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Lossless image compression using binary wavelet transform

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									Lossless image compression using binary wavelet
transform
H. Pan, W.-C. Siu and N.-F. Law

                 Abstract: A binary wavelet transform (BWT) has several distinct advantages over a real wavelet
                 transform when applied to binary data. No quantisation distortion is introduced and the transform is
                 completely invertible. Since all the operations involved are modulo-2 arithmetic, it is extremely
                 fast. The outstanding qualities of the BWT make it suitable for binary image-processing appli-
                 cations. The BWT, originally designed for binary images, is extended to the lossless compression
                 of grey-level images. An in-place implementation structure of the BWT is explored. Then, a simple
                 embedded lossless BWT-based image-coding algorithm called progressive partitioning binary
                 wavelet-tree coder (PPBWC) is proposed. The proposed algorithm is simple in concept and
                 implementation, but achieves promising lossless compression efficiency as compared with the con-
                 ventional bitplane scanning methods. Small alphabets in the arithmetic coding, non-causal adaptive
                 context modelling and source division are the major factors that contribute to the gain of com-
                 pression efficiency of the PPBWC. Experimental results show that the PPBWC outperforms
                 most of other embedded coders in terms of coding efficiency.




1     Introduction                                                         effects are introduced and the decomposition is completely
                                                                           invertible; (ii) it is extremely fast. As modulo-2 arithmetic is
Image compression is an important tool to reduce the band-                 equivalent to exclusive-or operations, the transform can be
width and storage requirements of practical image systems.                 performed using simple Boolean operations. Preliminary
Although most image-processing applications can tolerate                   researches [13, 15] indicate that BWT gives a compact rep-
some information loss, in several areas such as medicine,                  resentation of binary images and its application to lossless
satellite imagery, and legal imaging, lossless compression                 image compression is promising.
algorithms are still preferred. Usually, most of the efficient                 In this work, we study the lossless image-compression
lossless image compression methods like the context-based,                 algorithm using BWT. The aim of our study is to extend
adaptive lossless image codec (CALIC) [1] and the low-                     BWT, originally designed for binary images, to the lossless
complexity lossless compression for images (LOCO-I)                        compression of the grey-level image. A simple embedded
algorithm [2] are based on predictive techniques [1– 5] in                 lossless BWT-based image-coding algorithm, called pro-
the spatial domain. However, recently a number of competi-                 gressive partitioning binary wavelet-tree coder (PPBWC),
tive lossless image-compression techniques using wavelet                   which uses a joint bit scanning method and non-causal
transforms have been proposed [6– 10]. High coding effi-                    adaptive context modelling, is proposed. Because all calcu-
ciency, multiresolution image representation and rather                    lations in BWT are performed by Boolean operations, the
moderate computational complexity are the most striking                    PPBWC introduces no quantisation errors and is computa-
attributes of these wavelet-based coders.                                  tionally simple. Moreover, similar to the lifting scheme
   The real field wavelet transform is mostly concentrated                  [16] of the wavelet transform in the real field, an in-place
on real-valued functions in which the data to be analysed                  implementation structure of BWT is designed to further
are real and the arithmetic is performed in the real field.                 reduce the computational complexity of the PPBWC. Due
There have been several attempts to generalise the                         to its energy concentration ability, BWT is able to reduce
wavelet transform to finite fields in order to take into                     entropy in the transformed image significantly, which can
account image characteristics [11– 14]. Recently, Swanson                  help to decrease the bit budget. In the PPBWC, we adopt
and Tewfik [13] introduced a theory of binary wavelet trans-                a joint bit scanning approach to construct a prioritised quan-
form (BWT) for finite binary images over GF(2). BWT                         tisation scheme that generates a long zero-run sequence,
shares many of the important characteristics of the real                   especially in the high frequency sub-bands, thus leading
wavelet transform and also has several distinct advantages                 to high coding efficiency of the PPBWC. Different from
over it: (i) the entire decomposition is performed in                      other zerotree-based algorithms, our algorithm only pro-
GF(2), which means that the intermediate and transformed                   duces two symbols ‘0’ and ‘1’ instead of four (ZTR, POS,
data produced by the BWT are also binary. No quantisation                  NEG and IZ) symbols in embedded zerotree wavelet
                                                                           compression (EZW) [17]. Small alphabets allow an adap-
# The Institution of Engineering and Technology 2007                       tation algorithm with a short memory to learn quickly and
doi:10.1049/iet-ipr:20060195                                               can constantly track the changing symbols’ probabilities.
Paper first received 26th June 2006 and in revised form 24th May 2007       This also accounts for some of the effectiveness of the
The authors are with Center for Multimedia Signal Processing, Department   PPBWC. The bit streams generated by the joint bit scanning
of Electronic and Information Engineering, The Hong Kong Polytechnic
University, Hong Kong
                                                                           method are divided into two subsources, according to the
H. Pan is also with the Department of Automatic Control Engineering,       type of information they carry. By applying a non-causal
Southeast University, Nanjing, People’s Republic of China                  adaptive context modelling to each bit stream, the
E-mail: enwcsiu@polyu.edu.hk                                               PPBWC exploits self-similarity and the joint coefficient
IET Image Process., 2007, 1, (4), pp. 353– 362                                                                                         353
statistics in the wavelet spatial oriental tree; it also includes   signal. The upper half of y is the lowpass output whereas the
a part of the information that will be available in the future      lower half of y is the bandpass output.
for better conditional probability estimation. The ability to
look into forthcoming information generally reduces the             2.2        Constraints on filter design
uncertainty of encoding coefficients. This is another factor
of the high coding efficiency of the PPBWC.                          In order to guarantee that the BWT can perform a
   The rest of the paper is organised as follows. In the next       useful multiresolution decomposition and still be able to
section, we give a description of BWT, and some of the con-         reconstruct the original signal, the lowpass filter and the
straints on designing binary filters. A practical in-place           bandpass filter must satisfy three basic constraints [13]:
implementation is presented to reduce the transform complex-        bandwidth, vanishing moment and perfect reconstruction
ity in Section 3. Section 4 extends the theory of BWT from          constraints. These constraints are concluded as follows
the bi-level image to the grey-level image. An embedded loss-
less BWT-based image-compression algorithm, namely the                  X
                                                                        N À2               X
                                                                                           N À1                 X
                                                                                                                N À2                X
                                                                                                                                    N À1

PPBWC, is proposed in Section 5. Section 6 has experimental                     ci ¼ 0,             ci ¼ 1,              di ¼ 1,             di ¼ 1
                                                                    i¼0,even              i¼1,odd             i¼0,even             i¼1,odd
results and analysis. Properties of the PPBWC and future
work form Section 7.                                                                                                                             (5)
                                                                                                                          
2     Theory of BWT                                                                                                    N
                                                                                          d2iþ1 ¼ d2i 0          i                               (6)
                                                                                                                       2
Recently, a formulation of the wavelet transform of a vector
space over the finite field has been derived in [18]. Since it        Though the above three basic constraints guarantee that
does not require the existence of the Fourier transform, it         the BWT is invertible, they do not explicitly specify
can be extended to any finite field, including the binary             the form of the inverse filters. The form of the inverse
field [13]. Furthermore, this wavelet transform does not             filters is thus unconstrained such that it might change
depend on the field structure of binary numbers and non-             with signal length. In order to solve this signal
linear transforms can also be involved [19].                        length-dependency problem, Law and Siu [15] proposed
                                                                    another constraint, that is, the perpendicular constraint to
                                                                    achieve the length-independent property for the inverse
2.1   Formulation of BWT                                            binary filters. The perpendicular constraint relates the
                                                                    form of the forward filters to that of the inverse filters.
The construction of a two-band discrete orthonormal BWT
                                                                    With this constraint, the form of the inverse filters
can be viewed as equivalent to the design of a two-band
                                                                    remains unchanged after the up-sampling since the inverse
perfect reconstruction filter bank with added vanishing
                                                                    filters can be easily obtained by a simple zero-padding
moment conditions. In BWT, the input signal is simul-
                                                                    operation. By considering the BWT as linear operators on
taneously passed through the lowpass and the bandpass
                                                                    signals that are modelled as vectors over the binary
filters, which are then decimated by two to give an approxi-
                                                                    field, Kamstra [20] extended the results in [13, 15] by
mation component and a detailed component of the original
                                                                    describing a general mechanism for doubling the length of
signal. The two decimated signals may then be upsampled
                                                                    the forward filters and the inverse filters while retaining
and passed through the complementary inverse filters. The
                                                                    some characteristics of the original filters. Kamstra
results are summed to reconstruct the original signal. This
                                                                    demonstrated that the length of forward and inverse filters
is the same as the real field wavelet transform except that
                                                                    can be doubled by padding arbitrary vectors with the
the original signal and the transformed signal are recon-
                                                                    same length as the original filters, and the inverse filters
structed in the binary domain.
                                                                    can simply be calculated by multiplying known vectors
   Mathematically, an N Â N transform matrix T can be
                                                                    with known matrices. It is obvious that the zero-padding
constructed as follows
                                                                    approach used in [13, 15] is a special case of this general
                                                                  doubling mechanism.
                               C
                         T¼                             (1)            The filter design for the BWT is bounded by incorporat-
                               D
                                                                    ing all of these constraints. In [15, 21], it is shown that there
where                                                               are 4 and 32 different pairs of binary filters for filter lengths
                                                                    equal to 4 and 8, respectively. The length-8 binary filters are
            C ¼ (cjs¼0 , cjs¼2 , . . . , cjs¼N À2 )T ;              grouped into four classes depending upon the number of
                                                                    ‘1’s in the binary filters. Examples of the binary filters in
                 D ¼ (djs¼0 , djs¼2 , . . . , djs¼N À2 )T    (2)    each group are given in Table 1. The filter length has an
                                                                    important impact on the performance of a BWT-based com-
ajs¼k defines a vector with elements formed from a circular          pression system. It is shown in [19] that short filters perform
shifted sequence of a by k. AT is the transpose of A, and           better for synthetic images, such as character images, which
                                                                    comprise uniform areas with shape boundaries. In contrast,
             c ¼ {c0 , c1 , . . . , cN À2 , cN À1 }T ;              long filters perform well on images that are the dithered
                                                                    version of grey-level images, with a limited number of
                  d ¼ {d0 , d1 , . . . , dN À2 , dN À1 }T    (3)    grey levels. Usually, bi-level bitplanes of a natural grey-
                                                                    level image contain mostly area with a regular texture,
ci and di are the scaling and the wavelet filters, respectively.
                                                                    which can be effectively processed by long filters.
Then the BWT based on the circular convolution of binary
                                                                    Therefore, we choose length-8 binary filters in our work.
sequences with binary filters followed by decimation can be
defined as
                                                                    3          In-place implementation of BWT
                               y ¼ Tx                        (4)
                                                                    The lifting scheme [16] in the real field wavelet transform
where x is the original input signal and y is the transformed       enables an in-place implementation in the spatial domain,
354                                                                                               IET Image Process., Vol. 1, No. 4, December 2007
Table 1:       Filter groups of length-8 binary filters and the number of XOR operations in the forward BWT of each group

Groups                       Lowpass, bandpass                                                 No. of XOR operations in the forward BWT
                             Analysis filter             Synthesis filter                        Convolution case              In-place case

1                            f0, 1, 0, 0, 0, 0, 0, 0g   f1, 1, 0, 0, 0, 0, 0, 0g               1                             1
                             f1, 1, 0, 0, 0, 0, 0, 0g   f1, 0, 0, 0, 0, 0, 0, 0g
2                            f1, 1, 1, 0, 0, 0, 0, 0g   f0, 0, 1, 1, 0, 0, 0, 0g               3                             2
                             f1, 1, 0, 0, 0, 0, 0, 0g   f0, 1, 1, 1, 0, 0, 0, 0g
3                            f1, 1, 1, 1, 0, 0, 0, 1g   f0, 0, 0, 0, 0, 0, 1, 1g               5                             3
                             f1, 1, 0, 0, 0, 0, 0, 0g   f1, 0, 0, 0, 1, 1, 1, 1g
4                            f1, 1, 1, 1, 1, 1, 1, 0g   f0, 0, 0, 0, 0, 0, 1, 1g               7                             4
                             f1, 1, 0, 0, 0, 0, 0, 0g   f0, 1, 1, 1, 1, 1, 1, 1g




which greatly reduces the computational complexity and                                For the filters in group 2, there are three stages in the
saves the memory storage involved in the transform.                                forward BWT. Firstly, the operation is similar to that in
Similar to the case in the real field, the binary-field                              group 1 filters, that is, a subsampling operation and an
wavelet transform has a similar in-place implementation                            XOR operation between two neighbouring samples. The
architecture in which the transform can be conducted effi-                          even-indexed samples are then circularly shifted up by 1,
ciently in the binary field. Initially, the odd-indexed and                         while the odd-indexed samples remain unchanged.
the even-indexed samples of the original signal are split                          Finally, the even-indexed elements will have an XOR oper-
into two sequences. Those two sequences are then                                   ation with the odd-indexed elements to obtain the final
updated according to the filter coefficients from the                                results. The total number of XOR operations is reduced
lowpass and the bandpass filters. This structure is similar                         from three to two by using the in-place implementation
to the ‘split, update and predict’ procedure in the lifting                        structure. The operation of filters from group 3 and group
scheme of the real-field wavelet transform. The lowpass                             4 can also be implemented in the same way. By employing
output and the bandpass output are then interleaved to                             the in-place structure, the total number of XOR operations
produce the transformed output. The in-place implemen-                             is reduced as summarised in Table 1.
tations of the binary filters from group 1 to group 4 are illus-
trated in Fig.1, where È means a modulo-2 addition.                                4    Application of BWT to the grey-level image
   For the filter in group 1, the lowpass filter involves one
subsampling operation only, whereas the bandpass filter                             BWT is usually done only for bi-level images. A natural
involves an exclusive-or (XOR) operation between two                               way to generalise it to grey-level images is as follows.
neighbouring samples. Thus, the odd-indexed elements are                           First, separate a grey-level image into a series of bi-level
unchanged after the subsampling, and the even-indexed                              images using the bitplane decomposition method; second,
elements have an XOR operation with the neighbouring                               apply BWT to each individual bi-level bitplane image.
odd-indexed elements. Finally, the even-indexed elements                           For the bitplane decomposition, each pixel of a grey-level
and the odd-indexed elements are interchanged to give the                          image is represented by a K-bit binary code so that the grey-
desired outputs.                                                                   level image is decomposed into K images, each having only
                                                                                   two levels denoted by 0 or 1. Each two-level or 1-bit image




Fig. 1 In-place implementation of different binary filter groups
a   Binary   filter   of   group   1
b   Binary   filter   of   group   2
c   Binary   filter   of   group   3
d   Binary   filter   of   group   4

IET Image Process., Vol. 1, No. 4, December 2007                                                                                             355
is referred to as a bitplane, and the order of these bitplane      5   Progressive partitioning binary wavelet-tree
images starts from the most significant bitplane (MSB) to           coder
the least significant bitplane (LSB).
   Fig. 2 shows an 8-bit transformed image using the binary        In this section, we propose a simple embedded lossless
filters of group 1 discussed in Section 2.2; it also shows the      BWT-based image-compression algorithm called PPBWC,
bi-level image and the transformed image of the MSB of             which uses joint bit scanning and non-causal adaptive
‘Lena’. As a comparison, the integer (2, 2) wavelet trans-         context modelling. In the PPBWC, by progressively parti-
form [9] and Daub (9, 7) biorthogonal wavelet transform            tioning the binary wavelet coefficients in the spatial-
[22] are also applied to the ‘Lena’ image. The transformed         frequency domain, the coefficients are ordered on the
images of integer (2, 2) wavelet and Daub (9, 7) wavelet are       basis of their absolute range. The prioritised coefficients
shown in Fig. 3. It can be seen that high-sequency edge tran-      are quantised in a successive approximation manner to gen-
sitions detected by the integer (2, 2) kernel, Daub (9, 7)         erate a binary sequence that is divided into several sub-
kernel or the binary filter have mapped into high-sequency          sources and encoded by the arithmetic coder with a
regions in the transformed domain. Unlike the integer              non-causal adaptive context modelling. Like other
wavelet transform and the real field wavelet transform,             embedded coders such as the EZW [17] and set partitioning
which expand the range of the transformed coefficients,             in hierarchial trees (SPIHT) [23], the PPBWC adopts the
the BWT image maintains the range of the transformed               properties of progressive transmission and embeddedness.
coefficients from 0 to 255. In particular, the ranges of the        Besides, it eliminates zerotree analysis; therefore, the
transformed images using integer (2, 2) kernel and Daub            implementation and context modelling of the PPBWC are
(9, 7) kernel are ( – 191, 264) and ( – 627.3, 1967.8), respect-   simple.
ively. Most of the large value coefficients using integer (2,
2) kernel concentrate at the low frequency sub-bands. In           5.1     Joint bit scanning
contrast, most of the large value coefficients using the
binary filters correspond to high-sequency transition.              In the baseline JPEG, a discrete cosine transform (DCT)
Figs. 2 and 3 show the reconstructed images using the              coefficient in each transformed block has to be completely
three kernels. Both the binary filter and the integer (2, 2)        encoded before encoding the next coefficient in a zigzag
kernel involve only exact arithmetic calculations, and as a        scanning manner, as shown in Fig. 4a. Initially, bits of coef-
result, there is no loss in the reconstructed quality; the         ficient b1 denoted as {bitk21 , . . . , bit1 , bit0} in the binary
Daub (9, 7) kernel, however, involves floating-point                form are encoded from the most significant bit, that is,
calculations that give rise to an inexact arithmetic.              bitk21 , to the least significant bit, that is, bit0 , then bits of
                                                                   coefficients a1 and b2 are coded and so on. Such a
                                                                   coefficient-by-coefficient coding approach can be con-
                                                                   sidered as vertical bit scanning. Let us recall the concept
                                                                   known as embedded or layered coding, which was
                                                                   adopted by the EZW [17], SPIHT [23] and layered zero




Fig. 2 Original and binary wavelet transformed image of the ‘Lena’ image
a   Original grey-level image
b   Eight-bit binary wavelet transformed image
c   Binary image of the MSB
d   Transformed MSB image
e   Inverse transformed image

356                                                                                       IET Image Process., Vol. 1, No. 4, December 2007
Fig. 3 Wavelet transformed images and reconstructed images using two different kernels
a   Transformed image using Daub (9, 7) kernel
b   Reconstructed image using Daub (9, 7) kernel
c   Transformed image using integer (2, 2) kernel
d   Reconstructed image using integer (2, 2) kernel


coding (LZC) [24]. In these algorithms, the MSBs of all                5.2 Progressive partitioning binary wavelet-tree
coefficients are grouped together to form one layer and                 coder
encoded initially. The algorithm then moves to the layer
of the second significant bitplane and so on. The sign bit              In the PPC [25], the transformed coefficients C(i, j) are
is encoded after the first non-zero bit of the coefficient.              ordered in magnitude so that the largest coefficients,
Such a bitplane-scanning approach can be regarded as hori-             which contain the most energy, are first encoded.
zontal bit scanning.                                                   Magnitude ordering based on a prioritised coding scheme
   Inspired by the partitioning priority coding (PPC) for pro-         is suitable for progressive image transmission, since it
gressive DCT image compression [25] and in order to make               first codes and transmits the largest transformed coefficients
use of the embedded property of bitplane coding, we                    that capture the most important characteristics of the image.
suggest the construction of a prioritised quantisation                 The basic idea of the PPBWC is to divide the magnitude
scheme to generate long zero-runs especially in the high-              range of the transformed coefficients into several variable-
frequency sub-bands. This new scanning method is indi-                 sized partitions from the MSB to the LSB progressively.
cated in Fig. 4b. For each pass, it first scans bits horizon-           The lower bound of each partition is used as a control par-
tally. If a non-zero bit is encountered, which means that              ameter to handle the order-by-magnitude transmission. By
the current coefficient is identified as significant, then it             progressive partitioning and joint bit scanning of the
turns vertically and outputs the remaining bits of this signifi-        binary wavelet coefficients in the spatial-frequency
cant coefficient. It subsequently reverts to the original layer         domain, the coefficients are ordered based on their absolute
and continues to retrieve the remaining coefficients in the             range and quantised successively to form a binary sequence
horizontal direction again. The coefficients tagged as sig-             with a long zero-run.
nificant in previous passes will be skipped. For instance,                 Recall that in Section 4, we have applied BWT to each
in Fig. 4b, bit streams of the first and second passes are              bitplane of grey-level images and obtained the transformed
f11011011, 0, 0, 0, 0, 11010000, 0, 11001001, 0g and                   coefficients at all bitplanes. For an 8-bit grey-level image,
f1010001, 0, 1010101, 1001110, 0, 0g, respectively. Such               these binary transformed coefficients can be regarded as
an orientational switching between horizontal and vertical             the quantised transformed coefficients under a set of
bit scanning is known as joint bit scanning. Unlike vertical           thresholds f27, 26, 25, 24, 23, 22, 21, 20g. Thus, for the
bit scanning, which adopts zigzag scanning as the ordering             BWT coefficients, the bitplane partitioning is trivial.
controller, the ordering method in our joint bit scanning is           Initially, we begin the scanning process from the MSB, that
based on the absolute range. Therefore, the PPBWC sorts                is, bitplane 7 and decrease the bitplane by 1 at the end of
the coefficients more accurately according to their import-             each pass, until it reaches the LSB, that is, bitplane 0. In
ance. Furthermore, different from horizontal bit scanning,             each pass, the transformed coefficients are scanned in a
in which the most important coefficients can be fully                   zigzag manner, starting from the coarsest level of the
refined only until the last bitplane has been received, our             current bitplane. If the current coefficient value is equal to
scanning method refines the significant coefficients immedi-              one, it is identified as significant and a symbol ‘1’ is
ately when it is first found.                                           encoded to indicate its position and significance. The
                                                                       remaining bits of that coefficient at the remaining bitplanes
                                                                       are then refined. Otherwise, if the current coefficient value is
                                                                       equal to zero, a symbol ‘0’ is encoded to indicate its insig-
                                                                       nificance in the current pass. The coefficient labelled signifi-
                                                                       cant in the previous passes will never be encoded again.
                                                                          Note that the transformed coefficients of each bitplane are
                                                                       still binary, and take a value either ‘1’ or ‘0’. Therefore, it is
                                                                       fast in arithmetic coding, as the arithmetic coder only needs
                                                                       to maintain and update a single probability distribution table
                                                                       with two parameters. Moreover, every time a symbol is
                                                                       entered, the time it takes to calculate the new intervals
                                                                       becomes shorter. The reason is that because of the binary
                                                                       operations, there is only one endpoint change when a new
Fig. 4 Scanning of the coefficients                                     bit is entered. This accounts for some of the effectiveness
a Zigzag scanning of the encoding coefficients                          of the PPBWC.
b Joint bit scanning of the binary representation of the coefficients

IET Image Process., Vol. 1, No. 4, December 2007                                                                                     357
5.3   Non-causal adaptive context modelling                        the orientations at different sub-bands naturally lead to
                                                                   adaptive context modelling.
From the previous works [1, 2], we understand that a                  By using non-causal adaptive context modelling, not
proper context modelling, which is used to estimate the            only the past information, but also part of the future infor-
conditional probability function p(xiþ1jxi , . . . , x1) of xiþ1   mation is included. The ability to look into the future and
in a finite sequence fxi , . . . , x2 , x1g, is essential to        access information generally reduces the uncertainty of the
achieve high coding efficiency in the arithmetic coder.             encoding coefficient. Let us return to Fig. 4, which can be
Context modelling of the PPBWC is based on two funda-              used to illustrate the ability to look into future information
mental properties: (i) the magnitudes of coefficients,              of the PPBWC. If a zigzag scanning method is applied,
which tend to decay with a zigzag scan from a coarse to            coefficients b1 , a1 , b2 and a2 will be visited before c.
fine resolution level; (ii) the conventional parent – child         These coefficients form the past information of c, while
relationship (Fig. 4 of [17]) together with self-similarity        a3 , b3 , a4 and b4 form the future information of c.
and the joint coefficient statistics of the wavelet tree.           Fig. 4b shows joint bit scanning of the binary represen-
Unlike most of the well-known embedded coders that use             tation of coefficients. At the beginning, the encoder sets
the magnitudes of coefficients to form a context for con-           the significance status of each coefficient to be unknown.
ditional probability estimation, the PPBWC uses the signifi-        It updates the status of the coefficient during the encoding
cance status (significant, insignificant and unknown) of             process. For instance, in Fig. 4b, coefficient c will be
coefficients to estimate the conditional probability of the         encoded as significant in coding pass 2. When encoding
current coefficient C(i, j). It is observed that the significance    c, the significance status of b1 , a1 and a2 are already avail-
status of a wavelet coefficient C(i, j) has high correlation not    able, whereas for future information, only the values of a3
only with the spatial structures of its neighbouring coeffi-        and a4 are available. Since coefficients b2 , b3 and b4 are not
cients but also with the coefficients corresponding to              available in coding pass 2, their significance statuses
similar locations at the same scale level of different orien-      remain unknown. The decoding process is the same as
tations, namely, the cousin coefficients, and the parent coef-      the encoding process.
ficients. Fig. 5 demonstrates the context set U of the                 It has been proved that the entropy of a source and the
significant bits used in the PPBWC. Note that                       cost of coding are reduced if the source is partitioned into
                                                                   subsources with different probability distributions [25]. In
1. if C(i, j) exists in the horizontal orientation or the verti-   view of this, we divide the bit streams generated by the
cal orientation,                                                   PPBWC in each bitplane into two substreams, namely, the
                                                                   significant bit stream and the refinement bit stream, accord-
           U ¼ {ai , bi , P, DCji ¼ 1, 2, 3, 4}, or                ing to the type of information they carry. The significant bit
                                                                   stream is encoded and conditioned to different contexts.
                                                                   However, as the refinement bits of the transform coefficients
2. if C(i, j) exists in the diagonal orientation,                  are randomly distributed, the refinement bit stream is
                                                                   encoded with a single adaptive model. The magnitude-set
           U ¼ {ai , bi , P, HC, VCji ¼ 1, 2, 3, 4}                variable-length integer (MS-VLI) representation [7] is
                                                                   used to limit the number of modelling contexts of the
                                                                   significant bits in order to avoid ‘context dilution’ when
where fai , biji ¼ 1, 2, 3, 4g denote the neighbouring coeffi-      the count statistics spread over too many contexts, which
cients, HC, VC and DC denote the horizontal, vertical and          affects the accuracy of the corresponding conditional prob-
diagonal cousin coefficients, respectively, and P denotes           ability estimation. In the following part, MS(x) is used as an
the parent coefficient. Therefore, the significance status           operator of the MS-VLI representation.
of C(i, j)’s eight neighbours, cousins and one parent are             The first non-zero bit of C(i, j) is encoded conditioned to
included to form a non-causal context modelling for the            the local and the global variance of its neighbouring, cousin
conditional probability estimation. Such non-causal                and parent coefficients’ significance status. Since the human
context modelling is one of the factors that contributes to        visual system is sensitive to the horizontal and the vertical
the high coding efficiency of the PPBWC. In addition,               orientation, the conditioning context of significant bit mod-
                                                                   elling is defined as:


                                                                   1. If C(i, j) exists in the horizontal or the vertical sub-band

                                                                                                                             !
                                                                                             X
                                                                                             4              X
                                                                                                            4
                                                                              m ¼ MS 3             ai þ 2         bj þ 3DC           (7)
                                                                                             i¼1            j¼1




                                                                   2. If C(i, j) exists in the diagonal sub-band

                                                                                                                                 !
                                                                                       X
                                                                                       4              X
                                                                                                      4
                                                                          m ¼ MS 3           ai þ 2         bj þ 3(HC þ VC)          (8)
                                                                                       i¼1            j¼1




Fig. 5 Context members of the encoding coefficient C(i, j)
358                                                                                     IET Image Process., Vol. 1, No. 4, December 2007
Table 2:     Counts of different operations per pixel involved in all steps of PPBWC and CALIC

Algorithms                 Operations               ADD                  MUL              Shift           XOR             COM            ABS

PPBWC                      Step 1                     0                       0             0             0.5               0            0
                           Step 2                     0                       0             0             0                 1            0
                           Step 3                    11                       0             1             0                 4            0
                           Total                     11                       0             1             0.5               5            0
CALIC                      Total                     13                       1             5             0                17            3



and if a parent exists                                                            Step 3: Perform context modelling and context quantisation
                                                           !                      using (7) –(9).
                    X
                    4               X
                                    4
      m ¼ MS 3             ai þ 2         bj þ 3(DC þ P)       or                    The arithmetic and logic operations per pixel involved in
                     i¼1            j¼1
                                                                                  these steps are counted and tabulated in Table 2, where the
                                                                    !             group 1 binary filters are considered and symbols ADD,
                    X
                    4               X
                                    4
      m ¼ MS 3             ai þ 2         bj þ 3(HC þ VC þ P)           (9)       MUL, Shift, XOR, COM and ABS denote the addition, mul-
                     i¼1            j¼1                                           tiplication, bit shifting, XOR, comparison and absolute
                                                                                  operations, respectively. As a comparison, the number of
                                                                                  operations of the state-of-the-art lossless image-
                                                                                  compression algorithm, CALIC, is also listed.
By conditioning the distribution of significant bits of the
encoding coefficient C(i, j) on the significance status of its
neighbouring, cousin and parent coefficients, the significant                       6    Experimental results
bit model can be separated into classes of different var-
                                                                                  In this section, two of our experimental works are reported
iances. Thus, entropy coding of the significant bits of
                                                                                  to evaluate the performance of the proposed PPBWC algor-
C(i, j) using the estimated conditional probability
                                                                                  ithm in terms of coding efficiency and lossless compression
^
p(C(i, j)jm) improves coding efficiency over using the esti-
                                                                                  quality. First, the compactness of signal representation
                     ^
mated probability of p(C(i, j)).
                                                                                  resulting from the BWT using four groups of binary filters
5.4     Computational complexity of PPBWC                                         listed in Table 1 is shown. Then, we compare the PPBWC
                                                                                  with other state-of-the-art lossless image coders in terms
Compared with the previous context-based lossless image-                          of their lossless compression efficiency.
compression algorithms [1, 2, 26, 27], the PPBWC is
more efficient and also has a simpler structure. Since the                         6.1 Compactness of signal representation using
entropy-coding step is an independent module of source                            BWT coefficients
modelling in the PPBWC and is required by all lossless
image codecs, in the following, we only analyse the compu-                        In order to examine the capability of energy concentration
tational complexity of the preprocessing, context modelling                       in BWT, we perform a three-level BWT to the 256 Â 256
and context quantisation components of the PPBWC                                  ‘Lena’ image as shown in Fig. 2a using binary filters of
system, which determines the difference in execution                              group 1 – group 4. The transformed bi-level image of the
times among different algorithms.                                                 MSB using group 1 binary filters is shown in Fig. 2d. Of
   In the PPBWC, three major steps are taken by the                               the original 65 536 pixels, the original bi-level image at
encoder, besides the entropy coding of estimated con-                             the MSB has 20 049 non-zero pixels, whereas the trans-
ditional probability of significant bit streams and refinement                      formed image has only 4097 non-zero pixels. To measure
bit streams.                                                                      the compactness of signal representation, we use an
                                                                                  entropy-based cost function
Step 1: Preprocessing: apply BWT to each bi-level bitplane
image.                                                                                  H( p) ¼ Àp log 2( p) À (1 À p) log 2(1 À p)      (10)
Step 2: Get significant bit stream and refinement bit stream
via the joint bit scanning.                                                       where p is the probability of non-zero pixels in bi-level

Table 3: Summary of entropy values for the original bitplane images and the transformed bitplane images using the
filters from different groups

Bitplanes                    Original entropy                   Binary filters
                                                                Group 1                   Group 2               Group 3               Group 4

7                            0.8884                             0.3323                    0.3678                0.4078                0.4757
6                            0.9990                             0.4846                    0.5138                0.5469                0.6339
5                            0.9830                             0.6201                    0.7238                0.6881                0.7567
4                            0.9995                             0.7861                    0.8847                0.8837                0.9464
3                            1.0000                             0.9069                    0.9609                0.9770                0.9843
2                            1.0000                             0.9812                    0.9964                1.0000                0.9986
1                            1.0000                             0.9994                    1.0000                1.0000                0.9999
0                            1.0000                             1.0000                    1.0000                1.0000                1.0000


IET Image Process., Vol. 1, No. 4, December 2007                                                                                             359
Table 4: Comparison of PPBWC with various coders on natural images in terms of lossless compression efficiency
represented by bit per pixel (bpp)

Images                Embedded coders                                                                             Unembedded coders
                      PPBWC                  SPIHT              CREW                 ECECOW                       LJPEG                  CALIC

Lena                  4.145                  4.182              4.512                4.090                        4.694                  4.042
Airplane              3.915                  3.965              4.319                –                            4.403                  3.830
Boat                  4.077                  4.288              4.504                3.860                        4.663                  3.778
Pepper                4.551                  4.679              4.718                –                            4.978                  4.391
Barbara               4.677                  4.815              5.077                4.570                        5.491                  4.493
Mandrill              5.927                  6.160              6.212                –                            6.381                  5.987
Goldhill              4.544                  4.627              4.762                4.420                        4.985                  4.387
Hotel                 4.504                  4.613              4.885                4.380                        5.063                  4.224
Lax                   5.745                  5.850              5.970                –                            6.025                  5.625
Seismic               2.887                  2.894              2.924                –                            3.083                  2.960
Target                2.748                  2.646              2.887                –                            3.085                  2.291
Average               4.338                  4.439              4.615                4.264                        4.805                  4.183




bitplane images. The measure takes values between 0 and                 set and the second is a JPEG 2000 test set, including
1. A small value indicates a large discrepancy between                  images ‘Seismic’ and ‘Target’, where all images in the
the number of zero and non-zero coefficients, thus poten-                two sets are of 512 Â 512 resolution. The last set includes
tially standing for a more efficient compression represen-               twelve medical images with different sizes and types. All
tation. Table 3 summarises the entropy results when                     test images are 8-bit in depth. We compare the PPBWC
different binary filters are used. From the results, we see              approach using the group 1 binary filter with other
that the entropy is significantly reduced at bitplanes 7 – 5             state-of-the-art lossless image coders, such as SPIHT [23],
after the BWT for all binary filters. It indicates that the              CREW [8], ECECOW [26], LJPEG [28] and CALIC [1],
BWT image is an efficient representation of grey-level                   in terms of their lossless compression efficiency. Among
image compression. The entropy increases as the bitplane                them, CALIC and LJPEG are unembedded, predictive-
decreases, implying that the ability of energy compaction               coding algorithms without progressive transmission capa-
introduced by the BWT becomes less efficient for lower                   bility. In this experiment, the S þ P transformation [7] is
(less significant) bitplanes. Among all the binary filters, the           applied to test images to achieve a multiresolution represen-
entropies of the transformed image yielded by the group 1               tation for the SPIHT algorithm.
filters are always the lowest, which means that the                         Results of ECECOW are quoted from [26]. Since the
group 1 filter is more suitable for image compression than               ECECOW codec is not publicly available, some of the
other binary filters.                                                    test images cannot be compared. Table 4 and Table 5
                                                                        show that PPBWC outperforms SPIHT, CREW and
                                                                        LJPEG in compressing both natural and medical images,
6.2     Lossless compression quality of the PPBWC                       but is slightly inferior to CALIC and ECECOW (3.57 and
                                                                        1.71%, respectively) because of the complex context mod-
Three sets of test images are chosen to evaluate the pro-               elling, context selection, context quantisation and different
posed algorithm. The first is a JPEG standard test images


Table 5: Comparison of the PPBWC with various coders on medical images in terms of lossless compression efficiency
represented by bit per pixel (bpp)

Images                                  Type                    PPBWC                    SPIHT                   LJPEG                   CALIC

Mammogram 2048 Â 2048                   X-ray                   1.154                    1.274                   1.962                   1.095
X-ray 2048 Â 1680                       X-ray                   2.254                    2.312                   2.627                   2.199
Cr 1744 Â 2048                          X-ray                   3.231                    3.300                   3.592                   3.221
Mr_head_p 256 Â 256                     MRI                     4.035                    4.022                   4.539                   3.872
Mr_head_w 256 Â 228                     MRI                     3.764                    3.796                   4.436                   3.647
Mr_throat 256 Â 256                     MRI                     4.474                    4.520                   5.120                   4.452
Ct_abdomen_fr 512 Â 512                 CT                      1.824                    1.868                   2.062                   1.650
Ct_abdomen_tr 256 Â 256                 CT                      3.307                    3.297                   3.755                   3.141
Ct_head 512 Â 450                       CT                      2.163                    2.243                   2.751                   1.925
Angio 512 Â 512                         ANG                     3.105                    3.133                   3.487                   3.072
Echo 720 Â 504                          U-sound                 2.935                    3.012                   3.637                   2.706
Pet 414 Â 414                           PET                     4.542                    4.627                   4.134                   4.037
Average                                 –                       3.066                    3.117                   3.509                   2.918


360                                                                                              IET Image Process., Vol. 1, No. 4, December 2007
conditional entropy coding used in the CALIC and                  of this approach is not as good as that of others at this
ECECOW. A unique feature of CALIC and ECECOW is                   moment. In particular, for the 256 Â 256 resolution ‘Lena’
the use of a large number of error modelling contexts to          image, the PSNR of PPBWC is 0.303 and 0.208 dB inferior
condition a nonlinear predictor, which adapts to varying          to that of the SPIHT and the EBCOT [29] algorithms on
source statistics. High-order structures are exploited in         average, respectively, at a bit rate of less than 0.5 bpp. For
context modelling of both algorithms for further com-             the 256 Â 256 resolution ‘Barbara’ image, the PSNR of
pression gains. In particular, in CALIC, four error energy        PPBWC is 0.173 dB superior to that of the SPIHT and
contexts and 144 spatial texture contexts are combined            0.472 dB inferior to that of EBCOT on average, respectively,
together to form a total of 576 compound contexts in              at a bit rate of less than 0.5 bpp. In view of this, a further
context modelling. Whereas in our approach, only nine con-        improvement of the lossy compression performance while
ditional contexts are employed in significant bit modelling.       maintaining its lossless compression efficiency is an
Consequently, the computational complexity of context             emerging problem for further investigation.
modelling and context quantisation of the PPBWC is
much lower than that of the CALIC, which can be verified
from Table 2. For the maximum compression, ECECOW                 8     Acknowledgment
treats all the known bits of the encoded wavelet coefficient,
as well as the neighbouring coefficients in the current and        This work is supported by the Center for Multimedia Signal
parent sub-bands as potential modelling events. Actually,         Processing, Department of Electronic and Information
by using such complex contexts, the ECECOW proposes               Engineering, the Hong Kong Polytechnic University,
a statistical model of a very high order. In addition,            under the CERG grant (PolyU 5133/02E) of the Hong
PPBWC only uses a simple MS-VLI representation to quan-           Kong SAR Government.
tise the conditional contexts, which is similar to using
‘buckets’ for complexity reduction. In contrast, ECECOW
designs a more sophisticated off-line method to construct         9   References
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