IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 9, SEPTEMBER 2008 1555 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 direc- tional 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, in- tersubband, 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 adap- tive 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, redun- dant transform, sparse representation, wavelet transform. I. INTRODUCTION W AVELET-BASED image coding has witnessed great success in the past decade. Being separable, conven- tional 2-D discrete wavelet transform (DWT) efficiently cap- tures 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, 2008. 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 60528004), in part by the Distinguished Young Scholars of NSFC (Grant 60525111), and in part by the key project of NSFC (Grant 60432030). 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 100084, China (e-mail: 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 11201 USA (e-mail: yao@poly.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIP.2008.926159 representation. These tools can be classified into two categories according to the domain where they are designed: spatial-do- main multiscale directional transform (SMDT) and frequency- domain 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 direc- tions 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 cer- tain 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]. Var- ious FMDT-based image coding schemes have been proposed [15], [19], [25]–[29], [48]–[50]. For example, the bandelets- based 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 poten- tial 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 con- tourlet-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 1057-7149/$25.00 © 2008 IEEE