lower than that
indicated by the PSNR. The impact of the
boundary artifacts can be reduced by using postprocessing techniques, such as those employed in
reducing the blocking artifacts in low bit-rate
DCT-based JPEG and MPEG images. Part 2 of
the JPEG 2000 standard offers the option of using
single-sample overlap DWT (SSO-DWT), which
reduces edge artifacts at the tile boundaries.
5.1.2. Codeblock size
The codeblocks in JPEG 2000 are rectangular
with user-defined dimensions that are identical for
all subbands. Each dimension has to be a power of
two, and the total number of samples in a
codeblock cannot exceed 4096. Furthermore, when
Table 5
Comparison of R–D performance for different tile sizes with the (9, 7) filter-bank
Rate
(bits/pixel)
0.0625
0.125
0.25
0.5
1.0
2.0
Average PSNR in dB
No tiling
512 512
256 256
192 192
128 128
22.82
24.84
27.61
31.35
36.22
42.42
22.73
24.77
27.55
31.30
36.19
42.40
22.50
24.59
27.41
31.19
36.11
42.34
22.22
24.38
27.20
30.99
35.96
42.22
21.79
24.06
26.96
30.82
35.85
42.16
(0.09)
(0.07)
(0.06)
(0.05)
(0.03)
(0.02)
(0.32)
(0.25)
(0.20)
(0.16)
(0.11)
(0.08)
(0.60)
(0.46)
(0.41)
(0.36)
(0.26)
(0.20)
(1.03)
(0.78)
(0.65)
(0.53)
(0.37)
(0.26)
Table 6
Comparison of R–D performance for various codeblock sizes with the (9, 7) filter-bank
Rate
(bits/pixel)
0.0625
0.125
0.25
0.5
1.0
2.0
Average PSNR in dB
64 64
32 32
16 16
88
22.82
24.84
27.61
31.35
36.22
42.42
22.78
24.78
27.52
31.22
36.09
42.28
22.62
24.57
27.23
30.84
35.68
41.83
22.27
24.13
26.63
30.04
34.70
40.70
(0.04)
(0.06)
(0.09)
(0.13)
(0.13)
(0.14)
(0.20)
(0.27)
(0.38)
(0.51)
(0.54)
(0.59)
(0.55)
(0.71)
(0.98)
(1.31)
(1.52)
(1.72)
34
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
samples encoded. In addition, the pixels that lie on
the boundary of a codeblock have an incomplete
context since pixels from neighboring codeblocks
cannot be used in forming the coding contexts.
As the codeblock size decreases, the percentage
of boundary pixels with incomplete contexts
increases.
It can also be concluded from Table 6 that the
loss in compression performance with decreasing
codeblock size is more pronounced at higher bitrates. This can be explained as follows. When the
bit-rate is high, more coding passes are encoded,
and the inefficiencies due to model mismatch and
incomplete contexts add up. In comparison, at low
bit-rates, many codeblocks from the higher frequency subbands contribute no compressed data
to the compressed bit-stream. The JPEG 2000 bitstream syntax has provisions to signal this
information very efficiently, and for these codeblocks, a smaller size has almost no impact on the
coding efficiency. Moreover, these high frequency
subbands represent a large percentage of the total
number of codeblocks.
5.1.3. DWT filters
Part 1 of JPEG 2000 offers a choice of either the
(9, 7) or the (5, 3) filter-bank for lossy compression.
Fig. 22 compares the energy compaction of the
(9, 7) and the (5, 3) filter-banks graphically. Each
subband has been scaled with its L2-norm to reflect
its proper contribution to the overall energy.
Moreover, for better visualization of the subband
energies, the AC subbands of both images have
been scaled up by a factor of two, while the LL
subbands have been scaled down by a factor of
eight. It can be seen that the LL subband of the
(9, 7) filter-bank has a higher contrast, which
implies superior energy compaction.
Table 7 compares the R–D performance of the
two filter-banks. The (9, 7) filter-bank consistently
outperforms the (5, 3) filter-bank with the performance gap increasing with increasing bit-rate.
However, it should be noted that the (5, 3) filterbank can also perform lossless compression. Thus,
for a particular image, when the target bit-rate
equals the lossless bit-rate, the (5, 3) filter-bank
would result in zero MSE or infinite PSNR,
whereas the (9, 7) filter-bank would result in a
non-zero MSE. Thus for bit-rates in the range of
4.0 bits/pixel or above, the (5, 3) filter-bank
Table 7
Comparison of R–D performance of the irreversible (9, 7) and
the reversible (5, 3) filter-banks
Rate
(bits/pixel)
0.0625
0.125
0.25
0.5
1.0
2.0
Average PSNR in dB
Irreversible (9, 7)
Reversible (5, 3)
22.82
24.84
27.61
31.35
36.22
42.42
22.37
24.37
27.04
30.74
35.48
41.33
Fig. 22. Energy compaction comparison between the irreversible (5, 3) and (9, 7) filter-banks.
(0.45)
(0.47)
(0.57)
(0.61)
(0.74)
(1.09)
35
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
Table 8
Comparison of R–D performance for various levels of decomposition with the (9, 7) filter-bank
Rate
(bits/pixel)
0.0625
0.125
0.25
0.5
1.0
2.0
Average PSNR in dB
5 levels
4 levels
3 levels
2 levels
1 level
22.82
24.84
27.61
31.35
36.22
42.42
22.77
24.80
27.57
31.33
36.21
42.42
22.47
24.62
27.45
31.24
36.15
42.37
21.50
23.91
26.94
30.87
35.91
42.26
17.68
21.67
25.54
29.71
35.15
41.71
(0.05)
(0.04)
(0.04)
(0.02)
(0.01)
(0.01)
performs better than the (9, 7) filter-bank. In
addition, the 34 (5, 3) filter-bank has much less
computational complexity as it can be implemented using only integer arithmetic.
(0.35)
(0.22)
(0.16)
(0.11)
(0.07)
(0.05)
(1.30)
(0.93)
(0.67)
(0.48)
(0.31)
(0.16)
(5.12)
(3.17)
(2.07)
(1.64)
(1.07)
(0.71)
5.1.5. Lazy, parallel and lazy–parallel modes
As mentioned in Section 2.4, JPEG 2000
provides several entropy coding options that
facilitate the parallel processing of the quantized
coefficient bitplanes. The collection of these coding
options is termed the parallel mode. Another
option that reduces the computational complexity
of the entropy encoder (especially at high bit-rates)
is the lazy mode, where only the cleanup pass is
arithmetic encoded after the fourth most significant bitplane. Table 9 shows the R–D performance
of the parallel, lazy and lazy–parallel modes
relative to the reference implementation. It can
be seen that the loss in PSNR is generally
small (0.01–0.3 dB) and increases with increasing
bit-rate.
5.1.4. Wavelet decomposition levels
The number of decomposition levels affects the
coding efficiency of a JPEG 2000 encoder as well
as the number of resolutions at which an image
can be decompressed. In general, the number of
decomposition levels does not impact the computational complexity significantly because only the
LL band is further split at each level. Table 8
compares the R–D performance of the JPEG 2000
encoder for different numbers of decomposition
levels with the (9, 7) filter. Our simulations show
that the PSNRs resulting from five and eight levels
of decomposition are practically indistinguishable.
Thus, a 5-level decomposition is adequate even for
high-resolution images. Finally, the loss is greatest
at lower bit-rates and it tapers off with increasing
bit-rate.
5.1.6. Effect of multiple compression cycles
Table 10 examines the effect of multiple
compression cycles on PSNR where an image is
compressed and reconstructed multiple times to
the same bit-rate. Our reference implementation
Table 9
R–D performance of ‘‘lazy’’, ‘‘parallel’’ and ‘‘lazy-parallel’’ modes with the (9, 7) filter-bank
Rate
(bits/pixel)
0.0625
0.125
0.25
0.5
1.0
2.0
Average PSNR in dB
Reference
Lazy
22.82
24.84
27.61
31.35
36.22
42.42
22.81
24.82
27.57
31.28
36.10
42.28
(0.01)
(0.02)
(0.04)
(0.07)
(0.12)
(0.14)
Parallel
Lazy–parallel
22.76
24.76
27.49
31.19
36.03
42.22
22.75
24.74
27.46
31.14
35.94
42.12
(0.06)
(0.08)
(0.12)
(0.16)
(0.19)
(0.20)
(0.07)
(0.10)
(0.15)
(0.21)
(0.28)
(0.30)
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M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
Table 10
R–D performance of multiple compression cycles with the (9, 7) filter-bank
Rate
(bits/pixel)
Average PSNR in dB
0.0625
0.125
0.25
0.5
1.0
2.0
1 iteration
4 iterations
8 iterations
16 iterations
22.82
24.84
27.61
31.35
36.22
42.42
22.78
24.80
27.57
31.32
36.19
42.39
22.77
24.78
27.56
31.30
36.17
42.37
22.76
24.76
27.54
31.28
36.16
42.36
(0.04)
(0.04)
(0.04)
(0.03)
(0.03)
(0.03)
(0.05)
(0.06)
(0.05)
(0.05)
(0.05)
(0.05)
(0.06)
(0.08)
(0.07)
(0.07)
(0.06)
(0.06)
Table 11
R–D performance of multiple compression cycles with the (9, 7) filter-bank
Rate
(bits/pixel)
Average PSNR in dB
Reference
0.0625
0.125
0.25
0.5
1.0
2.0
22.82
24.84
27.61
31.35
36.22
42.42
No canvas coordinate system
Canvas coordinate system
4 iterations
16 iterations
4 iterations
16 iterations
21.14
22.74
25.16
28.61
33.30
39.26
18.58
20.30
22.75
26.40
31.29
37.08
22.78
24.80
27.57
31.32
36.19
42.39
22.76
24.76
27.54
31.28
36.16
42.36
(1.68)
(2.10)
(2.45)
(2.74)
(2.92)
(3.16)
with the (9, 7) filter was used in all cases. The postcompression R–D optimization engine is used to
achieve the desired bit-rate at each iteration. It can
be seen from the table that multiple compression
cycles cause very little degradation (0.03–0.08 dB)
in compression performance when the compression parameters are held constant.
Table 11 examines the effect of multiple
compression cycles when one image column is
cropped from the left side in between compression
cycles. Two scenarios are explored. In one case,
the image is always anchored at (0, 0) so that the
canvas coordinate system shifts by one column as
the image is cropped in between compression
cycles. This changes the alignment of the codeblocks. Furthermore, the column index for the
samples changes from odd to even and even to
odd, which results in a completely different set of
wavelet coefficients. In the other case, the anchoring point is shifted to preserve the codeblock
alignment using the canvas coordinate system.
(4.24)
(4.54)
(4.86)
(4.95)
(4.93)
(5.34)
(0.04)
(0.04)
(0.04)
(0.03)
(0.03)
(0.03)
(0.06)
(0.08)
(0.07)
(0.07)
(0.06)
(0.06)
In this case, only the wavelet coefficients near the
boundary of the image are affected by cropping.
From the table it can be seen that maintaining the
codeblock alignment leads to superior compression performance. More performance comparisons
can be found in [20].
5.1.7. JPEG 2000 versus JPEG baseline
Table 12 compares the R–D performance of
JPEG 2000 with JPEG baseline at equivalent bitrates for the reference test set. Our reference
implementation with the (9, 7) filter-bank was
used. The JPEG baseline PNSR results were
generated by iteratively compressing with JPEG
baseline to within 1% of the file size of the JPEG
2000-compressed image (including the file headers). The IJG code with the example luminance
q-table and a local Huffman table was used for
this purpose [57]. For at least one image from
our test set, rates of 0.0625 and 0.125 bits/pixel
were not achievable even when using a q-table
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M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
with all the entries set to the highest possible
value of 255; hence JPEG baseline results for those
rates are not listed in Table 12. It can be seen that
the use of JPEG 2000 results in about 2–4 dB
higher PSNR than JPEG baseline depending on
the bit-rate.
5.2. Lossless results
5.2.1. Reversible color transform (RCT)
It is well known that decorrelating the components of an image by applying a color transform
improves the coding efficiency. For example, RGB
images are routinely transformed into YCb Cr
before applying JPEG compression. In a similar
fashion, a lossless component transform can be
beneficial when used in conjunction with lossless
coding. Table 13 compares the performance of the
JPEG 2000 algorithm for lossless coding, with and
without applying the RCT transform. The results
are based on using the reversible (5, 3) filter-bank
with the reference set of compression parameters.
Table 12
R–D performance of JPEG2 000 and JPEG baseline for the
‘‘Lena’’ image
Rate
(bits/pixel)
0.0625
0.125
0.25
0.5
1.0
2.0
Average PSNR in dB
5.2.2. Lossless encoder options
Tables 14–17 summarize the lossless compression performance of Part 1 of the JPEG 2000
standard as a function of tile size, number
of decomposition levels, codeblock size, and
Table 14
Comparison of average lossless bit-rates (bits/pixel) for
different tile sizes
No tiling 512 512 256 256 128 128 64 64 32 32
4.797
4.801
4.811
4.850
5.015
5.551
Table 15
Comparison of average lossless bit-rates (bits/pixel) for
different number of decomposition levels
JPEG 2000
JPEG baseline
5 levels
4 levels
3 levels
2 levels
1 level
0 levels
22.82
24.84
27.61
31.35
36.22
42.42
F
F
25.65
28.65
32.56
38.24
4.797
4.798
4.802
4.818
4.887
5.350
Table 16
Comparison of average lossless bit-rates (bits/pixel) for
different codeblock sizes
Table 13
Comparison of lossless bit-rates for color images with and
without RCT
Image
Lena
Baboon
Bike
Woman
Instead of using our reference 8-bit test images,
we used the 24-bit color version of ‘‘Lena’’
and ‘‘Baboon’’ images (of size 512 512), in
addition to 24-bit versions of the ‘‘Bike’’ and
‘‘Woman’’ images. From the table it can be seen
that applying the RCT transform prior to lossless
compression results in savings of 0.16–2.39 bpp,
which is quite significant in the context of lossless
coding.
64 64
32 32
16 16
88
4.797
4.846
5.005
5.442
Bit-rate in bits/pixel
No RCT
RCT
13.789
18.759
13.937
13.892
13.622
18.103
11.962
11.502
Table 17
Comparison of average lossless bit-rates (bits/pixel) for ‘‘lazy’’,
‘‘parallel’’ and ‘‘lazy–parallel’’ modes
Reference
Lazy
Parallel
Lazy–parallel
4.797
4.799
4.863
4.844
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M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
lazy–parallel modes. The bit rates have been
averaged over the three test images ‘‘Cafe! ’’, ‘‘Bike’’
and ‘‘Woman’’ and the reversible (5, 3) filter-bank
has been used. A rather surprising finding is that
the average lossless performance difference between the one-level and five-level decompositions
is very small (o0.1 bpp). This suggests that the
three-pass bitplane entropy coding scheme and the
associated contexts efficiently exploit the redundancy of correlated samples. There is a small
(although significant) performance penalty when
using a codeblock size of 16 16 or smaller, or a
tile size of 64 64 or smaller. Finally, there is only
a slight decrease in coding efficiency when using
the ‘‘lazy’’, ‘‘parallel’’ or ‘‘lazy–parallel’’ modes.
Table 18 compares the effect of multiple layers
on the lossless coding efficiency. As mentioned in
Section 4.2.2, in order to facilitate bit-stream
truncation, it is desirable to construct as many
layers as possible. However, the number of packets
increases linearly with the number of layers, which
also increases the overhead associated with the
packet headers. As can be seen from the table, the
performance penalty for using 50 layers is small
for lossless compression. However, this penalty is
expected to increase at lower bit-rates [27].
Whereas, increasing the number of layers from 7
to 50 does not linearly increase the lossless bit-rate
since the header information for the increased
number of packets is coded more efficiently. In
particular, the percentage of codeblocks that do
not contribute to a given packet increases with the
number of layers, and the packet header syntax
allows this information to be coded very efficiently
using a single bit.
5.2.3. Lossless JPEG 2000 versus JPEG-LS
Table 19 compares the lossless performance of
JPEG 2000 with JPEG-LS [69]. Although the
JPEG-LS has only a small performance advantage
Table 18
Comparison of average lossless bit-rates (bits/pixel) for
different number of layers
1 layer
7 layer
50 layer
4.797
4.809
4.829
Table 19
Comparison of average lossless bit-rates (bits/pixel) for JPEG
2000 and JPEG-LS
JPEG 2000
JPEG-LS
4.797
4.633
(3.4%) over JPEG 2000 for the images considered
in this study, it has been shown that for certain
classes of imagery (e.g., the ‘‘cmpnd1’’ compound
document from the JPEG 2000 test set), the JPEGLS bit-rate is only 60% of that of JPEG 2000 [27].
5.3. Bitplane entropy coding results
In this section, we examine the redundancy
contained in the various bitplanes of the quantized
wavelet coefficients. These results were obtained by
quantizing the wavelet coefficients of the ‘‘Lena’’
image with the default quantization step-size for
VM8.6 (‘‘-step 1/128.0’’). Since ‘‘Lena’’ is an 8-bit
image, the actual step-size used for each band was
2.0 divided by the L2-norm of that band. This had
the effect that equal quantization errors in
each subband had roughly the same contribution
to the reconstructed image MSE. Hence, the
bitplanes in different subbands were aligned by
their LSBs. Eleven of the resulting bitplanes
were encoded starting with the most significant
bitplane.
One way to characterize the redundancy is to
count the number of bytes that are generated by
each sub-bitplane coding pass. The number of
bytes generated from each sub-bitplane coding
pass are not readily available unless each coding
pass is terminated. However, during post-compression R–D optimization, VM8.6 computes the
number of additional bytes needed to uniquely
decode each coding pass using a ‘‘near optimal
length calculation’’ algorithm [68]. It is not
guaranteed that the ‘‘near optimal length calculation’’ algorithm will determine the minimum
number of bytes needed for unique decoding.
Moreover, it is necessary to flush the MQ-coder
registers for estimation of the number of bytes.
This means that the estimated bytes for a coding
pass contain some data from the next coding pass,
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M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
which can lead to some unexpected results. With
these caveats in mind, Table 20 contains the
number of bytes generated from each sub-bitplane
coding pass. The estimated bytes for each coding
pass were summed across all the codeblocks in the
image to generate these entries.
During the encoding of the first bitplane, there is
only a cleanup pass and 36 coefficients turn
significant. All of these significant coefficients
belong to the 5LL subband. In the refinement
pass of the next bitplane, only these 36 coefficients
are refined. Surprisingly, the first refinement bit for
Table 20
Coded bytes resulting for sub-bitplane passes of ‘‘Lena’’ image
Bitplane number
‘‘Significance’’ bytes
‘‘Refinement’’ bytes
‘‘Clean up’’ bytes
1
2
3
4
5
6
7
8
9
10
11
0
18
38
78
224
551
1243
2315
4593
10720
25421
0
0
13
37
73
180
418
932
1925
3917
8808
21
24
57
156
383
748
1349
2570
5465
12779
5438
Total for current BP
21
42
108
271
680
1479
3010
5817
11983
27416
39667
Total for all BPs
21
63
171
442
1122
2601
5611
11428
23411
50827
90494
Fig. 23. Reconstructed Lena image after decoding bit-planes 1 through 9 (from left to right and top to bottom).
40
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
Table 21
Coding statistics resulting from the encoding of wavelet coefficient bitplanes of ‘‘Lena’’ image
BP
Compression ratio
Rate (bits/pixel)
PSNR (dB)
Percent refined
Percent significant
Percent insignificant
1
2
3
4
5
6
7
8
9
10
11
12483
4161
1533
593
233
101
47
23
11.2
5.16
2.90
0.000641
0.00192
0.00522
0.0135
0.0343
0.0792
0.170
0.348
0.714
1.55
2.76
16.16
18.85
21.45
23.74
26.47
29.39
32.54
35.70
38.87
43.12
49.00
0.00
0.01
0.05
0.11
0.23
0.57
1.32
2.91
6.01
12.34
28.12
0.01
0.04
0.06
0.12
0.32
0.75
1.59
3.10
6.33
15.78
25.08
99.99
99.95
99.89
99.77
99.43
98.68
97.09
93.99
87.66
71.88
46.80
all of these 36 coefficients are zero. Due to the fast
model adaptation of the MQ-coder, very few
refinement bits are generated for the second
bitplane. This, in conjunction with the possibility
of overestimating the number of bytes in the
cleanup pass of the first bitplane, leads to the
rather strange result that the refinement pass
for the second bitplane requires zero bytes.
It is also interesting that the number of bytes
needed to encode a given bitplane is usually
greater than the total number of bytes used to
encode all of the bitplanes prior to it (except for
bitplane 11).
Fig. 23 shows images reconstructed from the
first nine bitplanes, and Table 21 provides
the corresponding PSNRs. Table 21 also shows
the percentage of the coefficients that are refined
at each bitplane; the percentage of the coefficients
that are found to be significant at each bitplane; and the percentage of the coefficients
that remain insignificant after the completion
of the encoding of a bitplane. It is interesting
to note that about 72% of the coefficients still
remain insignificant after encoding the tenth
bitplane.
6. Additional features and Part 2 extensions
6.1. Region of interest (ROI) coding
In some applications, it might be desirable to
encode certain portions of the image (called the
region of interest or ROI) at a higher level of
quality relative to the rest of the image (called the
background). Alternatively, one might want to
prioritize the compressed data corresponding to
the ROI relative to the background so that it
appears earlier in the codestream. This feature is
desirable in progressive transmission in case of
early termination of the codestream.
Region of interest coding can be accomplished
by encoding the quantized wavelet coefficients
corresponding to the ROI with a higher precision
relative to the background, e.g., by scaling up the
ROI coefficients or scaling down the background
coefficients. A scaling based ROI encoding method
would generally proceed as follows [6]. First, the
ROI(s) are identified in the image domain. Next, a
binary mask in the wavelet domain, known as the
ROI mask, is generated. The ROI mask has a value
of one at those coefficients that contribute to the
reconstruction of the ROI and has a value of zero
elsewhere. The shape of the ROI mask is
determined by the image domain ROI as well as
the wavelet filter-bank, and it can be computed in
an efficient manner for most regular ROI shapes
[29]. Prior to entropy coding, the bitplanes of the
coefficients belonging to the ROI mask are shifted
up (or the background bitplanes are shifted
down8) by a desired amount that can vary from
8
The main idea is to store the magnitude bits of the quantized
coefficients in the most significant part of the implementation
register so that any potential precision overflow would only
impact the LSB of the background coefficients.
41
one ROI to another within the same image. The
ROI shape information (in the image domain) and
the scaling factor used for each ROI is also
encoded and included in the codestream. In
general, the overhead associated with the encoding
of an arbitrary shaped ROI might be large unless
the ROI has a regular shape, e.g., a rectangle or a
circle, which can be described with a small set of
parameters. At the decoder, the ROI shape and
scaling factors are decoded, and the quantized
wavelet coefficients within each ROI (or background) coefficient are scaled to their original
values.
The procedure described above requires the
generation of an ROI mask at both the encoder
and decoder, as well as the encoding and decoding
of the ROI shape information. This increased
complexity is balanced by the flexibility to encode
ROIs with multiple qualities and to control the
quality differential between the ROI and the
background. To minimize decoder complexity
while still providing ROI capability, JPEG 2000
Part 1 has adopted a specific implementation of
the scaling based ROI approach known as the
Maxshift method [12].
In the Maxshift method, the ROI mask is
generated in the wavelet domain, and all wavelet
coefficients that belong to the background are
examined and the coefficient with the largest
magnitude is identified. Next, a value s is
determined such that 2s is larger than the largest
magnitude background coefficient, and all bitplanes of the background coefficients are shifted
down by s bits. This insures that the smallest nonzero ROI coefficient is still larger than the largest
background coefficient as shown in Fig. 24. The
presence of ROI is signaled to the decoder by a
marker segment and the value of s is included in
the codestream. The decoder first entropy decodes
all the wavelet coefficients. Those coefficients
whose values are less than 2s belong to the
background and are scaled up to their original
value. In the Maxshift method, the decoder is not
required to generate an ROI mask or to decode
any ROI shape information. Furthermore, the
encoder can encode any arbitrary shape ROI
within each subband, and it does not need to
encode the ROI shape information (although
Coefficient Value
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
Background
ROI Coefficients
Background
Fig. 24. Maxshift method of ROI coding in Part 1.
it may still need to generate an ROI mask). The
main disadvantage of the Maxshift method is
that ROIs with multiple quality differentials
cannot be encoded.
In the Maxshift method, the ROI coefficients are
prioritized in the codestream so that they are
received (decoded) before the background. However, if the entire codestream is decoded, the
background pixels will eventually be reconstructed
to the same level of quality as that of the ROI. In
certain applications, it may be desirable to encode
the ROI to a higher level of quality than the
background even after the entire codestream has
been decoded. The complete separation of the ROI
and background bitplanes in the Maxshift method
can be used to achieve this purpose. For example,
all the wavelet coefficients are quantized to the
precision desired for the ROI. The ROI coefficients
are encoded first, followed by the encoding of the
background coefficients in one or more layers. By
discarding a number of layers corresponding to the
background coefficients, any desired level of
quality can be achieved for the background.
Since the encoding of the ROI and the background coefficients in the Maxshift method are
completely disjoint processes, it might seem that
the ROI needs to be completely decoded before
any background information is reconstructed.
However, this limitation can be circumvented to
some extent. For example, if the data is organized
in the resolution progressive mode, the ROI data is
decoded first followed by the background data for
each resolution. As a result, at the start of
decoding for each resolution, the reconstructed
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M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
image will contain all the background data
corresponding to the lower resolutions. Alternatively, due to the flexibility in defining the ROI
shape for each subband, the ROI mask at each
resolution or subband can be modified to include
some background information. For example, the
entire LL subband can be included in the ROI
mask to provide low resolution information about
the background in the reconstructed image.
Experiments show that for the lossless coding of
images with ROIs, the Maxshift method increases
the bit rate by 1–8% (depending on the image size
and the ROI size and shape) compared to the
lossless coding of the image without ROI [12]. This
is a relatively small cost for achieving the ROI
functionality.
6.2. Error resilience
Many emerging applications of the JPEG 2000
standard require the delivery of the compressed
data over communications channels with different
error characteristics. For example, wireless communication channels are susceptible to random
and burst channel errors, while internet communication is prone to data loss due to traffic
congestion. To improve the transmission performance of JPEG 2000 in error prone environments,
Part 1 of the standard provides several options for
error resilience. The error resilience tools are based
on different approaches such as compressed data
partitioning and resynchronization, error detection, and Quality of Service (QoS) transmission
based on priority. The error resilience bit-stream
syntax and tools are provided both at the entropy
coding level and the packet level [24,28].
As discussed before, one of the main differences
between the JPEG 2000 coder and previous
embedded wavelet coders is in the independent
encoding of the codeblocks. Among the many
advantages of this approach is improved error
resilience, since any errors in the bit-stream,
corresponding to a codeblock will be contained
within that codeblock. In addition, certain entropy
coding options described in Section 2.4.3 can be
used to improve error resilience. For example, the
arithmetic coder can be terminated at the end of
each coding pass and the context probability
models can be reset. The optional lazy mode
allows the bypassing of the arithmetic coder for
the first two coding passes of each bitplane and
can help protect against catastrophic error propagation that is characteristic of all variable-length
coding schemes. Finally, JPEG 2000 provides for
the insertion of error resilience segmentation
symbols at the end of the cleanup pass of each
bitplane that can serve as error detection. The
segmentation symbol is a binary ‘‘1010’’ symbol
whose presence is signaled in the marker segments.
It is coded with the uniform arithmetic coding
context, and its correct decoding at the end of each
bitplane confirms the correctness of the decompressed data corresponding to that bitplane.
If the segmentation symbol is not decoded
correctly, the data for that bitplane and all
the subsequent bitplanes corresponding to that
codeblock should be discarded. This is because
the data encoded in the subsequent coding
passes of that codeblock depend on the previously
coded data.
Error resilience at the packet level can be
achieved by using resynchronization markers,
which provide for spatial partitioning and resynchronization. This marker is placed in front of
each packet in a tile, and it numbers the packets
sequentially starting at zero. Also, the packet
headers can be moved to either the main header
(for all tiles) or the tile header to create what is
known as short packets. In a QoS transmission
environment, these headers can be protected
more heavily than the rest of the data. If there
are errors present in the packet compressed
data, the packet headers can still be associated
with the correct packet by using the sequence
number included in the resynchronization
marker. The combination of these error resilience
tools can often provide adequate protection
in some of the most demanding error-prone
environments.
6.3. File format
Most digital imaging standards provide a file
format structure to encapsulate the coded image
data. While the codestream specifies the compressed image, the file format serves to provide
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
useful information about the characteristics of the
image and its proper use and display. Sometimes
the file format includes redundant information
that is also included in the codestream, but such
information is useful in that it allows trivial
manipulation of the file without any knowledge
of the codestream syntax. A minimal file format,
such as the one used in the JPEG baseline system,
includes general information about the number of
image components, their corresponding resolutions and bit depths, etc. However, two important
components of a more comprehensive file format
are colorspace and metadata. Without this information, an application might not know how to use
or display an image properly. The colorspace
defines how the decoded component values relate
to real world spectral information (e.g., sRGB or
YCb Cr ), while the metadata provides additional
information about the image. For example,
metadata can be used to describe how the image
was created (e.g., the camera type or photographer’s name) as well as describe how the image
should be used (e.g., IPRs related to the image,
default display resolution, etc.). It also provides
the opportunity to extract information about
an image without the need to decode it, which
enables fast text-based search in databases. The
SPIFF file format defined in Part 3 extensions
of the existing JPEG standard [56] was targeted
at 8-bit per component sRGB and YCb Cr
images, and there was limited capability for
metadata. The file format defined by the JPEG
2000 standard is much more flexible with respect
to both the colorspace specification and the
metadata embedding.
Part 1 of the JPEG 2000 standard defines a file
format referred to as JP2. Although this file format
is an optional part of the standard, it is expected
to be used by many applications. It provides a
flexible, but restricted, set of data structures to
describe the coded image data. In order to balance
flexibility with interoperability, the JP2 format
defines two methods of colorspace specification.
One method (known as the Enumerated
method) limits flexibility, but provides a high
degree of interoperability by directly specifying
only two colorspaces, sRGB and gray scale
(with YCb Cr support being added through an
43
amendment). Another method known as the
Restricted ICC (International Color Consortium
[53]) method, allows for the specification of a
colorspace using a subset of standard ICC
profiles, referred to in the ICC specification as
three-channel matrix-based and monochrome
input profiles. These profiles, which specify a
transformation from the reconstructed codevalues to the profile connection space (PCS),
contain at most three 1-D look-up tables followed
by a 3 3 matrix. These profile types were chosen
because of their simplicity. The Restricted ICC
method can simply be thought of as a data
structure that specifies a set of colorspace transformation equations. Finally, the JP2 file format
also allows for displaying palletized images, i.e.,
single component images where the value of the
single component represents an index into a
palette of colors.
The JP2 file format also defines two mechanisms
for defining and embedding metadata in a
compressed file. The first method uses a universal
unique identifier (UUID) while the second method
uses XML [54]. For both methods, the individual
blocks of metadata can be embedded almost
anywhere in the file. Although very few metadata
fields have been defined in the JP2 file format, its
basic architecture provides a strong foundation for
extension.
Part 2 of the standard defines extensions to
the JP2 file format, encapsulated in an extended
file format called JPX. These extensions increase
the colorspace flexibility by providing more
enumerated color spaces (and also allows
vendors to register additional values for colorspaces) as well as providing support for all
ICC profiles. They also add the capability for
specifying a combination of multiple images
using composition or animation, and add a
large number of metadata fields to specify
image history, content, characterization and
IPR.
6.4. Part 2: extensions
Decisions that were made by the JPEG 2000
committee about which technologies to include
in Part 1 of the JPEG 2000 standard depended
44
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
on a number of factors including coding efficiency, computational complexity, and performance for a generic class of images. In addition,
there was a strong desire to keep Part 1 free of
IPR issues. Most of the technologies that were
excluded from Part 1 of the JPEG 2000 standard
due to the aforementioned reasons have been
included in Part 2. In addition, the file format
has been extended as described in Section 6.3.
A special feature of the technologies included
in Part 2 is the ability to adapt the compression
parameters to a specific class of images. Part 2
became a Final Draft International Standard
(FDIS) in July of 2001 and is expected to
become an International Standard (IS) in
November of 2001. The following is a brief
description of some of the technologies that are
included in Part 2.
6.4.1. Generalized offsets
In Part 1 of the JPEG 2000 standard, unsigned
image components with a bit-depth of B bits are
shifted down by 2B1 : In Part 2, the default offset
is the same as in Part 1, but a generalized offset
maybe specified for every image component. This
offset is applied before applying any component
transformation. For images with sharply peaked
histograms, using a generalized offset can result in
significantly improved compression performance.
When generalized offsets are being used, care must
be taken to adjust the number of guard bits to
prevent overflows.
6.4.2. Variable scalar quantization offset
This option extends the default scalar quantization method of Part 1 to allow deadzones of
different widths for each subband when using
floating-point filters. The size of the deadzone is
specified by the parameter nzb ; which must lie in
the half-open range of [1, 1). Given the quantizer
step-size Db of a particular subband, the size of
its deadzone is given by 2(1nzb) Db. A value of
nzb ¼ 0 corresponds to a deadzone width that is
twice the step-size (as in JPEG 2000 Part 1), while
a value of nzb ¼ 0:5 corresponds to a uniform
quantizer as is used in the existing JPEG standard.
As shown in [46], the resulting parameterized
family of the deadzone quantizers still maintains
the embedded property described in Section 2.3. In
particular, if an Mb-bit quantizer index resulting
from a step-size of Db is transmitted progressively
starting with the MSB and proceeding to the
LSB, the resulting index after decoding only Nb
bits is identical to that obtained by using a
quantizer with a step-size of Db 2Mb Nb and a
deadzone parameter of (nzb =2Mb Nb ). For nonzero
values of nzb ; the deadzone width rapidly converges to twice the step-size with decreasing Nb ;
while for a value of nzb ¼ 0; the width of the
deadzone always remains at twice the step-size and
the resulting embedded quantizers have exactly the
same structure.
6.4.3. Trellis coded quantization
Trellis coded quantization (TCQ) [7,21,25] is a
form of spatially varying scalar quantization with
delayed-decision coding. The wavelet coefficients
contained in each codeblock are scanned in the
same order as described in Section 2.4.2, and each
coefficient is quantized using one of four separate
scalar quantizers that have an approximately
uniform structure. The specific choice of the scalar
quantizer is governed by the restrictions imposed
by a finite state machine that is represented as a
trellis with eight states. The optimal sequence of
the states (i.e., the sequence of the quantizer
indices for a particular codeblock) is determined
at the encoder by using the Viterbi algorithm.
Visual testing of TCQ with the ISO test images has
shown a significant improvement in the reconstructed quality of low contrast image features
such as textures, skin tones, road features, cloth,
woodgrain, and fruit surfaces [11]. The tradeoff is
that the computational complexity is significantly
increased by using TCQ.
6.4.4. Visual masking
Masking refers to the decreased visibility of a
signal due to the presence of a suprathreshold
background signal. When applying visual masking
to the encoding of the wavelet coefficients, the
amplitude of the coefficients is considered to be
the background (mask), while the quantization
error is considered to be the signal, whose visibility
should be minimized. In Part 2 of JPEG 2000,
the masking properties of the visual system are
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
exploited by applying a non-linearity to the
wavelet coefficients prior to quantization. The
characteristics of the non-linearity may depend
on the amplitude of the wavelet coefficient
being quantized (referred to as self-contrast masking) as well as the quantized amplitude of the
neighboring wavelet coefficients (referred to as
neighborhood masking). The use of visual
masking typically improves the reconstructed
image quality for low-resolution displays for
which the visual system CSF is essentially flat. A
key area of improvement is in low amplitude
textures such as skin, and the improvement
typically becomes greater as the image becomes
more complex [51,52]. Another area of improvement is in the appearance of edges with zero
transition width in digitally generated graphics
images. Finally, in certain applications where
images are compressed to a fixed size, the
use of visual masking often creates more consistent image quality with variations in image
content.
6.4.5. Arbitrary decomposition of tile-components
In Part 2 of the JPEG 2000 standard, it is
possible to specify an arbitrary wavelet decomposition for each tile-component. First, the tilecomponent is decomposed using the octave
decomposition that is allowed in Part 1. Then,
the subbands resulting from the octave decomposition may be split further. Different subbands
may be split further to different depths. Subbands
can also be split to different depths in the
horizontal and vertical directions, thus allowing subbands to have differing sub-sampling
factors in the horizontal and vertical directions.
This mode is useful when the intent is to optimize
the wavelet decomposition to a particular class
of images or even for an individual image [35].
Another application is in memory conservation,
where the number of vertical decompositions might be less than horizontal to
reduce the line buffering required for the wavelet
transform.
6.4.6. Transformation of images
Part 1 of the JPEG 2000 standard permits
the use of only two wavelet decomposition
45
filter-banks, the irreversible (9, 7) filter-bank
and the reversible (5, 3) filter-bank. In Part 2
of the standard, arbitrary user specified
wavelet decomposition filters [8,9] are permitted,
and their category (even- or odd-length), type
(irreversible or reversible), and weights are
signaled in the codestream. This allows
the optimization of the filter coefficients for
a particular class of images.
6.4.7. Single sample overlap discrete wavelet
transform (SSO-DWT)
When non-overlapping tiles are used, the
artifacts at the tile boundaries can be objectionable at low bit-rates. Part 2 of the standard
allows the use of tiles that overlap by a single
row and column to eliminate the tile-boundary
artifacts. The advantage is that the single
sample overlap wavelet transformation can
still be carried out in a block-based fashion,
requiring a much smaller amount of memory
than performing a wavelet transformation of the
entire image.
6.4.8. Multiple component transformations
Part 1 of the JPEG 2000 standard allows the use
of only two inter-component color transformations to decorrelate the components; the ICT that
is used with irreversible wavelet filters, and the
RCT that is used with reversible wavelet filters.
Both of these transformations are designed for
three-component RGB input images, so their
utility is limited when the image components
belong to a different color space or when there
are more than three components (e.g., LANDSAT
images with six components or CMYK images
with four components). Part 2 of the standard
provides two general approaches for decorrelating
multi-component data. One approach is a generalized method of forming linear combinations of
components to reduce their correlation. This may
include a linear predictive transform to remove
recursive dependencies (e.g., Gramm–Schmidt
procedure), or a decorrelating transform (e.g.,
KLT). Another approach is using a default or
a user-specified one-dimensional wavelet transformation in the component direction to decorrelate
the components.
46
M. Rabbani, R. Joshi / Signal Processing: Image Communication 17 (2002) 3–48
6.4.9. Non-linear transformation
Part 2 of the JPEG 2000 standard offers two
ways of non-linear transformation of the component samples before any inter-component transform is applied to increase coding efficiency. The
two non-linear transformations are gamma-style
and look up table (LUT) style. This feature is
especially useful when the image components are
in the linear intensity domain, but it is desirable to
bring them to a perceptually uniform domain for
compression that is more efficient. For example,
the output of a 12-bit linear sensor or scanner can
be transformed to 8 bits using a gamma or a
logarithmic function.
6.4.10. Extensions to region of interest (ROI)
coding
Part 1 of the JPEG 2000 standard includes
limited ROI capability provided by the Maxshift
method as described in Section 6.1. The Maxshift
method has the advantage that it does not require
the transmission of the ROI shape and it can
accommodate arbitrary-shaped ROIs. On the
other hand, it cannot arbitrarily control the
quality of each ROI with respect to the background. Part 2 extends the ROI capability by
allowing the wavelet coefficients corresponding to
a given ROI to be scaled by an arbitrary scaling
factor. However, this necessitates the sending of
the ROI information explicitly to the decoder,
which adds to the decoder complexity. Part 2
supports only rectangular and elliptic ROIs, and
the ROI mask is constructed at the decoder based
on the shape information included in the codestream.
Acknowledgements
The authors would like to thank Paul Jones for
his careful review of the entire manuscript and
many helpful suggestions that significantly improved the clarity of its presentation. The authors
would also like to thank Brian Banister for
generating the bitplane results in Section 5.3, and
Scott Houchin for providing the material on file
format in Section 6.3.
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