IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 9, SEPTEMBER

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1 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 9, SEPTEMBER Image Coding Using Dual-Tree Discrete Wavelet Transform Jingyu Yang, Yao Wang, Fellow, IEEE, Wenli Xu, and Qionghai Dai, Senior Member, IEEE Abstract In this paper, we explore the application of 2-D dual-tree discrete wavelet transform (DDWT), which is a directional and redundant transform, for image coding. Three methods for sparsifying DDWT coefficients, i.e., matching pursuit, basis pursuit, and noise shaping, are compared. We found that noise shaping achieves the best nonlinear approximation efficiency with the lowest computational complexity. The interscale, intersubband, and intrasubband dependency among the DDWT coefficients are analyzed. Three subband coding methods, i.e., SPIHT, EBCOT, and TCE, are evaluated for coding DDWT coefficients. Experimental results show that TCE has the best performance. In spite of the redundancy of the transform, our DDWT TCE scheme outperforms JPEG2000 up to 0.70 db at low bit rates and is comparable to JPEG2000 at high bit rates. The DDWT TCE scheme also outperforms two other image coders that are based on directional filter banks. To further improve coding efficiency, we extend the DDWT to an anisotropic dual-tree discrete wavelet packets (ADDWP), which incorporates adaptive and anisotropic decomposition into DDWT. The ADDWP subbands are coded with TCE coder. Experimental results show that ADDWP TCE provides up to 1.47 db improvement over the DDWT TCE scheme, outperforming JPEG2000 up to 2.00 db. Reconstructed images of our coding schemes are visually more appealing compared with DWT-based coding schemes thanks to the directionality of wavelets. Index Terms Anisotropic decomposition, image coding, redundant transform, sparse representation, wavelet transform. I. INTRODUCTION WAVELET-BASED image coding has witnessed great success in the past decade. Being separable, conventional 2-D discrete wavelet transform (DWT) efficiently captures point singularities, but fails to capture 1-D singularities, such as edges and contours in images that are not aligned with the horizontal or vertical direction [1]. Therefore, 2-D DWT cannot provide efficient approximation for directional features of images which in turn affects the performance of DWT-based coding schemes. Many multiscale tools have been invented to boost image coding performance by incorporating directional Manuscript received November 26, 2007; revised April 23, First published July 9, 2008; last published August 13, 2008 (projected). This work was supported in part by the Joint Research Fund for Overseas Chinese Young Scholars of NSFC (Grant ), in part by the Distinguished Young Scholars of NSFC (Grant ), and in part by the key project of NSFC (Grant ). The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Kai-Kuang Ma. J. Yang, W. Xu, and Q. Dai are with the Department of Automation, Tsinghua University, Beijing , China ( yangjy03@mails.tsinghua.edu.cn; xuwl@mail.tsinghua.edu.cn; qhdai@tsinghua.edu.cn). Y. Wang is with the Department of Electrical and Computer Engineering, Polytechnic University, Brooklyn, NY USA ( yao@poly.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TIP representation. These tools can be classified into two categories according to the domain where they are designed: spatial-domain multiscale directional transform (SMDT) and frequencydomain multiscale directional transform (FMDT). SMDTs are mainly designed under the lifting framework [2] since it is very convenient to introduce adaptivity into lifting steps while guaranteeing perfect reconstruction (PR). Relevant issues such as invertibility, stability, and artifacts are investigated in several early adaptive lifting frameworks [3] [5]. Realizing that filtering along the elongated direction of edge-like discontinuities helps to annihilate large wavelet coefficients, adaptive directional lifting schemes choose to warp its filtering directions of lifting steps to the orientation of directional discontinuities. The selection of filtering direction is an optimization problem, e.g., to minimize the energy of prediction error. Generally, work on SMDT falls into two types: SMDT without side information and SMDT with side information. In SMDT without side information, filtering directions in lifting steps are determined by casual samples so that they can be recovered with the same procedure at the decoder side [6], [7]. However, it cannot be applied to scalable coding since mismatch occurs when the reconstructions of spatially causal samples at the decoder are not the same as the ones at the encoder due to truncation of bitstream in low bit rate decoding. In SMDT with side information, the selected filtering directions are sent to the decoder together with coefficients as side information. Therefore, it is free of the direction mismatch problem for scalable coding. Schemes of this kind are fully developed, and show significant coding performance over JPEG2000. Representatives can be found in [8] [11]. In FMDT, the basis functions of each subband orient at a certain direction, overcoming the poor directionality of 2-D DWT. Representative work includes curvelets [12], contourlets [13], bandelets [15], directionlets [16], multiscale directional filter banks (DFB) [17] [19], and complex wavelets [20], [21]. Various FMDT-based image coding schemes have been proposed [15], [19], [25] [29], [48] [50]. For example, the bandeletsbased coder in [15] outperforms a DWT-based coder (with CDF filter bank and the same subband coding method) by about 0.5 db for Lena and 1.5 db for Barbara, which shows its potential for image coding. A contourlet-based image coding scheme in [25] is visually competitive to DWT-based coder in spite of its redundancy. Hybrid schemes of combining contourlets and wavelets are proposed to eliminate the redundancy, and show about 0.5 db improvement for images of rich directional features [26], [27]. However, the improvement of current contourlet-based image coding schemes over JPEG2000 is marginal so far. This may be due to the aliasing effects of contourlet filter bank. New aliasing-free filter banks may help to further improve the coding performance [14]. In [19], a hybrid image coding /$ IEEE

2 1556 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 9, SEPTEMBER 2008 scheme based on wavelet and DFB shows competitive coding performance compared with JPEG2000. In [28], to code coefficients of the pyramidal DFB, morphological operations are employed to cluster significant coefficients and context models for entropy coding are also designed. This coder gains 0.4 db on average for Barbara over JPEG2000 at low bit rate range. Further improvement of 0.3 db can be achieved by using a quincunx directional filter bank [29]. Exploration of image coding using these FMDT tools is just at its early stage. In this paper, we propose image coding schemes based on a complex wavelet transform, the 2-D dual-tree wavelet transform (DDWT) [20], [21]. Our work falls into the second category, i.e., the FMDT-based image coding. With good directionality and shift invariance, DDWT has been successfully used in many applications such as image retrieval [22], denoising [23], and watermarking [24]. To employ DDWT for image coding is much more challenging since it is a redundant transform. It seems contradictory to the goal of compression which is to reduce whatever redundancy as much as possible while the redundant transform itself introduces redundancy. However, if coefficients of a redundant transform are sparse enough, compression can benefit from the introduced directionality of the transform. In related work, it has been shown that 3-D DDWT-based video coding can outperform DWT-based video coding, when both do not use motion compensation [44] [47]. Encouraged by the coding performance of 3-D DDWT for video coding, we propose to use 2-D DDWT for image coding. Kingsbury and Reeves, in their early work on DDWT [34], have shown that 2-D DDWT gives slightly higher PSNR than 2-D DWT at the same estimated entropy of quantized coefficients. To our best knowledge, no real coding result has been reported in literature before preliminary results of our work [48]. To get sparse representation with DDWT, three algorithms, matching pursuit (MP) [31], basis pursuit (BP) [32], and noise shaping (NS) [33], are compared in terms of nonlinear approximation (NLA) efficiency. We found that NS achieves the highest NLA with the lowest computational complexity. To examine the characteristics of DDWT coefficients, we investigate the interscale, intersubband, and intrasubband dependency among DDWT coefficients via information-theoretic analysis. After NS, DDWT coefficients are substantially decorrelated. Interscale dependency and intersubband dependency are rather weak while there is still residual intrasubband dependency for images with complicated texture, such as Barbara. This suggests that intrasubband dependency should be exploited when coding DDWT coefficients. We consider three methods for coding DDWT coefficients: SPIHT [51], EBCOT [52], and TCE [54]. EBCOT and TCE exploit intrasubband dependency while SPIHT focus on interscale dependency. Experimental results show that DDWT with TCE provides the best coding performance, and outperforms JPEG2000 at low bit rates and is comparable to JPEG2000 at higher bit rates. This is very encouraging as previous image codecs using redundant systems usually gain at low bit rates but lose at high bit rates [40]. Our image coding scheme also outperforms two FMDT-based image coders: HWD [19] and nuqdfb [29]. Realizing that anisotropic basis functions can better capture directional features of images, better performance (about 0.3 db) can be achieved by applying anisotropic decomposition on DDWT [49], [50]. However, both anisotropic decomposition structures in [49] and [50] are fixed and, thus, lack adaptivity to image content. To further improve coding efficiency, we incorporate adaptivity into DDWT and increase the orientations of DDWT wavelets by performing adaptive anisotropic wavelet packet decomposition on DDWT subbands, which is referred to as anisotropic dual-tree discrete wavelet packets (ADDWP). Anisotropic decomposition is iteratively performed on each DDWT subband to generate all possible decomposition structures which can be described with a quadtree. Then decomposition structures (which corresponds to basis functions) are selected among a large number of candidates to adapt to image characteristics. Notice that best basis selection methods for orthonormal wavelet packets, such as in [41] and [42], are no longer suitable here since DDWT is a redundant transform. The basis selection procedure is divided into two steps to reduce the search space. The quadtree of each DDWT subband is first pruned to minimize the norm cost function. Given the selected decomposition structure, we then employ noise shaping to achieve sparser representation. The proposed ADDWP does not introduce extra redundancy and its coefficient statistics are similar to those of DDWT. It is shown that ADDWP has higher NLA efficiency and captures directional structures with fewer coefficients. For its application to image coding, each ADDWP subband is coded with the TCE coder. Our ADDWP TCE scheme shows better performance compared with the DDWT-based schemes, and outperforms JPEG2000 up to 2 db for images with rich directional features. Compared with two SMDT-based image coders: oriented wavelet transform (OWT) in [9] and adaptive directional lifting (ADL) in [10], the ADDWP TCE scheme provides better performance at low bit rates while shows comparable performance at high bit rates. The reconstructed images of our DDWT-based and ADDWP-based schemes are visually more appealing than those of DWT-based methods thanks to the effectiveness of directional basis functions. The rest of this paper is organized as follows. Section II gives a brief introduction of 2-D DDWT as well as the three methods to achieve sparser representation for DDWT. The nonlinear approximation efficiency of different methods for sparsifying the DDWT coefficients are also compared in Section II. In Section III, dependency between DDWT coefficients is analyzed via an information-theoretic approach and coding results of DDWT-based image coding schemes are presented. In Section IV, we propose 2-D ADDWP and compare the performance of ADDWP-based coding scheme with other state-of-the-art coding methods. Finally, conclusions of our work are drawn in Section V. II. TWO-DIMENSIONAL DDWT AND ITS SPARSE REPRESENTATION A. Two-Dimensional DDWT DDWT [20] brings the advantages of directionality and shift invariance which, thus, provides efficient representation for directional features in images, such as edges and contours. It is an overcomplete transform with redundancy of for m-dimensional signals. Either the real part or the imaginary part

3 YANG et al.: IMAGE CODING USING DUAL-TREE DISCRETE WAVELET TRANSFORM 1557 Let be the vector form of an input image (via vectorization). Let represent the analysis matrix of DDWT while the synthesis matrix. Each column in is a basis function of DDWT. The perfect reconstruction requires that DDWT coefficients should satisfy (1) Fig. 1. (a) Frequency tiling of 2-D DWT and (b) frequency tiling of 2-D DDWT; (c) three wavelets of 2-D DWT and (d) six wavelets of 2-D DDWT. of DDWT can be used as a stand-alone transform since they both guarantee perfect reconstruction. Only the real part of 2-D DDWT is taken in our work to reduce the introduced redundancy from to. The real part of DDWT is simply referred to as DDWT hereafter, unless otherwise stated. Typical wavelets of six 2-D DDWT subbands together with their ideal spectrum supports are illustrated in Fig. 1. For convenient reference hereafter, subband refers to the subband whose wavelets is marked with in Fig. 1,. Those of 2-D DWT are also shown for comparison. It can be seen that wavelets of subband of 2-D DWT are ambiguous in directionality, mixing 45 and together. However, 2-D DDWT is free from this problem. Each DDWT wavelet function has a unique direction, oriented at, respectively. The design of DDWT is fully described in [20] and [21]. B. Transform Setups For the experiments in the following sections, all test images of are 6-level decomposed with DWT or DDWT. For DWT, the CDF filter bank is employed. With respect to DDWT, there is no strong constraint on the filters for the first level decomposition. We choose the CDF filter bank since this filter bank shows excellent performance, and is widely used in image coding. The 6-tap Qshift filters [20, Table II] are used for the remaining stages of DDWT. The coding results of JPEG2000 are produced by Kakadu software [57] with the CDF filter bank. C. Sparse Representation With 2-D DDWT 1) Problem Formulation: 2-D DDWT is a redundant transform with twice number of coefficients as critically sampled 2-D DWT. The challenge of using a redundant transform in coding applications is to select basis functions to provide a sparse representation in which only a small portion of coefficients have large magnitude. This problem is actually to optimize a measurement of coefficient sparseness subject to a set of underdetermined linear equations. Obviously, this is a set of underdetermined linear equations since there are more columns of than its rows. Therefore, there are infinite solutions for (1). The one given by DDWT forward transform usually is not sparse enough as shown later. Unfortunately, to find the sparsest solution (with the fewest number of nonzero coefficients, i.e., the smallest norm) for (1) is NP-hard [30]. 2) Methods To Achieve Sparsity: A number of algorithms turn to find suboptimal solutions. Two well-known algorithms are basis pursuit (BP) [32] and matching pursuit (MP) [31]. In BP, the solution that has the minimum norm is picked. And the optimization problem can be attacked by linear programming. The computational complexity is too high for large scale problem arising in image coding. MP is a pure greedy algorithm. Two sequences are generated in MP: one is the approximation of the signal and the other is the residue between the original signal and the approximation. At each iteration, the basis function that best matches the residual signal is selected. The approximation and residue are updated with the selected basis functions. Due to its inherent greedy nature, MP can be trapped into a bad suboptimum. Another alternative is the iterative projection-based noise shaping method (NS) proposed by Kingsbury [33], [34]. NS can be considered as a special case of the projected Landweber algorithm [35]. The iterative thresholding algorithm also shares the same framework, and is further investigated especially on its connection to inverse problem with sparse constrain [36], [37]. The basic idea of NS is to modify the components of DDWT coefficients that lie in the null space of DDWT s synthesis matrix. Let denotes the range space of while its orthogonal complementary space is. Note that is also the null space of synthesis matrix, where. So, a solution that lies in, e.g.,, plus any component in will give the same reconstruction as itself, i.e.,. One would naturally like to seek a proper in which makes as sparse as possible while perfect reconstruction is naturally guaranteed. NS introduces components of into with an iterative algorithm. More specifically, at each step, coefficients are quantized via thresholding. The reconstructed error in image domain will be projected onto, and then added back to the quantized coefficients. This procedure continues until converging. To accelerate convergence, the reconstructed error image is multiplied by a gain factor which is empirically set to 1.6 as suggested in [33]. Compared with using a fixed threshold for all iterations, the sparseness of coefficients is significantly improved by starting with a large initial threshold, and decreasing

4 1558 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 9, SEPTEMBER 2008 Fig. 3. Examples of subband images: (a) LH band of DWT, (b) S subband of DDWT, and (c) S subband of DDWT NS at the finest scale. Top row: Barbara. Bottom row: Lena. Fig. 2. Nonlinear approximation capability of DWT and DDWT for (a) Barbara and (b) Lena. DWT and DDWT refer to using the largest M largest coefficients directly. DDWT MP, DDWT BP, and DDWT NS refer to using the M largest coefficients obtained by the MP, BP, and NS methods, respectively. the threshold to a target one in each iteration. It is shown that the initial threshold, target threshold, and decrement affect the coding performance [33]. We found that the performance does not vary with the initial threshold and the decrement when the initial threshold is large enough and the decrement is sufficiently small. Therefore, the initial threshold is set to 256 while the decrement is set to 1. For the coding experiments, the best target threshold is chosen for each image and bit rate over a set of possible threshold values. Specifically, the candidate thresholds are 128, 64, 32, 16, and 8. 3) NLA Comparison: In general, largest coefficients are kept to approximate original signals in NLA whereas the first coefficients are used in linear approximation. Approximation error is usually measured by the mean square error (MSE) or peak signal-to-noise ratio (PSNR). To evaluate the capability in sparsifying DDWT coefficients of above three methods, we compare the NLA efficiency of their obtained DDWT coefficients by plotting the pairs of approximation error versus (the number of selected coefficients). As a bench mark, we also present the results obtained from using largest M coefficients of the DWT and DDWT coefficients directly. It can be seen in Fig. 2 that the NLA of original DDWT coefficients is poorer than DWT. All the three methods show significant improvement in terms of NLA capability over the original DDWT. Compared with DWT, BP and MP have better NLA performance only when the number of retained coefficients is in low to medium range. DDWT with NS (referred to as DDWT NS hereafter) gives the best results, and outperforms DWT consistently over the whole interested range. The superiority of NS becomes more prominent as M increases. DDWT NS provides up to 2.50 db over DWT for Barbara when the same number of coefficients are used. To obtain the same reconstruction quality, DDWT NS needs up to 30% fewer coefficients. Subband images of DDWT and DWT at the finest scale are shown in Fig. 3. Subband coefficients are normalized to the range of [0, 255] for illustration purpose. It can be observed that DDWT subbands are indeed much sparser after NS. Experimental results for other subbands and test images are similar. The experiments are carried out on a desktop with a Intel Pentium IV 2.80-GHz CPU and 512-MB RAM. For BP, we use the Atomizer [58] which is based on Wavelab [59]. C codes of 2-D DDWT, including the forward transform and inverse transform, are compiled with mex so that it can be called by Atomizer. As to the computational complexity, the time consumed by BP and NS are 5.5 h and 1.5 min, respectively. For MP, the consumed time are determined by the number of iterations. For example, the running time is 1.1 h for 5000 iterations with reconstruction quality of db. We can conclude that NS performs the best in terms of both sparseness and computational complexity among the three methods. It is, hence, employed in the following coding experiments. In the remainder of this paper, we simply refer to the DDWT NS coefficients as the DDWT coefficients, unless specified otherwise. III. IMAGE CODING USING 2-D DDWT In this section, we investigate image coding using DDWT. The dependency characteristics of DDWT coefficients are exploited using mutual information. Then three representative wavelet subband coding methods are briefly reviewed. Coding results of DDWT coefficients with these methods are reported and compared with JPEG2000 as well as two FMDT-based image coding schemes.

5 YANG et al.: IMAGE CODING USING DUAL-TREE DISCRETE WAVELET TRANSFORM 1559 TABLE I ESTIMATED MUTUAL INFORMATION BETWEEN CURRENT COEFFICIENT X AND ITS PARENT PX,COUSINS CX, AND NEIGHBORS NX A. Dependency Analysis of DDWT Coefficients Exploiting dependency among coefficients is a crucial technique to achieve high compression performance in popular DWT-based image coding schemes, such as SPIHT [51], EBCOT [52], and TCE [54]. In this subsection, we examine the intrasubband, intersubband, and interscale dependency among DDWT coefficients using an information-theoretic measure. Mutual information is a good criterion to measure how much information one random variable tells about another. Mutual information of three pairs, i.e.,, and, are evaluated, where denotes a coefficient, represents its parent (the coefficient at the same spatial location at the next coarser subband), contains its cousins (the coefficients at the same spatial location of other subbands at the same scale), and denotes its eight neighbors (the neighboring coefficients within the same subband). Therefore, and measures interscale, intersubband, and intrasubband correlation, respectively. Note that a coefficient has multiple cousins and neighbors. To estimate mutual information between a random variable and multiple random variables suffers the dilution problem due to lack of enough large number of samples which, thus, severely affects the estimation accuracy. Following the method in [39], we reduce the dimensionality with a sufficient statistic. We estimate instead of, where is considered to be a sufficient statistic of, and is set to equal weights. Estimated mutual information for DWT coefficients and DDWT NS coefficients of two test images, Lena and Barbara, are reported in Table I. It can be seen that and for DDWT coefficients are almost zero. Intrasubband correlation is still large for some subbands of images with rich textures such as Barbara. However, it is significantly smaller than that for DWT coefficients. Results for other natural test images are quite similar to those reported here. We can conclude that the intersubband correlation and interscale correlation are very weak, and the intrasubband correlation is significantly reduced than that of DWT but still exists for highly textured images. B. Subband Coding Methods All state-of-the-art techniques for coding wavelet coefficients utilize bit-plane coding. Fractional bit-plane coding techniques that consist of multiple passes are adopted to produce scalable bitstream. The key issue to achieve optimal R-D decay is to identify significant coefficients and code the location information efficiently [38]. We examine the application of three representative bit-plane coding methods for DDWT NS coefficients: treestructured based SPIHT [51], block-based EBCOT [52] that is used in JPEG2000, and subband-based TCE [54]. SPIHT mainly exploits interscale correlation of wavelet coefficients. Orientation trees is introduced to group significant coefficients at each bitplane. This elegant algorithm is based on the fact that if a parent is insignificant, its children are also insignificant with high probability since the magnitude of wavelet coefficients decays exponentially from the coarsest scale to the finest scale. If all descendants of a node are insignificant, they are marked as a zerotree and efficiently represented with only one symbol. EBCOT focuses on utilization of intrasubband correlation. Each subband is partitioned into nonoverlapped blocks, and each block is independently coded with three passes: significance, refinement, and cleanup. This fractional bit-plane technique generates granular embedded bitstream with R-D optimized truncation. For each pass, symbols are coded with adaptive context-based arithmetic coder. Orientation of subband are taken into account in context modeling. The TCE coding scheme [54] is an extended version of the subband coding method that uses a one-order recursive Tarp filter [53], by introducing the fractional bit-plane coding technique. The core of this scheme is to code location information with an arithmetic coder driven by a probability model that runs on the bit-plane level. The probability that current symbol is one is estimated by three Tarp filtering steps. The Tarp filtering is within each subband; hence, intrasubband dependency is exploited. C. Coding Results of DDWT Coefficients Two-dimensional DDWT has six subbands at each scale instead of three for 2-D DWT. To apply SPIHT to DDWT coefficients, orientation trees for DDWT subbands are constructed in the same way as for DWT subbands. Hence, DDWT has twice the number of orientation trees as DWT. For EBCOT and TCE, each subband is coded individually. Coding performances of above three coding schemes together with JPEG2000 are shown in Fig. 4. DDWT TCE gains up to 0.86 db over JPEG2000 while outperforming other two methods. The coding gain of DDWT TCE over DWT mainly comes from directionality of DDWT wavelets and the effectiveness of noise shaping to select the proper DDWT wavelets. As shown in Fig. 5, DDWT TCE also gives better visual quality than JPEG2000 thanks to thanks to the capability of DDWT in representing directional features. Among the three DDWT-based methods, the biggest gap of coding efficiency is 1 db (DDWT TCE over DDWT EBCOT at 0.8 bpp for Lena). This indicates that subband coding method is very crucial for coding efficiency. Compared with DDWT SPIHT, DDWT TCE shows 0.42 db gain on average for Barbara. This is because intrasubband correlation, which

6 1560 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 9, SEPTEMBER 2008 Fig. 6. Illustration of (a) isotropic refinement and (b) anisotropic refinement. Fig. 7. Resulting wavelets of anisotropic decomposition on the 75 and their idealized spectrum supports. subband Fig. 4. R-D performance for DDWT TCE, DDWT SPIHT, DDWT EBCOT, JPEG2000 for (a) Lena and (b) Barbara Fig. 8. Illustration of quadtree of anisotropic decomposition. (a) A complete quadtree, (b) a pruned quadtree, and (c) the corresponding decomposition structure. Fig. 5. Visual comparison between DDWT TCE and JPEG2000 with enlarged patches of reconstructed images at 0.2 bpp. Top row: Lena. Bottom row: Barbara. Left: original images. Middle: results of JPEG2000. Right: results of DDWT TCE. DDWT TCE mainly exploits, is much more stronger than the other two kinds of correlation as indicated in Table I. DDWT SPIHT works better for Lena than Barbara since Lena has slightly more interscale correlation. DDWT EBCOT performs the worst although it exploits intrasubband correlation. We have adapted the context models used by EBCOT to the characteristics of the DDWT NS coefficients. However, such adaptation does not lead to performance improvement. This may be due to the fact that subband coefficients are partitioned into stripes with the height of four samples and each stripe is coded independently with EBCOT. This structure limits its efficiency in coding highly sparse coefficients despite the use of context-based arithmetic coding. Through the comparison among these three methods, we identified TCE to be a more suitable coding method for DDWT coefficients. We also compare the DDWT TCE scheme with two FMDTbased methods: HWD [19] and nuqdfb [29]. The basic idea of these two schemes is to decompose image with DFB so that directional features can be separated into directional subbands. In HWD, DFB is applied to high-pass subbands of DWT to obtain directional basis functions while the low-pass subband is iteratively decomposed using traditional wavelet filter bank. The resulting subbands are coded with a SPIHT-like algorithm. In nuqdfb, an image is decomposed by a nonuniform quincunx directional filter bank for the first two scales, and the remaining scales employ traditional wavelet filter bank. There are two steps

7 YANG et al.: IMAGE CODING USING DUAL-TREE DISCRETE WAVELET TRANSFORM 1561 Fig. 9. NLA curves of 2-D ADDWP coefficients and 2-D DDWT coefficients for Barbara. TABLE II R-D PERFORMANCE COMPARISON FOR BARBARA bit rates. This dismisses the previously held impression that image compression with redundant representation only shows good performance at very low bit rates. As to computational complexity of the DDWT-based schemes, the encoder and decoder are asymmetric since NS in the encoding is computationally demanding. To demonstrate this, we run encoding and decoding with Lena at 1.0 bpp on a desktop with a Intel Pentium IV 3.0-GHz CPU and 1.5-GB RAM. For NS, we used a target threshold of 8. DDWT TCE takes s and 1.17 s for encoding and decoding, respectively. For encoding, the running time for forward transform of DDWT, NS, and TCE subband encoding are 0.06 s (0.13%), s (98.02%), and 0.86 s (1.85%), respectively; for decoding, the running time for inverse transform of DDWT and TCE subband decoding are 0.09 s (8.13%) and 1.06 s (91.87%), respectively. The computational complexity of the encoder is dominated by the NS procedure, and is considerably higher than that of the DWT-based encoder; while the computational complexity of the decoder is dominated by the TCE subband decoding, and is about 2 times more than that of the DWT-based TCE decoder (due to the 2:1 redundancy of DDWT coefficients). The DDWT transform is implemented via convolution in current implementation. The encoder can be accelerated (about twice faster [2]) via lifting implementation of DDWT transform since the computation in each iteration of NS involves primarily the DDWT transform. TABLE III R-D PERFORMANCE COMPARISON FOR LENA. (DATA FOR LENA IS NOT AVAILABLE WITH HWD) in subband coding of nuqdfb. In the first step, a progressive morphological dilation algorithm is used to cluster significant coefficients. In the second step, significant coefficients that have not been processed are coded with the Tarp filter method. As shown in Tables II and III, DDWT TCE outperforms HWD and nuqdfb for most cases. Note that directional transforms in both HWD and nuqdfb are nonredundant while DDWT are redundant. Many literatures address the importance of nonredundancy in transform design for the potential application to image coding. However, the performance of our scheme is a good example to show that, for image coding, redundant transforms can outperform nonredundant transforms with effective techniques to sparsify the coefficients and to code the resulting coefficients. It is interesting to compare our scheme with other schemes using redundant system. A representative work is the image coding scheme proposed in [40], which is based on MP expansion over a redundant contour-like dictionary. This scheme shows better performance at very low bit rate over JPEG2000, but suffers performance degradation as the bit rate increases as shown in Fig. 5 in [40]. It is worth pointing out that our DDWT TCE scheme is competitive to JPEG2000 even at high IV. TWO-DIMENSIONAL ADDWP AND ITS APPLICATION TO IMAGE CODING In the last section, DDWT TCE achieves better coding efficiency than JPEG2000 and two FMDT-based image coding schemes. In this section, we extend the DDWT to an anisotropic dual-tree discrete wavelet packets (ADDWP). A basis selection method is also developed to choose decomposition structures that well adapt to image content. Significant improvement of coding performance over DDWT TCE demonstrates the effectiveness of ADDWP. A. Two-Dimensional ADDWP Although being directional, DDWT wavelet functions are isotropic since their spectrum supports are square as shown in Fig. 1(a) and (c). Isotropic basis functions fail to efficiently represent 1-D singularities that elongate at varying orientations. For example, in Fig. 6(a), an edge is successively approximated by multiscale directional but isotropic basis functions such as DDWT wavelets. The square support of isotropic basis function cannot adapt to the elongated shape of the edge. As a result, the edge will intersect with many basis functions which, thus, produces many large coefficients. Anisotropy of basis function is advocated to overcome this shortcoming [12], [13], [40]. As shown in Fig. 6(b), anisotropic refinement provides much more concise representation for the edge. Another drawback of DDWT is the lack of adaptivity. Note that the success of SMDTs for image coding is to align filtering directions with the orientations of directional features. However, DDWT has only six fixed orientations. Edges or contours in natural images can have various orientations. Therefore, edges which lie in between directions of two DDWT wavelets cannot be efficiently represented.

8 1562 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 9, SEPTEMBER 2008 Fig. 10. (a) Patch of Barbara, and reconstruction with 1500 largest wavelet coefficients for (b) DDWT and (c) ADDWP. Fig. 11. Comparison of coding performance between the proposed ADDWP-based scheme, DDWT-based scheme, and JPEG2000 for nine test images. Images in (b), (e), and (h) are from image database of Computer Vision Group at University of Granada (available on Specifically, image in (h) is indexed as among several aerial images in the database. To overcome these two shortcomings of DDWT, we propose a new transform, called 2-D ADDWP. The basic idea is to perform anisotropic wavelet packet decomposition on DDWT subbands in order to obtain anisotropic basis functions and also increase the orientations of basis functions. Then the decomposition structures are selected so that the basis functions can adapt to image characteristics. These two steps are described in the following subsections. 1) Anisotropic Wavelet Packet Decomposition on DDWT: In isotropic wavelet packet decomposition, horizontal (row) decomposition is always followed by vertical (column) decomposition or vice versa. As a result, the spectrum supports of resulting subbands are square. On the contrary, in anisotropic wavelet packet decomposition, subbands are allowed to be decomposed only vertically or horizontally. Anisotropic decomposition on DWT subbands provides basis functions with different aspect ratios which are, thus, anisotropic [43]. However,

9 YANG et al.: IMAGE CODING USING DUAL-TREE DISCRETE WAVELET TRANSFORM 1563 the directions of these basis functions are still horizontal, vertical, or diagonal. The idealized spectrum supports of DDWT wavelets are symmetric with respect to the origin (as shown in Fig. 1), which makes DDWT wavelets directional. This property still holds for resulting basis functions of further decomposition with real filter bank DDWT subbands, which provides the directionality of DDWT wavelet packets. If the wavelet packet decomposition is allowed to be anisotropic, resulting basis functions are anisotropic as well. Moreover, the finer is the division in the frequency plane, the more orientations will the resulting basis functions have. That is, performing anisotropic decomposition on DDWT subbands not only generates anisotropic basis functions, but also increases the number of orientations of directional basis functions. Since the employed filter bank is critically-sampled, no extra redundancy is introduced in anisotropic wavelet packet decomposition. For example, performing vertical decomposition on subband in Fig. 1(c) produces two new subbands: one low-pass subband and one high-pass subband. Each of them has the same width as the original subband while the height is reduced to the half of the original subband. The basis functions and corresponding idealized spectrum supports are illustrated in Fig. 7. It can be observed that the resulting basis functions have different directions than the original before anisotropic decomposition. Also they are anisotropic since their supports are rectangular rather than square. Similarly, performing horizontal decomposition on subband obtains two other anisotropic basis functions, as well. Anisotropic wavelet packets is obtained by iteratively applying anisotropic decomposition on DDWT subbands. The wavelet packet decomposition on each DDWT subband can be described by a quadtree as shown in Fig. 8(a). The root of the quadtree represents a DDWT subband. Each node in the quadtree has four children, two of them come from horizontal decomposition while the other two are produced by vertical decomposition. Suppose that an image is decomposed with 2-D DDWT up to scales, producing two low-pass subbands and high-pass subbands, where and. To reduce the complexity of the quadtree pruning described in the next section, the finest frequency resolution of anisotropic wavelet packets along both directions is limited to on the 2-D frequency plane (normalized to [0 1] [0 1]). In other words, the decomposition is allowed only when the width (height) of the subband is greater than that of the DDWT low-pass subband. Therefore, the maximum level of anisotropic decomposition on along both directions is. Then the maximum depth of the quadtree for is (including the root). 2) Adaptive Basis Selection: The problem at hand is how to prune the generated quadtrees so that the selected basis can well adapt to the image characteristics, which is often called best basis selection. Suppose that there are possible bases for wavelet packet decomposition. Each basis contains basis functions of length, i.e.,. These bases form a basis dictionary. Under basis, an image of length can be represented as where denotes the coefficients. The best basis and the corresponding coefficients in the basis dictionary is determined by (2) where is an additive function that measures the cost of coefficients, e.g., norm. For orthonormal wavelet packets, the number of basis functions in is equal to the signal length, i.e.,. And coefficients of under each basis can be determined as. In this case, problem (2) is also called best orthogonal basis (BOB). To search the optimum exhaustively is infeasible for practical applications. Coifman and Wickerhauser proposed a fast algorithm that prunes the decomposition tree from bottom to top using entropy as a cost function [41]. In [42], Ramchandran and Vetterli formulated the basis selection problem in a R-D sense which is solved using a similar pruning strategy. Both these methods depend heavily on the orthogonality of underlying wavelet packets. However, the basis selection methods for orthogonal wavelet packets are not suitable for the proposed anisotropic dual-tree wavelet packets where the bases are not orthogonal due to the redundancy of, i.e.,. In orthogonal wavelet packets, the coefficients can be uniquely determined by the basis. However, in the redundant wavelet packets, the reconstruction equation does not have a unique solution given a basis. Therefore, both and are to be optimized in (2) which, thus, dramatically enlarges the entire search space. Careful design of searching algorithm is required to ensure practical implementation while achieving a good-enough suboptimum. There are two approaches to reduce the search space. One is to first select an optimal basis in for each and then search optimal coefficients given the optimal basis. Straightforwardly, in our setting, for are the coefficients produced by decomposing DDWT subbands up to the decomposition structures associated with. The other is to conduct the optimization vice versa. If coefficients are first optimized, the obtained coefficients will be very sparse. It is difficult to select an good basis from the very sparse coefficients. Therefore, we employ the first approach. The first step of our method is to seek an optimal basis in so that the corresponding feasible coefficients have the minimum cost. The feasible coefficients under basis are obtained when performing the anisotropic wavelet packet decomposition. Each basis corresponds to quadtrees of specific decomposition structures on DDWT subbands. Therefore, solving the optimization problem is equivalent to pruning the quadtrees properly so that the obtained decomposition structures on DDWT subbands have the minimum cost of the wavelet packet coefficients. We use norm as the cost function since it is an effective sparseness measurement of coefficients [32]. Let be the quadtree of subband, where denotes the th node on level of the quadtree, the depth of. Then the child nodes generated by horizontal decomposition on are (low-pass) and (high-pass) and those generated by vertical decomposition are (low-pass) and (high-pass). Let be the decomposition coefficients of. The cost for node, is measured as which is calculated when performing anisotropic wavelet packet decomposition. Let be the subtree of with as its root, then the cost of is the sum of the cost of all its leaves, i.e.,,

10 1564 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 9, SEPTEMBER 2008 where is the index set of the leaves of. Denote as the cost of no decomposition for the cost of horizontal decomposition and the cost of vertical decomposition. If no decomposition is performed on, then will become a leaf, and, hence, the cost of no decomposition is the cost of this node, i.e., ; if is decomposed horizontally, the cost of horizontal decomposition is the cost of two associated subtrees and ; similarly, if is decomposed vertically, the cost of vertical decomposition is the cost of two associated subtrees and. Therefore,, and are calculated as Let be the decision of decomposition for, where, and stand for no decomposition, vertical decomposition, and horizontal decomposition, respectively. Among the three possible decisions for each node, the one that gives the minimum cost will be selected. Each quadtree is pruned from bottom to up as the following Algorithm 1. Algorithm 1 Quadtree Pruning Initialize: ; while do for to do Calculate, and according to (3) if } then ; ; Delete all its subtrees,, and ; else if then ; ; Delete subtrees and ; else ; ; Delete subtrees and ; end if end for end while After traversing with algorithm 1, the quadtree is pruned and the optimal decision for each retained node is determined. And after pruning the quadtrees of all DDWT subbands, the best basis is obtained. Note that horizontal decomposition and vertical decomposition cannot be applied on a node simultaneously. Each node in the pruned version of the quadtree can only (3) have two children left at most. That is, a quadtree is pruned into a binary tree rather than still a quadtree as for isotropic wavelet packet decomposition in [42]. A possible pruned version of the quadtree in Fig. 8(a) is illustrated in Fig. 8(b), as well as the resulting decomposition structure in Fig. 8(c). The second step is to search the optimal coefficients given the selected (redundant) basis from the first step. This is actually the problem we meet in Section II-C. When the cost function is the norm, it is a NP-hard optimization problem, thus cannot be efficiently solved. When the cost function is norm, it becomes BP whose computational complexity is too high for the scale of our problem. In Section II, it has been shown that NS outperforms BP and MP in terms of both NLA of resulting coefficients and computational complexity. Therefore, we use NS for this step as for sparse representation of DDWT in Section II. With the two steps described above, the adaptive basis selection algorithm of 2-D ADDWP can be summarized as Algorithm 2. Algorithm 2 Basis Selection Algorithm of 2-D ADDWP Step 1: Decompose the input image up to level with 2-D DDWT. Step 2: Perform anisotropic wavelet packet decomposition on each DDWT subban. Step 3: Search the optimal decomposition structures by pruning quadtrees with algorithm 1. Step 4: Obtain sparse representation with NS under the selected decomposition structures. The CDF filter bank is employed for anisotropic decomposition. Experiments for image coding in the following section also use this configuration. As shown in Fig. 9, ADDWP has higher NLA than DDWT, which verifies its effectiveness of image representation. To further visualize their capability in capturing directional features, we transform Barbara with DDWT and ADDWP, respectively, and each retains 1500 largest wavelet coefficients for reconstruction. The bottom-right patches of reconstructed images are shown in Fig. 10. It can be observed that more directional textures of Barbara s trousers can be recovered by ADDWP with the same number of retained coefficients compared with DDWT. The improvement of image representation efficiency can be translated into the improvement of coding performance as shown in the next section. B. Image Coding Using 2-D ADDWP In this section, we present image coding results using 2-D ADDWP. Mutual information analysis suggests that only intrasubband dependency significantly exits for regions with rich texture. This dependency characteristics of ADDWP coefficients is the same as that of DDWT coefficients reported in Section III-A, and, thus, the corresponding results are not shown here. With the experience of previous image coding for DDWT coefficients, we code each ADDWP subband using TCE coder. Nine test images are employed for coding experiments.

11 YANG et al.: IMAGE CODING USING DUAL-TREE DISCRETE WAVELET TRANSFORM 1565 TABLE IV COMPARISON OF CODING PERFORMANCE (PSNR) AMONG OWT [9], ADL [10], AND ADDWP TCE The structures of pruned quadtrees are sent to the decoder as side information to ensure correct reconstruction. Each nonleaf node of a pruned quadtree uses two bits to describe the decomposition decision: 00, 01, and 10 indicate no decomposition, vertical decomposition, and horizontal decomposition, respectively. Then the decomposition decisions of a pruned quadtree are coded with adaptive arithmetic coding in the depth-first order. At the decoder side, the structure of the pruned quadtree is recovered by decoding the decomposition decisions in the depth-first order. Coding performance of our coding schemes and JPEG2000 are shown in Fig. 11. ADDWP TCE shows significant coding performance improvement over DDWT TCE, e.g., up to 1.47 db for Barbara, 2.00 db for Elaine, 0.60 db for Bike, 0.40 db for Aerial, and 0.30 db for Boat. Since ADDWP TCE employs the same subband coding method as DDWT TCE, the only difference between the two schemes is on the transform. Therefore, the coding gain is due to the proposed adaptive anisotropic wavelet packet decomposition which provides adaptive and anisotropic wavelets and increases the number of wavelet orientations. Generally speaking, the improvement by ADDWP TCE is larger for images that contain rich directional features, such as Barbara and Bike, and is smaller for images that have large smooth regions such as Lena and Boat. This also verifies the effectiveness of the proposed ADDWP in capturing directional features of images. Compared with DWT-based coding scheme, ADDWP TCE outperforms JPEG2000 by up to 2.00 db for Barbara, 1.52 db for Elaine, 0.82 db for Lena, 0.94 db for Bike, 0.47 db for Aerial, and 0.40 db for Boat. This comparison shows that significant performance improvement over the DWT-based scheme (JPEG2000) can be achieved by designing redundant transforms of appealing properties (such as directionality, anisotropy, and adaptivity) and effective basis selection procedure among redundant basis functions. We also compare our ADDWP TCE method with two recently developed SMDT-based image coding schemes: OWT [9] and ADL [10]. These two schemes are based on the adaptive lifting framework and send the selected filtering directions as side information. Coding results of OWT (with EBCOT for subband coding) are generated with the codec provided on the website of OWT [60]. Coding results of ADL with the best setup (CDF filters and quarter pixel accuracy in prediction and update) are taken from [10]. The overhead of adaptive anisotropic wavelet packets in ADDWP TCE ( bpp) is smaller than that of OWT ( bpp) and ADL ( bpp). This allows more bits are allocated to ADDWP coefficients, leading to higher reconstruction quality especially at low bit rates. As listed in Table IV, ADDWP TCE shows better performance than OWT and ADL at low bit rates and comparable results at higher bit rates. To show improvement on visual quality, patches of original images and reconstructed images by ADDWP TCE and JPEG2000 are shown in Fig. 12. For the space reason, five of nine test images are shown here. For each test image, we present two patches of reconstructed images which is recovered at 0.1, 0.2, 0.4 bpp, respectively. It can be observed that ADDWP TCE not only achieves better coding efficiency in terms of PSNR, but also provides better visual quality for reconstructed images at the same bit rates. For example, we cannot distinguish the texture orientation of the scarf in reconstructed Barbara by JPEG2000 at 0.1 and 0.2 bpp. Our scheme preserves directional pattern more truthfully. The same phenomenon also can be observed about the line patterns in Barbara s trousers. This, in turn, confirms that ADDWP has stronger capability in capturing directional features thanks to the additional directions representable by ADDWP. Since wavelets are compactly supported on both spatial domain and frequency domain, the truncation of small coefficients will cause artifacts known as Gibbs oscillation. In image processing, this phenomenon is called ringing artifacts, occurring around sharp edges. ADDWP also suffers from this. However, due to different properties of wavelets, the artifacts for ADDWP representation show different visual quality compared with DWT. For DWT, the wavelets in subbands mix direction of and 45 together and appear like a checkerboard as shown in Fig. 1(b). Therefore, distorted edges are sawtoothed besides ringing. For ADDWP, each wavelet function are directional and anisotropic. Distorted edges are smooth and ringing effects are along the geometry flow of the edges, which is visually more appealing. The visual difference can be observed in the spokes of Bike, the backstay of Boat, and the shape of Pentagon. The ADDWP TCE encoder requires about three times as much computation as the DDWT TCE encoder whereas the computational complexity of the ADDWP TCE decoder is about 10% higher than that of the DDWT TCE decoder. For example, under the same running environment and setting, ADDWP TCE consumes s and 1.31 s for encoding and decoding Lena at 1.0 bpp respectively while the DDWT TCE schemes takes s and 1.17 s, respectively. In the encoding of ADDWP TCE, the anisotropic wavelet packet decomposition (including quadtree generating and quadtree pruning) needs s (12.36%) while the NS needs s (86.89%). These two steps consume 99.25% of the overall computation of the encoding. Obviously, the computational complexity of ADDWP TCE encoder is much higher than that of most DWT-based image encoders whereas the decoder side only

12 1566 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 9, SEPTEMBER Fig. 12. Comparison of visual quality between ADDWP and JPEG2000 for five test images, i.e., Barbara, Boat, Bike, Pentagon, and Aerial from top to bottom. The left column places original images. For each test image, two patches of the reconstructed images by ADDWP TCE (top row) and JPEG2000 (bottom row) at 0.1, 0.2, and 0.4 bpp, respectively. 2 needs slightly more computation compared with DWT-based decoder. Our scheme is suitable for applications where offline encoding is allowed to achieve high coding efficiency. C. Further Discussions Many researches, including our work, show that image coding schemes using directional wavelet transforms show better coding performance than DWT-based methods. Par- ticularly, image coding schemes using adaptive directional transforms, such as OWT [9], ADL [10], and the proposed ADDWP TCE, outperform those using nonadaptive directional transforms such as contourlet-based schemes [25] [27], DFB-based schemes [18], [19], [28], [29], and our DDWT-based methods. The main reason is that adaptive transforms can much better adapt to the image characteristics than nonadaptive directional transforms. All SMDTs are

13 YANG et al.: IMAGE CODING USING DUAL-TREE DISCRETE WAVELET TRANSFORM 1567 adaptive transforms and most of FMDTs are nonadaptive such as curvelets, contourlets, multiscale DFBs, and complex wavelets. Note that adaptive directional transforms cannot afford pixel-by-pixel adaptivity since overhead of side information will deteriorate overall coding performance. To make a tradeoff, a group of pixels, e.g., 8 8 blocks or blocks with variable sizes as in [10] and [11], share one filtering direction, which is quite similar to the scenarios of nonadaptive directional transforms where a set of subband coefficients share the same orientation. Improvement is possible if adaptivity is properly introduced into nonadaptive directional transforms. Our research is an effort in this direction and indeed shows significant coding gain. The idea can be extended to other nonadaptive directional transforms, which is to be exploited in future work. It is also worthwhile to note that other recently developed coders, such as [55], that do not use directional transforms, have also shown promising performance. The coding scheme in [55] employs a block-based discrete cosine transform (DCT) with pre- and postfiltering [56], rather than a directional transform. This coder achieves higher coding performance than JPEG2000 for highly texture images while maintaining low computational complexity. This suggests that other ways to boost coding efficiency are possible besides employing directional transforms. V. CONCLUSION In this paper, we investigate image coding based on 2-D DDWT, which is a directional and redundant transform. Kingsbury s noise shaping method is employed to sparsify the DDWT coefficients, to enable efficient compression. Our dependency analysis reveals that the DDWT coefficients after noise shaping are highly uncorrelated. We examine the application of SPIHT, EBCOT, and TCE for coding the DDWT coefficients and find that TCE performs the best. In spite of the redundancy of the transform, DDWT TCE outperforms JPEG2000 at low bit rates and gives comparable performance at high bit rates. DDWT TCE was also found to yield better coding efficiency than two other methods that use directional transforms. We extend DDWT to ADDWP by performing adaptive anisotropic wavelet packet decomposition on DDWT subbands to improve representation efficiency for directional features of images. The wavelets of ADDWP are anisotropic and the number of wavelet orientations is dramatically increased. The anisotropic wavelet packets (described by quadtrees) are selected using the norm as criterion for good adaptation to image characteristics. Then NS under the chosen wavelet packets is applied to achieve sparser representation. Using TCE for subband coding, ADDWP TCE shows significant improvement over DDWT TCE. It outperforms JPEG2000 by up to 2 db for images with rich directional features. Compared with two image coding schemes, OWT and ADL, which also use adaptive and directional transforms, ADDWP TCE shows better coding performance at low bit rates and is only slightly inferior at high bit rates. Thanks to its directionality, anisotropy and adaptivity, ADDWP preserves geometric structures more faithfully and, thus, provides better visual quality especially at low bit rates. Nonredundant transforms have been considered to be generally more preferable than redundant transforms in transform coding. The reason is that there are more coefficients to be coded for redundant transforms, which will quite likely degrade the coding performance. Our research is a good example to show that redundant transforms can outperform nonredundant transforms with effective techniques to sparsify the coefficients and to code the resulting coefficients. 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Liang, and C. Tu, Lapped transform via time-domain pre- and post-filtering, IEEE Trans. Signal Process., vol. 51, no. 6, pp , Jun [57] Kakadu, [Online]. Available: [58] Atomizer, [Online]. Available: atomizer/ [59] Wavelab, [Online]. Available: wavelab/ [60] Oriented Wavelet Transform, [Online]. Available: temics/equipe/chappelier/owavelets/ Jingyu Yang received the B.S. degree from the Beijing University of Posts and Telecommunications, Beijing, China, in He is currently pursuing the Ph.D. Degree at the Department of Automation, Tsinghua University, Beijing, China. His research interests include image/video coding, denoising, and analysis.

15 YANG et al.: IMAGE CODING USING DUAL-TREE DISCRETE WAVELET TRANSFORM 1569 Yao Wang (M 90 SM 98 F 04) received the B.S. and M.S. degrees in electronic engineering from Tsinghua University, Beijing, China, in 1983 and 1985, respectively, and the Ph.D. degree in electrical and computer engineering from University of California, Santa Barbara, in Since 1990, she has been with the faculty of Polytechnic University, Brooklyn, NY, and is presently a Professor of electrical and computer engineering. She was on sabbatical leave from Princeton University, Princeton, NJ, in 1998, and from the Thomson Corporate Research, Princeton, from She was a Consultant with AT&T Laboratories-Research, formerly AT&T Bell Laboratories, from 1992 to Her research areas include video communications, multimedia signal processing, and medical imaging. She is the leading author of a textbook titled Video Processing and Communications (Englewood Cliffs, NJ: Prentice-Hall, 2002) and has published over 150 papers in journals and conference proceedings. Dr. Wang has served as an Associate Editor for the IEEE TRANSACTIONS ON MULTIMEDIA and the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY. She received the New York City Mayor s Award for Excellence in Science and Technology in the Young Investigator Category in year She was elected Fellow of the IEEE in 2004 for contributions to video processing and communications. She is a co-winner of the IEEE Communications Society Leonard G. Abraham Prize Paper Award in the Field of Communications Systems in She received the Oversea Outstanding Young Investigator Award from the National Natural Science Foundation of China (NSFC) in 2005 and the Yangtze River Scholar Award from the Ministry of Education of China in Wenli Xu received the B.S. degree in electrical engineering and the M.E. degree in automatic control engineering from Tsinghua University, Beijing, China, in 1970 and 1980, respectively, and the Ph.D. degree in electrical and computer engineering from the University of Colorado, Boulder, in He is currently a Professor at Tsinghua University and Director of the Chinese Association of Automation. His research interests are mainly in the areas of automatic control and computer vision. Qionghai Dai (SM 05) received the B.S. degree in mathematics from Shanxi Normal University, China, in 1987, and the M.E. and Ph.D. degrees in computer science and automation from Northeastern University, China, in 1994 and 1996, respectively. Since 1997, he has been with the faculty of Tsinghua University, Beijing, China, and is currently a Professor and the Director of the Broadband Networks and Digital Media Laboratory. His research areas include signal processing, broad-band networks, video processing, and communication.