PERFORMANCE ANALYSIS OF EMBEDDED-WAVELET IMAGE CODERS
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PERFORMANCE ANALYSIS OF EMBEDDED-WAVELET CODERSShih-Hsuan Yang and Wu-Jie LiaoDepartment of Computer Science and Information EngineeringNational Taipei University of Technology1, Sec. 3, Chung-Hsiao E. Rd.Email: shyang@.twABSTRACTIn this paper, we analyze the design issues for the SPIHT (set partitioning in hierarchical trees) coding, one of the most prestigious embedded-wavelet-based algorithms in the literature. Equipped with the multiresolution decomposition, progressive scalar quantization and adaptive arithmetic coding, SPIHT generates highly compact scalable bitstreams suitable for real-time multimedia applications. The design parameters at each stage of SPIHT greatly influence its performance in terms of compression efficiency and computational complexity. We first evaluate two important classes of wavelet filters, orthogonal and biorthogonal. Orthogonal filters are energy preserving while biorthogonal linear-phase filters allow symmetric extension across boundary. We investigate the benefits from energy compaction, energy conservation, and symmetric extension, respectively. Second, the magnitude of biorthogonal wavelet coefficients may not faithfully reflect their actual significance. We explore a scaling scheme in quantizationthat minimizes the overall mean square error. The contribution of entropy coding is measured at last.1. INTRODUCTIONCompression lays the basis for the processing, transmission, and storage of multimedia data. A picture is worth a thousand words, but full utilization of pictorial information is impossible without compression. The most successful image coders adopt the transform-coding structure shown in Fig. 1. A linear transformation such as the discrete cosine transform (DCT) or discrete wavelet transform (DWT) converts the pixels into uncorrelated and condensed transform coefficients. The quantizer adequately divides the coefficient space into disjoint cells and the transform coefficients are reconstructed by a representative value within the cell. Quantization is thus lossy in nature; under a specified distortion requirement it aims to minimize the bit rate (or entropy) of the output symbols. The entropy coder at the last stage finds the most economical binary representation for the quantization symbol sequence. The baseline JPEG standard follows this structure that combines DCT, perceptually weighted scalar quantization, and Huffman coding of zig-zag scanned symbols. In 1993, Shapiro’s embedded zerotree wavelets (EZW) coding [1] established a new transform-coding paradigm with DWT, successive scalar quantization, and arithmetic coding. The essential novelty of EZW is the introduction of the “zerotrees” (a group of insignificant wavelet coefficients pertaining to the same spatial location and orientation). Of the various improvements of EZW, the setpartitioning in hierarchical trees (SPIHT) coding [2] is the most renowned. Because of its excellent compression performance and implementation elegancy, SPIHT has become one of the de facto standard coding algorithms in the image coding/processing/transmission community.Many researchers have investigated the design issues of EZW, SPIHT, and other DWT-based image coders.Before the introduction of SPIHT, Villasenor et al. [3] evaluated the compression efficiency of biorthogonal wavelet filters in terms of the Holder regularity and the impulse and step response properties, where a simple adaptive scalar quantization scheme with an optimized bit-allocation procedure was used as the coding platform. Li et al. [4] examined several wavelet filters and extension methods for EZW. Adams and Kossentini [5] evaluated a wide range of reversible DWT kernels under the JPEG2000 framework. Unser and Blu [6] investigated the mathematical properties of the Daubechies 9/7 and LeGall 5/3 wavelets pertaining to their compression performance. Woods and Naveen [7] derived the optimal bit allocation for non-orthogonal transforms. Moulin [8] derived a multiscale relaxation algorithm to improve the coding performance of non-orthogonal wavelet coding. Liu and Moulin [9] employed the mutual information to model the interscale and intrascale dependencies between wavelet coefficients. In [10], Xiong et al. showed that DWT outperforms DCT within 1 dB under the same embedded coding structure. The parent-child coding gain of SPIHT was quantified in [11]. He and Mitra [12] presented a unified analysis framework for the transform coding, where a new rate-distortion model in terms of the zeros of the quantized coefficients was developed. Finally, more sophisticated quantization schemessuch as vector quantization (VQ) and trellis-coded quantization (TCQ), have been incorporated into SPIHT [13]-[15].This paper comprehensively investigates the factors crucial to the performance of SPIHT. Although SPIHT was implemented mostly with a conventional set of parameters (e.g., Daubechies 9/7 wavelet with symmetric data extension), this setup may not be appropriate for all applications. It is thus important to investigate how to attain the desired performance with the available resources. Furthermore, this investigation offers an insight into the modern wavelet coders. In the following sections, we first explore the essential properties of wavelet transforms, including the orthogonality, energy-compacting capability, and symmetry. Since a biorthogonal wavelet transform distorts the magnitude of wavelet coefficients, we examine the effects of a scaling scheme to the quantization efficiency. The effect of the arithmetic coding is presented at last.2. WAVELET TRANSFORMS AND SPIHT2.1 The SPIHT CodingThe encoding process of SPIHT is summarized in Fig. 2. The DWT converts the pixels into wavelet coefficients, which are organized as the spatial-orientation trees depicted in Fig. 3. SPIHT adopts a two-pass scalar deadzone quantizer. The first pass, sorting pass, identifies the significant coefficients with respect to a threshold and gives their sign. Thepositions of significant coefficients are recorded in the list of significant pixels (LSP). Insignificant spatial-orientation trees (i.e., zerotrees) are recorded in the list of insignificant sets (LIS), and the other isolated insignificant coefficients are indexed in the list of insignificant pixels (LIP). The second pass, refinement pass, narrows the quantization level by a half for all the entries in the LSP excluding those newly added in the last iteration. The two passes are repeated with a halved threshold in the next iteration, until a specified rate or distortion constraint has been reached. An optional arithmetic coder can be used to generate even more compact bit streams. However, the arithmetic coding incurs intensive computation and reduces error robustness for the otherwise elegant SPIHT algorithm [2]. Without being otherwise specified, the coding results given in this paper are exempt from the arithmetic coding.2.2 Discrete wavelet transform (DWT)Transformation plays an essential role in image processing. Transformation can be regarded as an approximation of a signal with a new set of basis functions. The new space (called the transform domain) manifests itself in the decorrelating capability, space-frequency localization, energy compaction, and/or other properties desirable for further processing. In contrast to the conventional Fourier analysis, the wavelet transform reveals both transient and stationary characteristics of a signal under a multiresolution framework [16]. For discrete-time signals, the wavelet transform can be realized with the filter-bank structure shown in Fig. 4, where h0[n] and h1[n] correspond to the scaling (lowpass) coefficients and wavelet (highpass) coefficients, respectively. A p-leveldecomposition along the lowpass subbands creates p +1 subbands H1, H2, … , H p and L p , where the L p subband stands for the base (approximation) component and the H subbands represent the details at various scales. Perfect reconstruction can be built from the constituent subbands by upsampling and interpolative filtering with the synthesis filters g 0[n ] and g 1[n ]. For two-dimensional signals such as images, DWT is mostly independently performed along rows and columns. Observe a two-level decomposition of the Lena image shown in Fig. 3(a). The resulting DWT coefficients demonstrate two important facts that support the zerotree coding. First, an overwhelming majority of energy concentrates in the lowpass subbands. This property is termed “energy compaction.” Second, there exists obvious correlation between parent and child nodes.2.3 Properties of wavelet filtersThe multiresolution analysis requires the lowpass filter h 0[n ] to satisfy the following scaling equation2][0=∑n n h (1)An orthonormal wavelet of length N +1 further satisfies][)1(][01n N h n h n −−=, ][][00n h n g −=, ][][11n h n g −=(2)and the energy conservation condition 1][][][][21202120====∑∑∑∑nn n n n g n g n h n h (3)Biorthogonal wavelets have two sets of complementary bases that satisfy][)1(][10n h n g n −−=, ][)1(][01n h n g n −−=. (4)Orthonormal transformation implies norm preserving; effective quantization can thus be directly applied to the transform coefficients. Several near-orthogonality measures for biorthogonal wavelets have been proposed in the literature, mostly based on the norm-preserving property [6], [17-19]. Let,][200∑=n n h w .][211∑=n n h w(5)It can be shown that w 0 and w 1 are the weighting factors of the mean-square quantization error to the lowpass and highpass subbands, respectively [18]. In fact, w 0 and w 1 are related to the Riesz constants upon considering the reconstruction error in the frequency domain [6, 18]. In this paper, we adopt the near-orthogonality measure (NOM) defined in[17]:}.,max{NOM 10w w = (6)NOM upper bounds the multiplicative distortion to the quantization error introduced by non-orthogonality. Clearly, orthogonal wavelets have the NOM value equal to 1. A larger deviation of NOM from 1 indicates less orthogonality. Recently, a more elaborate model for measuring orthogonality has been proposed in [19]. Their derivation was based on the eigen-analysis of the discrete hyper-wavelet transform.In this paper, we examine a wide variety of wavelet bases commonly referred to in the image-coding community. According to the data type and orthogonality, these wavelet bases naturally fall into three categories:A. Orthonormal wavelets: Haar (D2), D4, D6, D8.B. Floating-point biorthogonal filters: 9/7D and 10/18.C. Integer biorthogonal wavelets: 5/3, 9/7M, 5/11A, 5/11C, 13/7C, 13/7T, and9/7WY.Haar is the simplest nontrivial wavelet, which takes the sum and difference of input samples for approximation and detail, respectively. D4, D6, and D8 are the Daubechies orthonormal wavelets with compact support and maximal number of vanishing moments [20]. The 9/7D wavelet is an odd-symmetric filter derived from an orthonormal mother wavelet [21], while 10/18 is a longer even-symmetric filter [22]. Integer biorthogonal wavelets [5] are fixed-point approximations to their parent real counterparts; they can be implemented in the lifting framework without costly floating-point operations [5, 23, 24]. The 9/7WY wavelet [25] is a recently derived integer filter that is similar to 9/7D but with much less computational burden. Moreover, transformation through integer wavelets is reversible and suitable for a unified lossy and lossless codec [26]. The 5/3 and 9/7D filters have been adopted in the JPEG-2000 standard [27].The analysis filters h0[n] and h1[n] of the wavelets under study are listed in Table 1. For biorthogonal wavelets, only half of the coefficients are given since the other half can be deduced from symmetry. The NOM values of the biorthogonal filters are listed in Table 2. The 9/7D and 9/7WY wavelets are much closer to orthogonality than the others. For Category C wavelets, the scaling factor 2 in (1) is rescaled to 1 in Table 1 to facilitate integer operations. We evaluated the time complexity of these wavelet transforms on PC (Pentium 4). The results are given in Table 3, where the number indicates the ratio of the required processing time with respect to the simplest Haarwavelet with convolution. It should be reminded that this complexity measure is hardware dependent. No hardware acceleration such as the SIMD (single instruction multiple data) technique was involved in our evaluation. Nevertheless, it is clear that the convolution-based 10/18 filter is far more complex than the others even with the floating-point support of the Pentium CPU. In contrast, the integer biorthogonal wavelets are very attractive in practice.3. OPTIMIZED WAVELET TRANSFORMS FOR THE SPIHT CODINGIn this section, we investigate the parameters of the SPIHT coding crucial to compression efficiency. Table 4 lists the coding performance of SPIHT at four bit rates (1/8, 1/4, 1/2, and 1 bpp). Four 512×512 gray-level images Lena, Baboon, Pepper, and F16 (Fig. 5) selected from the USC image databases [28] are tested. A 5-level DWT with each of the aforementioned filters is employed for multiresolution decomposition. The visual quality is objectively measured by the peak signal-to-noise ratio (PSNR). For an image X = (x 1, x 2, …, x M ) and its distorted version Y = (y 1, y 2, …, y M ), the PSNR is computed as follows.)(1255log 10PSNR 12210⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−=∑=M i i i y x M (7)Considerable PSNR gap (up to 3dB) is observed when different wavelets are employed, especially for smooth images such as Lena and Pepper. Among the examined filters, the9/7D, 9/7WY and 10/18 filters generally achieve the best PSNR performance. Filters of shorter taps compromise on their compression efficiency. It is also noted that the biorthogonal wavelets are substantially better than the orthogonal ones; the 5/3 filter outperforms the much more complex D8 filter in many cases.It was conjectured that energy compaction might be the dominant factor for the compression performance of wavelet transform. We measure the energy compaction in two ways, approximation spectral significance (in spectral domain) and reconstruction error (in spatial domain), which are both listed in Table 5. The approximation spectral significance is the percentage of the sum of squared coefficients within the approximation subband (LL5) to the sum of squared coefficients of all subbands. The reconstruction error is defined as the mean square error (MSE) when only the LL5 subband is decoded (the other subbands are filled with 0). In the view of the SPIHT coding, the former primarily affects the formation of zerotrees whereas the latter is related to the quantization error. The 9/7D and 9/7WY wavelets possess the best approximation spectral significance while the 10/18 wavelet possesses the least reconstruction error. This partially explains the excellent performance of these two wavelets. The 9/7D and 9/7WY may also benefit from it’s being near orthonormal [6]. However, the coding performance is not solely determined by energy compaction and orthogonality. For example, higher approximation spectral significance of orthogonal wavelets does not translate into better coding performance.To further distinguish between orthogonal and biorthogonal wavelets, it is of interest to know the advantage obtained by symmetric extension. Filtering with finite-length sequences causes data expansion. A universal solution to this problem is the circularconvolution together with periodic extension. However, this approach introduces boundary artifacts; the aliasing components introduced by periodic extension makes compression less efficient. To circumvent this difficulty, symmetric extension of input samples have been developed for linear-phase filters [29]. Imposing the linear-phase constraint, however, usually breaks the orthogonality of the filter. The only real-valued orthogonal linear-phase wavelet with compact support is the trivial Haar filter. Good linear-phase biorthogonal FIR wavelets (categories B and C) are thereby designed. Note that the method of data extension should be in conformity with the type of filter’s symmetry (Fig. 6). Similar consideration should be also borne in mind for down-sampling and up-sampling. The results shown in Table 4 were obtained with the best extension method, i.e., periodic extension for orthogonal wavelets and appropriate symmetric extension for biorthogonal wavelets. In Table 6, the coding results with both symmetric and periodic extension are given for comparison. Symmetric extension provides a substantial edge over periodic extension at low rates for low-activity images (Lena and Pepper). Compared to the orthogonal wavelets of similar length, the distinguished 5/3, 9/7D (and 9/7WY) filters stand out not solely owing to symmetry. Unser and Blu [6] attributed the success of these filters to the better approximation for smooth regions of images [6]. The number of decomposition levels (p) is another factor that influences the performance of wavelet-based image coders. The PSNR performance of SPIHT for p = 4 and p = 6 under a similar test environment of Table 4 are given in Tables 7 and 8, respectively. When a very scarce bit budget is available, encoding down to the higher-level nodes is beneficial because these nodes refer to a larger area. As a11consequence, a nontrivial increase is observed from p = 4 to p = 5 at low bit rates. Moving one level further (from p = 5 to p = 6) makes negligible contribution because the LL6 subband contains few coefficients relative to the available bit budget.4. QUANTIZATION AND ENTROPY COING OF SPIHTDivide-and-conquer is the basic philosophy behind the transform coding. An adequate transformation manages to remove the inter-pixel correlation and arrange the transform coefficients in a prioritized order. Simple scalar quantization can thus be effortlessly performed on the resulting transform coefficients. An optimal encoder produces the most economical representation of wavelet coefficients in the order of their relative importance. In SPIHT, the magnitude of wavelet coefficients is the prime indicator of their significance. However, this indicator may not faithfully reflect the actual importance in the rate-distortion sense, especially when non-orthogonal transformation is employed. In this paper, we investigate an optimized quantization scheme by scaling subbands. Adjusting the magnitude of wavelet coefficients may alter their quantized value and encoding priority. We define the scaling factor K, by which the coefficients of highpass filter h1[n] are multiplied. A smaller scaling factor implies an emphasis on lowpassB Bcoefficients upon quantization and a better change of forming zerotrees. The best choice of K makes a compromise between the quantization error and the number of bits required for specifying the significance map. Table 9 gives the optimal scaling factor (using exhaustive search) and the resulting PSNR gain (cf. Table 4). For a quantization scheme12that does not take the parent-child correlation into consideration, scaling can provide substantial coding gain [30]. However, for SPIHT the PSNR increase is negligible (less than 0.21 dB) and insensitive to the scaling factor. Our conjecture is that the zerotree coding of SPIHT has encoded the wavelet coefficients in the order of their relative importance. Both the approximation and detail information is well preserved even with the skewed coefficients. The arithmetic coding can further squeeze the quantization symbols. However, the arithmetic coding involves more intensive computation. Our simulation under the PC environment indicates that the overall computation time approximately increases by 50% with the arithmetic coding. The PSNR gain of the arithmetic coding for SPIHT is listed in Table 10 (cf. Table 4). A nontrivial but limited PSNR gain (about 0.5 dB) is achieved and this margin is relatively consistent across the filters.5. CONCLUSIONWe have investigated the factors crucial to the performance of the prestigious SPIHT coding. We first explored the influence of wavelet filters, data extension types, and decomposition levels. Second, a scaling scheme for restoring the energy distortion of wavelet subbands was investigated. Finally, the effect of the entropy coding was examined. A comprehensive evaluation in terms of PSNR and time complexity was made at four bit rates for four test images. The results of this paper establish the guidelines for implementing wavelet-based codecs.136. ACKNOWLEDGEMENTThis work was supported by the National Science Council, R. O. China, under the contract number NSC 92-2218-E-027-016. The authors want to thank the anonymous reviewers for their helpful comments.REFFERENCES[1] J. M. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Signal Processing 41(12), 3445-3462 (1993). [2] A. Said and W. A. Pearlman, “A new, fast, and efficient image codec based on set partitioning in hierarchical trees, ” IEEE Trans. Circuits Syst. Video Technol. 6(3), 243-250 (1996). [3] J. D. Villasenor, B. Belzer, and J. Liao, “Wavelet filter evalution for image compression,” IEEE Trans. Image Processing 4(8), 1053-1060 (1995). [4] J. Li, P.-Y. Cheng, and C.-C. J. 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Gopinath, and H. Guo, Introduction to Wavelets and Wavelet Transforms: A Primer, Prentice-Hall, Inc. (1998). [17] F. Moreau de Saint Martin, A. Cohen, and P. Siohan, “A measure of near-orthogonality of PR biorthogonal filter banks,” Proc. Acoustics, Speech, Signal Processing, 1480-1483 (1995). [18] B. Usevitch, “Optimal bit allocation for biorthogonal wavelet coding,” Proc. Data Compression Conf., Snowbird, UT, 387-395 (1996). [19] G. Wang and W. Yuan, “Optimal model for biorthogonal wavelet filters,” Opt. Eng., 42(2), 350-356 (2003). [20] I. Daubechies, Ten Lectures on Wavelets, SIAM, CBMS series (1992). [21] M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Processing 1(2), pp.205-220, Apr. 1992. [22] M. J. Tsai, J. D. Villasenor, and F. Chen, “Stack-run image coding,” IEEE Trans. Circuits Syst. Video Technol. 6(5), 519-521 (1996). [23] I. Daubechies and W. Sweldens, “Factoring wavelet transforms into lifting steps,” J. Fourier Anal. Appl. 4(3), 247-269 (1998). [24] A. R. Calderbank, I. Daubechies, W. Sweldens, and B.-L. Yeo, “Wavelet transforms that map integers to integers,” Appl. Comput. Harmon. Anal. 5, 332-369 (1998). [25] G. Wang and W. Yuan, “Generic 9-7-tap wavelets filters and their performance studies on image compression,” Acta Electron. Sinica, 29(1), 130-132 (2001). IEEE Int. Conf.16[26] J. Reichel, G. Menegas, M. J. Nadenau, and M. Kunt, “Integer wavelet transform for embedded lossy to lossless image compression,” IEEE Trans. Image Processing 10(3), 383-392 (2001). [27] B. E. Usevitch, “A tutorial on modern lossy wavelet image compression: foundations of JPEG 2000,” IEEE Signal Processing Mag. 18(5), 22-35 (2001). [28] USC image database, /services/database/Database.html [29] H. Kiya, K. Nishikawa, and M. Iwahashi, “A development of symmetric extension method for subband image coding,” IEEE Trans. Image Processing 3(1), 78-81 (1994). [30] Shih-Hsuan Yang and Wu-Jie Liao, “Optimal scaling of wavelet-based image coders,” National Symposium on Telecommunication, Keelung, Taiwan, (2004).17BIOGRAPHIESShih-Hsuan Yang received the B.S. degree in electrical engineering from the National Taiwan University in 1987. He obtained the M.S. and Ph.D. degrees in electrical engineering and computer science from the University of Michigan, Ann Arbor, in 1990 and 1994, respectively. Since 1994, he has been a faculty member of the National Taipei University of Technology, Taiwan. He is currently an associate professor of Computer Science and Information Engineering. His major research interests include image and video coding, multimedia transmission, data hiding, and information theory.Wu-Jie Liao was born in Yunlin, Taiwan, in 1979. He received the B.S. and M.S. degrees from the National Taipei University of Technology in 2002 and 2004, respectively. Currently he is working with the Primax Electronics Ltd., Taipei, Taiwan, for developing multifunction peripherals.18Table 1. Analysis filters of the wavelets under study. The negative indexes apply to categories B and C. An extra scaling factor of 2 is needed for Category C wavelets to conform to the scaling equation.Category & Name Haar h0[n] (D2) h1[n] h0[n] D4 h1[n] A h0[n] D6 h1[n] h0[n] D8 h1[n] h0[n] 9/7D h1[n] B h0[n] 10/18 h1[n] h0[n] 5/3 h1[n] h0[n] 9/7M h1[n] h [n] 5/11A 0 h1[n] C h0[n] 5/11C h1[n] h [n] 13/7C 0 h1[n] h [n] 13/7T 0 h1[n] h [n] 9/7WY 0 h1[n]B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B0 0.7071 0.7071 0.48296 0.12941 0.33267 -0.03523 0.23038 0.01060 0.85267 0.78849 0.75891 0.62336 3/4 1 23/32 1 3/4 63/64 3/4 31/32 41/64 1 87/128 1 19/32 9/81 (-1) 0.7071 -0.7071 0.83652 0.22414 0.80689 -0.08544 0.71485 0.03288 0.37740 -0.41809 0.07679 -0.16337 1/4 -1/2 1/4 -9/16 1/4 -67/128 1/4 -35/64 5/16 -9/16 9/32 -9/16 43/160 -19/322 (-2)3(-3)Filter Index 4 (-4) 5 (–5)6(-6)7(-7)8(-8)0.22414 -0.83652 0.45988 0.13501 0.63088 -0.03084 -0.11062-0.12941 0.48296 -0.13501 0.45988 -0.02798 -0.18703 -0.02385-0.08544 -0.80689 -0.18703 0.02798 0.037830.03523 0.33267 0.03084 0.03288 -0.01060 0.63088 -0.71485 0.23038-0.04069 0.06454 -0.15753 8.2e-5 0.02885 -0.08566 0.01377 0.03083 0.00253 -0.00945 2.7e-6 -1/8 -1/8 0 -1/8 0 -1/8 0 -31/256 0 -63/512 0 -12/160 -1/16 0 1/16 7/256 7/128 -1/16 1/16 -1/32 1/16 -3/160 3/32 1/640.000951/128 1/64 7/128 9/256 9/320-1/256 -1/128 0 0-1/256 -1/512Table 2: NOM (near-orthogonality measure) of the biorthogonal wavelets under study9/7D 1.040 10/18 1.215 5/3 1.438 9/7M 1.347 5/11A 1.438 5/11C 1.438 13/7C 1.310 13/7T 1.300 9/7WY 1.021Table 3. Relative time complexity of the wavelet transforms under study.D2 1.00 D4 1.92 D6 2.92 D8 4.03 9/7D 3.94 10/18 7.58 5/3 1.03 9/7M 1.17 5/11A 1.39 5/11C 1.40 13/7C 1.21 13/7T 1.26 9/7WY 1.6119。
