基于上下文的图像压缩技术研究
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摘要
随着数字通信、多媒体技术的飞速发展,图像压缩编码已成为信号传输及存储中的一个关键环节。目前数字图像压缩编码方法的种类繁多,近些年,基于上下文的图像压缩算法成为一个新的研究热点,它因能获得高的压缩性能、好的重构质量而在图像压缩领域受到广泛应用。
本文对基于上下文的图像压缩算法做了进一步的研究,主要包括以下几方面的工作:
首先,本文针对目前基于下采样的压缩方法中存在的复杂度高及边缘重构质量差的问题,提出了一种低码率下基于自适应下采样和交叠变换的图像压缩方法,在编码端根据图像的DCT变换系数自适应地选择图像的平滑区域下采样,而在解码端使用简单的Cubic插值重构被下采样的平滑区域。另外,考虑到DCT变换只能去除块内的相关性,我们加入了交叠变换以去除块间的相关性。实验结果表明,该算法不仅具有高的压缩性能并且拥有低的复杂度。
其次,针对传统的四叉树编码方法使用固定的编码浏览顺序及相等长度的码字表示一个重要块,本文提出了带有上下文权值和率失真优化的可变长图像编码方法,能够根据建立的上下文权值预测模型估计子块的重要性程度,调整子块的编码顺序,从而得到一种有效的可变长编码。另外,可利用上下文权值对率失真进行优化,使重要信息能被尽早编码,以进一步改善压缩性能。
最后,本文提出了一种带有基于上下文的反量化技术的图像编码方法,在解码端能够根据前边已经解码出来的系数估计变换系数的分布情况,按照估计出来的分布曲线寻找待解码系数的最佳重建点,从而避免使用均值量化器。文中所提出的基于上下文的反量化技术使我们的压缩方法满足一种非对称的压缩方案,保持编码端的复杂度不变,而在计算能力更强的解码端采用更高复杂度的反量化器代替均值量化器,以提高整体的压缩性能。
With the rapid development of digital communication and multimedia technology, image compression has played an important role in signal transmission and storage. At present, there are many kinds of image compression. In recent years, the compression method based on context has become a new study point. This method is greatly used in the field of image compression because of its high compression performance and clear restored image.
In this dissertation, the further research about the image compression based on context is made. It includes the following three aspects.
Firstly, in order to improve the drawbacks of the current downsample-based compression having high complexity and bad reconstruction for edge region, we propose an image compression with adaptive downsampling and lapped transform at the low bit rates. At the encoder side, codec can choose adaptively the smooth area of images to downsample according to the DCT coefficients; and at the decoder side, codec uses simple Cubic interpolation to restore the downsampled area. In addition, in view of that DCT can not remove the cross-block correlation, so lapped transform is adopted in this new method. Experimental results show that the method has not only the high compression performance but also the low complexity.
Secondly, aimed at the problem that the conventional quad-tree coding uses fixed scanning order and equal-length bit to code a significant block, a new variable-length quad-tree coding method with context weight and rate-distortion optimization is proposed in this paper. The method can adjust the scanning order of sub-blocks using their significance degrees which are predicted by the proposed context weight model, and spend different-length bits to represent different cases. In addition, the context weight model can also be used in rate-distortion optimization in order to further improve the compression performance.
Finally, we propose a new image coding method with inverse quantization method based on context. This method is to estimate the distribution of transformed coefficients by the coefficients which have been decoded at the decoder side, and then find the optimal reconstructed point of the coefficients that will be decoded using the estimated distribution of transformed coefficients. This method makes us avoid using mean quantizer. With the inverse quantization technology based on context, our coding system satisfies a dissymmetrical compression scheme which keeps the complexity of the encoder invariable and uses the complex quantization method at the decoder side that has powerful computational ability to increase the compression performance.
本文对基于上下文的图像压缩算法做了进一步的研究,主要包括以下几方面的工作:
首先,本文针对目前基于下采样的压缩方法中存在的复杂度高及边缘重构质量差的问题,提出了一种低码率下基于自适应下采样和交叠变换的图像压缩方法,在编码端根据图像的DCT变换系数自适应地选择图像的平滑区域下采样,而在解码端使用简单的Cubic插值重构被下采样的平滑区域。另外,考虑到DCT变换只能去除块内的相关性,我们加入了交叠变换以去除块间的相关性。实验结果表明,该算法不仅具有高的压缩性能并且拥有低的复杂度。
其次,针对传统的四叉树编码方法使用固定的编码浏览顺序及相等长度的码字表示一个重要块,本文提出了带有上下文权值和率失真优化的可变长图像编码方法,能够根据建立的上下文权值预测模型估计子块的重要性程度,调整子块的编码顺序,从而得到一种有效的可变长编码。另外,可利用上下文权值对率失真进行优化,使重要信息能被尽早编码,以进一步改善压缩性能。
最后,本文提出了一种带有基于上下文的反量化技术的图像编码方法,在解码端能够根据前边已经解码出来的系数估计变换系数的分布情况,按照估计出来的分布曲线寻找待解码系数的最佳重建点,从而避免使用均值量化器。文中所提出的基于上下文的反量化技术使我们的压缩方法满足一种非对称的压缩方案,保持编码端的复杂度不变,而在计算能力更强的解码端采用更高复杂度的反量化器代替均值量化器,以提高整体的压缩性能。
With the rapid development of digital communication and multimedia technology, image compression has played an important role in signal transmission and storage. At present, there are many kinds of image compression. In recent years, the compression method based on context has become a new study point. This method is greatly used in the field of image compression because of its high compression performance and clear restored image.
In this dissertation, the further research about the image compression based on context is made. It includes the following three aspects.
Firstly, in order to improve the drawbacks of the current downsample-based compression having high complexity and bad reconstruction for edge region, we propose an image compression with adaptive downsampling and lapped transform at the low bit rates. At the encoder side, codec can choose adaptively the smooth area of images to downsample according to the DCT coefficients; and at the decoder side, codec uses simple Cubic interpolation to restore the downsampled area. In addition, in view of that DCT can not remove the cross-block correlation, so lapped transform is adopted in this new method. Experimental results show that the method has not only the high compression performance but also the low complexity.
Secondly, aimed at the problem that the conventional quad-tree coding uses fixed scanning order and equal-length bit to code a significant block, a new variable-length quad-tree coding method with context weight and rate-distortion optimization is proposed in this paper. The method can adjust the scanning order of sub-blocks using their significance degrees which are predicted by the proposed context weight model, and spend different-length bits to represent different cases. In addition, the context weight model can also be used in rate-distortion optimization in order to further improve the compression performance.
Finally, we propose a new image coding method with inverse quantization method based on context. This method is to estimate the distribution of transformed coefficients by the coefficients which have been decoded at the decoder side, and then find the optimal reconstructed point of the coefficients that will be decoded using the estimated distribution of transformed coefficients. This method makes us avoid using mean quantizer. With the inverse quantization technology based on context, our coding system satisfies a dissymmetrical compression scheme which keeps the complexity of the encoder invariable and uses the complex quantization method at the decoder side that has powerful computational ability to increase the compression performance.
引文
[1]冈萨雷斯.数字图像处理.第2版.北京:电子工业出版社,2003.3.
[2] Kunt M, Ikonomopoluos A, Kocher M. Second Generation Image Coding Techniques. Proceedings of the IEEE. 1985. pp.549-574.
[3]王汇源.数字图像通信原理与技术.第1版.北京:国防工业出版社,2000。
[4]阮秋琦.数字图像处理学.第2版.北京:电子工业出版社,2007.
[5]吴成柯,戴善荣,陆心如.图像通信.第1版.西安:西安电子科技大学出版社,1990.
[6] Labaw C. Airborne Imaging Spectrometer: All Advanced Concept Instrument. Proceedings of SPIE Conference on Infrared Technology. 1983. pp. 68-74.
[7] Proter W M. A Systern Overview of the Airborne Visible/infrared Imaging Spectrometer(AVIRIS). Proceedings of SPIE Conference Oil Image Spectroscopy. 1987. pp. 166-174.
[8] Richard L J. HYDICE: all Airbome System for Hyperspectral Imaging. Proceedings of SPIE Conference on Imaging Spectrometry of the Terrestrial Environment. 1993. pp. 173-179.
[9] Gandhi P P. JPEG-based Image Compression for Low-bit-rate Coding. Proc. SPIE Still-Image Compression II Conf.. R. L. Stevenson, A. I. Drukarev, and T. R. Gardos, Eds., vol. 2669, 1996. pp. 82–94.
[10] Bezy J L, Delwart S, Rast M. ESA Medium Resolution Imaging Spectrometer MERIS. Proceedings of SPIE Conference on Earth Observing Systems. 1998. pp. 594-604.
[11] Shapiro J M. Embedded Image Coding using Zerotrees of Wavelet Coefficients. IEEE Trans. on Signal Processing. vol. 41, Dec 1993. pp. 3445–3462.
[12] Said A, Pearlman W A. A New, Fast and Efficient Image Codec based on Set Partitioning in Hierarchical Trees. IEEE Trans. Circuits and Systems for Video Technology. vol. 6, June 1996. pp. 243–250.
[13] Islam A, Pearlman W A. Embedded and Efficient Low-complexity Hierarchical Image Coder. SPIE Conf. on Visual Communications and Image Processing'99. San Jose, CA, USA, vol. 3653, 1999. pp. 294-305.
[14] Hsiang S T, Woods J W. Embedded Image Coding Using Zeroblock of Subband/Wavelet Coefficients and Context Modeling. Circuits and Systems, 2000. Proceedings. ISCAS 2000 Geneva. The 2000 IEEE International Symposium on,2000. pp. 662-665.
[15] Information Technology-jpeg 2000 Image Coding System: Iso/iec fdis15444-1:2000, Tech. Rep., August 2000.
[16] Taubman D, Zakhor A. Multi-rate 3-D Subband Coding of Video. IEEE Transactions on Image Processing, 1994, pp. 572-588.
[17] Bilgin A, Sementilli P J, and Marcellin M W. Progressive Image Coding using Trellis Coded Quantization. IEEE Trans. Image Processing. vol. 8, no. 11, 1999. pp. 1638-1643.
[18] Ramchandran K, Ortega A, and Vetterli M. Bit Allocation for Dependent Quantization with Applications to Multiresolution and MPEG Video Coders. IEEE Trans. on Image Processing. vol. 3, no. 10, 1994. pp. 533-545.
[19] Sriram P, Marcellin M W. Image Coding using Wavelet Transforms and Entropy—Constrained Trellis Quantization. IEEE Trans. on Image Processing. vol. 23, no. 4, l995. pp. 725-733.
[20] Xiong Z, Ramchandran K, and Orchard M T. Space-frequency Quantization for Wavelet Image Coding. IEEE Trans. on Image Processing. vol. 6, no. 5,1997. pp. 677-693.
[21] Xiong Z, Ramchandran K, and Orchard M T, Wavelet Packet Image Coding Using Space-frequency Quantization. IEEE Trans. on Image Processing. vol. 7, no. 6, 1998. pp. 892-898.
[22] Winger L L, Venetsanopoulos A N. Stack-Run Coding with Space Frequency Quantization. ICIP, Santa Barbara. vol. 1, 1997. pp. 620-623.
[23]秦前清,杨宗凯.实用小波分析.第1版.西安:西安电子科技大学出版社,2002.页码:14-25.
[24] Ahmed N, Natarajan T, Rao K R. Discrete Cosine Transfom. IEEE Transactions on vol. C-23, Jan. 1974. pp. 90-93.
[25] Sikora T, Makai B. Shape-adaptive DCT for Generic Coding of Video. IEEE Trans. Circuits Syst. Video Technol.. vol. 5, Feb. 1995. pp. 59-62.
[26] Sikora T. Low Complexity Shape-adaptive DCT for Coding of Arbitrarily Shaped Image Segments. Signal Process.: Image Commun., vol. 7, 1995. pp. 381-395.
[27] Zhang X, Wu X, and Wu F. Image Coding on Quincunx Lattice with Adaptive Lifting and Interpolation. 2007 Data Compression Conference, 2007. pp. 193-202.
[28] Zhang X, Wu X. Can Lower Resolution Be Better. 2008 Data Compression Conference, 2008. pp. 302-311.
[29] Li X, Orchard M T. New Edge-Directed Interpolation. IEEE Trans. on ImageProcessing, vol. 10, no. 10, 2001. pp. 1521-1527.
[30] Zhang L, Wu X. An Edge-guided Image Interpolation Algorithm via Directional Filtering and Data Fusion. IEEE Trans. Image Processing, vol. 15, no. 8, Aug. 2006. pp. 2226-2238.
[31] Wu X, and Zhang X. Image Interpolation using Texture Orientation Map and Kernel Fisher Discriminant. Proc. IEEE Int. Conf, on Image Processing, vol. 1, Sept. 2005. pp. 49-52.
[32] Liu D, Sun X, Wu F, Li S, and Zhang Y. Image Compression With Edge-Based Inpainting. IEEE Trans. on Circuits and Systems for Video Technology, vol. 17, no. 10, 2007. pp. 1273-1287.
[33] Lin W, and Dong L. Adaptive Downsampling to Improve Image Compression at Low Bit Rates. IEEE Trans. on Image Processing, vol. 15, no. 9, 2006. pp. 2513-2521.
[34] Keys R. Cubic Convolution Interpolation for Digital Image Processing. IEEE Trans. on Acoustics, Speech, and Signal Processing, vol.assp-29, no.6, 1981. pp. 1153-1160.
[35] Tran T D, Liang J, and Tu C. Lapped Transform via Time-domain Pre- and Post-processing. IEEE Trans. on Signal Processing, vol. 51, no. 6, 2003. pp. 1557-1571.
[36] Gall D L, Tabatai A. Sub-band Coding of Digital Images using Symmetric Short Kernel Filters and Arithmetic Coding Techniques. Proceedings of IEEE International Conference on Acoustics, Speech, Signal Processing, 1988. pp. 761-764.
[37] Xiong Z, Guleryuz O, and Orchard M T. A DCT-based Embedded Image Coder. in: IEEE Transactions on Signal Processing, vol. 3, 1996. pp. 289-290.
[38] Lu Z, and Pearlman W A. Wavelet Video Coding of Video Object by Object-Based SPECK Algorithm. Picture Coding Symposium (PCS-2001), Seoul, Korea, 2001. pp. 413-416.
[39] Han J, Kim H. Modified Cubic Convolution Scaler with Minimum Loss of Information. Opt. Eng. 40(4), April 2001. pp. 540-546.
[40] Meijeringa E. Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing. proc. Of the IEEE, vol. 90, no. 3, Mar 2002. pp. 319-342.
[41] Meijeringa E, Unser M. A Note on Cubic Convolution Interpolation. IEEE Trans. on Image Processing, vol. 12, no. 4, April 2003. pp. 477-479.
[42] Wu X. High-Order Context Modeling and Embedded Conditional Entropy Coding of Wavelet Coefficients for Image Compression. Proceedings of 31st Asilomar Conference on Signals and Computers, vol. 2, 1997. pp. 1378-1382.
[43] Chrysafis C, Ortega A. Efficient Context-based Entropy Coding for Lossy Wavelet Image Compression. Proceedings of 1997 Data Compression Conference, 1997. pp. 241-250.
[44] Li Jin, Lei Shawmin. An Embedded Still Image Coder with Rate-distortion Optimization. IEEE Trans. on image processing, vol. 8, no. 7, 1999. pp. 913-924.
[45] Kim T, Kim H M, Tsai P, and Acharya T. Memory Efficient Progressive Rate-Distortion Algorithm for JPEG 2000. IEEE Trans. On Circuits and System for Video technology. vol. 15, no. 1, Jan 2005. pp. 181-187.
[46] Fang H, Wang Y, Huang C, Chen L. High-Performance JPEG 2000 Encoder with Rate-Distortion Optimization. IEEE Trans. On Multimedia, vol. 8, no. 4, Aug 2006. pp. 645-653.
[47] Antonini M, Barlaud M, Mathieu P, and Daubechies I. Image Coding using Wavelet Transform. IEEE Trans. Image Processing, vol. 1, Apr 1992. pp. 205-220.
[48] Ding W, Wu F, Wu X, Li S. Adaptive Directional Lifting-Based Wavelet Transform for Image Coding. IEEE Trans. On Image Processing, vol. 16, 2007. pp. 416-427.
[49] LoPresto S, Ramchandran K. Image Coding based on Mixture Modeling of Wavelet Coefficients and a Fast Estimation-Quantization Framework. Data Compression Conference, March 1997. pp. 1-10.
[50] Yoo Y, Ortega A, Yu B. Image Subband Coding Using Context-Based Classification and Adaptive Quantization. IEEE Trans. On image processing, vol. 8, no. 12, Dec. 1999. pp. 1702-1715.
[51] Xiong Z, Ramchandran K, and Orchard M. Space-frequency Quantization for Wavelet Image Coding. IEEE Trans. Image Processing, vol. 6, May 1997. pp. 677-693.
[2] Kunt M, Ikonomopoluos A, Kocher M. Second Generation Image Coding Techniques. Proceedings of the IEEE. 1985. pp.549-574.
[3]王汇源.数字图像通信原理与技术.第1版.北京:国防工业出版社,2000。
[4]阮秋琦.数字图像处理学.第2版.北京:电子工业出版社,2007.
[5]吴成柯,戴善荣,陆心如.图像通信.第1版.西安:西安电子科技大学出版社,1990.
[6] Labaw C. Airborne Imaging Spectrometer: All Advanced Concept Instrument. Proceedings of SPIE Conference on Infrared Technology. 1983. pp. 68-74.
[7] Proter W M. A Systern Overview of the Airborne Visible/infrared Imaging Spectrometer(AVIRIS). Proceedings of SPIE Conference Oil Image Spectroscopy. 1987. pp. 166-174.
[8] Richard L J. HYDICE: all Airbome System for Hyperspectral Imaging. Proceedings of SPIE Conference on Imaging Spectrometry of the Terrestrial Environment. 1993. pp. 173-179.
[9] Gandhi P P. JPEG-based Image Compression for Low-bit-rate Coding. Proc. SPIE Still-Image Compression II Conf.. R. L. Stevenson, A. I. Drukarev, and T. R. Gardos, Eds., vol. 2669, 1996. pp. 82–94.
[10] Bezy J L, Delwart S, Rast M. ESA Medium Resolution Imaging Spectrometer MERIS. Proceedings of SPIE Conference on Earth Observing Systems. 1998. pp. 594-604.
[11] Shapiro J M. Embedded Image Coding using Zerotrees of Wavelet Coefficients. IEEE Trans. on Signal Processing. vol. 41, Dec 1993. pp. 3445–3462.
[12] Said A, Pearlman W A. A New, Fast and Efficient Image Codec based on Set Partitioning in Hierarchical Trees. IEEE Trans. Circuits and Systems for Video Technology. vol. 6, June 1996. pp. 243–250.
[13] Islam A, Pearlman W A. Embedded and Efficient Low-complexity Hierarchical Image Coder. SPIE Conf. on Visual Communications and Image Processing'99. San Jose, CA, USA, vol. 3653, 1999. pp. 294-305.
[14] Hsiang S T, Woods J W. Embedded Image Coding Using Zeroblock of Subband/Wavelet Coefficients and Context Modeling. Circuits and Systems, 2000. Proceedings. ISCAS 2000 Geneva. The 2000 IEEE International Symposium on,2000. pp. 662-665.
[15] Information Technology-jpeg 2000 Image Coding System: Iso/iec fdis15444-1:2000, Tech. Rep., August 2000.
[16] Taubman D, Zakhor A. Multi-rate 3-D Subband Coding of Video. IEEE Transactions on Image Processing, 1994, pp. 572-588.
[17] Bilgin A, Sementilli P J, and Marcellin M W. Progressive Image Coding using Trellis Coded Quantization. IEEE Trans. Image Processing. vol. 8, no. 11, 1999. pp. 1638-1643.
[18] Ramchandran K, Ortega A, and Vetterli M. Bit Allocation for Dependent Quantization with Applications to Multiresolution and MPEG Video Coders. IEEE Trans. on Image Processing. vol. 3, no. 10, 1994. pp. 533-545.
[19] Sriram P, Marcellin M W. Image Coding using Wavelet Transforms and Entropy—Constrained Trellis Quantization. IEEE Trans. on Image Processing. vol. 23, no. 4, l995. pp. 725-733.
[20] Xiong Z, Ramchandran K, and Orchard M T. Space-frequency Quantization for Wavelet Image Coding. IEEE Trans. on Image Processing. vol. 6, no. 5,1997. pp. 677-693.
[21] Xiong Z, Ramchandran K, and Orchard M T, Wavelet Packet Image Coding Using Space-frequency Quantization. IEEE Trans. on Image Processing. vol. 7, no. 6, 1998. pp. 892-898.
[22] Winger L L, Venetsanopoulos A N. Stack-Run Coding with Space Frequency Quantization. ICIP, Santa Barbara. vol. 1, 1997. pp. 620-623.
[23]秦前清,杨宗凯.实用小波分析.第1版.西安:西安电子科技大学出版社,2002.页码:14-25.
[24] Ahmed N, Natarajan T, Rao K R. Discrete Cosine Transfom. IEEE Transactions on vol. C-23, Jan. 1974. pp. 90-93.
[25] Sikora T, Makai B. Shape-adaptive DCT for Generic Coding of Video. IEEE Trans. Circuits Syst. Video Technol.. vol. 5, Feb. 1995. pp. 59-62.
[26] Sikora T. Low Complexity Shape-adaptive DCT for Coding of Arbitrarily Shaped Image Segments. Signal Process.: Image Commun., vol. 7, 1995. pp. 381-395.
[27] Zhang X, Wu X, and Wu F. Image Coding on Quincunx Lattice with Adaptive Lifting and Interpolation. 2007 Data Compression Conference, 2007. pp. 193-202.
[28] Zhang X, Wu X. Can Lower Resolution Be Better. 2008 Data Compression Conference, 2008. pp. 302-311.
[29] Li X, Orchard M T. New Edge-Directed Interpolation. IEEE Trans. on ImageProcessing, vol. 10, no. 10, 2001. pp. 1521-1527.
[30] Zhang L, Wu X. An Edge-guided Image Interpolation Algorithm via Directional Filtering and Data Fusion. IEEE Trans. Image Processing, vol. 15, no. 8, Aug. 2006. pp. 2226-2238.
[31] Wu X, and Zhang X. Image Interpolation using Texture Orientation Map and Kernel Fisher Discriminant. Proc. IEEE Int. Conf, on Image Processing, vol. 1, Sept. 2005. pp. 49-52.
[32] Liu D, Sun X, Wu F, Li S, and Zhang Y. Image Compression With Edge-Based Inpainting. IEEE Trans. on Circuits and Systems for Video Technology, vol. 17, no. 10, 2007. pp. 1273-1287.
[33] Lin W, and Dong L. Adaptive Downsampling to Improve Image Compression at Low Bit Rates. IEEE Trans. on Image Processing, vol. 15, no. 9, 2006. pp. 2513-2521.
[34] Keys R. Cubic Convolution Interpolation for Digital Image Processing. IEEE Trans. on Acoustics, Speech, and Signal Processing, vol.assp-29, no.6, 1981. pp. 1153-1160.
[35] Tran T D, Liang J, and Tu C. Lapped Transform via Time-domain Pre- and Post-processing. IEEE Trans. on Signal Processing, vol. 51, no. 6, 2003. pp. 1557-1571.
[36] Gall D L, Tabatai A. Sub-band Coding of Digital Images using Symmetric Short Kernel Filters and Arithmetic Coding Techniques. Proceedings of IEEE International Conference on Acoustics, Speech, Signal Processing, 1988. pp. 761-764.
[37] Xiong Z, Guleryuz O, and Orchard M T. A DCT-based Embedded Image Coder. in: IEEE Transactions on Signal Processing, vol. 3, 1996. pp. 289-290.
[38] Lu Z, and Pearlman W A. Wavelet Video Coding of Video Object by Object-Based SPECK Algorithm. Picture Coding Symposium (PCS-2001), Seoul, Korea, 2001. pp. 413-416.
[39] Han J, Kim H. Modified Cubic Convolution Scaler with Minimum Loss of Information. Opt. Eng. 40(4), April 2001. pp. 540-546.
[40] Meijeringa E. Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing. proc. Of the IEEE, vol. 90, no. 3, Mar 2002. pp. 319-342.
[41] Meijeringa E, Unser M. A Note on Cubic Convolution Interpolation. IEEE Trans. on Image Processing, vol. 12, no. 4, April 2003. pp. 477-479.
[42] Wu X. High-Order Context Modeling and Embedded Conditional Entropy Coding of Wavelet Coefficients for Image Compression. Proceedings of 31st Asilomar Conference on Signals and Computers, vol. 2, 1997. pp. 1378-1382.
[43] Chrysafis C, Ortega A. Efficient Context-based Entropy Coding for Lossy Wavelet Image Compression. Proceedings of 1997 Data Compression Conference, 1997. pp. 241-250.
[44] Li Jin, Lei Shawmin. An Embedded Still Image Coder with Rate-distortion Optimization. IEEE Trans. on image processing, vol. 8, no. 7, 1999. pp. 913-924.
[45] Kim T, Kim H M, Tsai P, and Acharya T. Memory Efficient Progressive Rate-Distortion Algorithm for JPEG 2000. IEEE Trans. On Circuits and System for Video technology. vol. 15, no. 1, Jan 2005. pp. 181-187.
[46] Fang H, Wang Y, Huang C, Chen L. High-Performance JPEG 2000 Encoder with Rate-Distortion Optimization. IEEE Trans. On Multimedia, vol. 8, no. 4, Aug 2006. pp. 645-653.
[47] Antonini M, Barlaud M, Mathieu P, and Daubechies I. Image Coding using Wavelet Transform. IEEE Trans. Image Processing, vol. 1, Apr 1992. pp. 205-220.
[48] Ding W, Wu F, Wu X, Li S. Adaptive Directional Lifting-Based Wavelet Transform for Image Coding. IEEE Trans. On Image Processing, vol. 16, 2007. pp. 416-427.
[49] LoPresto S, Ramchandran K. Image Coding based on Mixture Modeling of Wavelet Coefficients and a Fast Estimation-Quantization Framework. Data Compression Conference, March 1997. pp. 1-10.
[50] Yoo Y, Ortega A, Yu B. Image Subband Coding Using Context-Based Classification and Adaptive Quantization. IEEE Trans. On image processing, vol. 8, no. 12, Dec. 1999. pp. 1702-1715.
[51] Xiong Z, Ramchandran K, and Orchard M. Space-frequency Quantization for Wavelet Image Coding. IEEE Trans. Image Processing, vol. 6, May 1997. pp. 677-693.