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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. 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