基于小波变换的图像压缩方法研究
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摘要
在小波变换图像压缩方法中，小波基选择的好坏，直接决定小波系数的性质，从而影响压缩过程中后继的其它处理，影响最终的压缩比和图像重建质量；小波变换的实现方法决定着计算的复杂度、数据的恢复性能等，从而影响压缩时间和图像重建质量。
本文围绕小波变换图像压缩方法这一课题，对小波基的选取、小波变换的实现、小波系数的优化、自适应小波的构造、IWT的性能分析及优化实现进行了深入有价值的研究，所做的工作及创新成果有：
(1) 概述了图像压缩的原理、压缩系统的结构及性能评价指标、图像压缩的发展现状。
(2) 评述了从连续小波变换到R小波变换的去冗余过程、紧支撑双正交小波的构造、基于第一代小波的函数正交分解(Mallat算法)与双正交分解(推广的Mallat算法)、小波变换的时－频分析特性及多分辨分析特性在图像压缩中的应用。
(3) 根据近些年发表的文章，分析并系统整理了第二代小波变换的理论与实现方法，分析了第二代小波变换的优势及这些优势在图像压缩中的应用。
(4) 分析了图像小波变换后小波系数的特征，讨论了优化小波系数的小波基选择问题。
(5) 提出了构造自适应小波变换以适应图像局部特征的思想，给出了两种自适应小波的构造方法，并分析了所构造小波在图像压缩中的性能。两种构造方法分别是：
a) 构造自适应预测小波变换的方法。
b) 构造自适应提升小波变换的方法。
(6) 提出了在小波变换前加平滑预处理以削弱图像局部特征的思想，通过平滑预处理可优化小波系数，有效提高压缩比。
(7) 对IWT的性能及产生原因进行了分析，真对其在有损压缩中劣于DWT的特点，提出了两种改进方案。
a) 优化因式分解，以减少取整误差的引入。
b) 扩幅，以效屏蔽取整误差。
In the wavelet image compression system，the choice of wavelet basis directly determines the statistical characteristics of wavelet coefficients, thus it not only affects subsequent processes, but also affects the final compression ratio and the the quality of reconstructed image. The method of realizing wavelet transform determines computational complexity, whether the transformed data can be lossless recovered or not, etc., accordingly it can affect the time spending on compression and the quality of reconstructed image.
In this paper, the author has performed comprehensive research on the method of wavelet basis selection, wavelet transform realization, wavelet coefficients' optimization, adaptive wavelet's construction, the performance analysis and optimal realization of IWT. The work and innovative results are as follows:
1.Have analyzed principles of image compression, the structure of compression system, methods of evaluating compression system's performance and present methods of image compression.
2.Have summarized the process from continuous wavelet transform to R wavelet transform to gradually eliminate redundancy, the method of constructing biorthogonal and compactly supported wavelets, function's decomposition based on orthogonal or biorthogonal basis, the time-frequency characterization of the first generation wavelet transform and the application of the multi-resolution characterization in image compression.
3. Have analyzed and systemically summarized principles and realizing methods of the second generation wavelet, have analyzed advantages of the second generation wavelet transform and their applications in image compression.
4.Have analyzed properties of wavelet coefficients obtained from an image through wavelet transform, and discussed how to select wavelet basis to optimize wavelet coefficients.
5. Have put forward the idea of constructing adaptive wavelet transform to adapt to an image's local characters, given two methods of constructing adaptive wavelet and analyzed the performance of adaptive wavelets in image compression. Two methods are:
a) the method of constructing adaptive predict wavelet.
b) the method of constructing adaptive update wavelet.
6. Have put forward the idea of introducing smooth process before wavelet transform to weaken an image's local characters. Experiments show the method can optimize wavelet coefficients and increase compression ratio.
7. Have analyzed the performance of IWT and reasons resulting in the performance, and put forward two methods to improve its performance inferior to DWT in lossy compression. Two methods are
a) optimizing factorization to decrease the error of rounding operations.
b) amplifying range to avoid the error of rounding operations.
本文围绕小波变换图像压缩方法这一课题，对小波基的选取、小波变换的实现、小波系数的优化、自适应小波的构造、IWT的性能分析及优化实现进行了深入有价值的研究，所做的工作及创新成果有：
(1) 概述了图像压缩的原理、压缩系统的结构及性能评价指标、图像压缩的发展现状。
(2) 评述了从连续小波变换到R小波变换的去冗余过程、紧支撑双正交小波的构造、基于第一代小波的函数正交分解(Mallat算法)与双正交分解(推广的Mallat算法)、小波变换的时－频分析特性及多分辨分析特性在图像压缩中的应用。
(3) 根据近些年发表的文章，分析并系统整理了第二代小波变换的理论与实现方法，分析了第二代小波变换的优势及这些优势在图像压缩中的应用。
(4) 分析了图像小波变换后小波系数的特征，讨论了优化小波系数的小波基选择问题。
(5) 提出了构造自适应小波变换以适应图像局部特征的思想，给出了两种自适应小波的构造方法，并分析了所构造小波在图像压缩中的性能。两种构造方法分别是：
a) 构造自适应预测小波变换的方法。
b) 构造自适应提升小波变换的方法。
(6) 提出了在小波变换前加平滑预处理以削弱图像局部特征的思想，通过平滑预处理可优化小波系数，有效提高压缩比。
(7) 对IWT的性能及产生原因进行了分析，真对其在有损压缩中劣于DWT的特点，提出了两种改进方案。
a) 优化因式分解，以减少取整误差的引入。
b) 扩幅，以效屏蔽取整误差。
In the wavelet image compression system，the choice of wavelet basis directly determines the statistical characteristics of wavelet coefficients, thus it not only affects subsequent processes, but also affects the final compression ratio and the the quality of reconstructed image. The method of realizing wavelet transform determines computational complexity, whether the transformed data can be lossless recovered or not, etc., accordingly it can affect the time spending on compression and the quality of reconstructed image.
In this paper, the author has performed comprehensive research on the method of wavelet basis selection, wavelet transform realization, wavelet coefficients' optimization, adaptive wavelet's construction, the performance analysis and optimal realization of IWT. The work and innovative results are as follows:
1.Have analyzed principles of image compression, the structure of compression system, methods of evaluating compression system's performance and present methods of image compression.
2.Have summarized the process from continuous wavelet transform to R wavelet transform to gradually eliminate redundancy, the method of constructing biorthogonal and compactly supported wavelets, function's decomposition based on orthogonal or biorthogonal basis, the time-frequency characterization of the first generation wavelet transform and the application of the multi-resolution characterization in image compression.
3. Have analyzed and systemically summarized principles and realizing methods of the second generation wavelet, have analyzed advantages of the second generation wavelet transform and their applications in image compression.
4.Have analyzed properties of wavelet coefficients obtained from an image through wavelet transform, and discussed how to select wavelet basis to optimize wavelet coefficients.
5. Have put forward the idea of constructing adaptive wavelet transform to adapt to an image's local characters, given two methods of constructing adaptive wavelet and analyzed the performance of adaptive wavelets in image compression. Two methods are:
a) the method of constructing adaptive predict wavelet.
b) the method of constructing adaptive update wavelet.
6. Have put forward the idea of introducing smooth process before wavelet transform to weaken an image's local characters. Experiments show the method can optimize wavelet coefficients and increase compression ratio.
7. Have analyzed the performance of IWT and reasons resulting in the performance, and put forward two methods to improve its performance inferior to DWT in lossy compression. Two methods are
a) optimizing factorization to decrease the error of rounding operations.
b) amplifying range to avoid the error of rounding operations.
引文
[1] Jerry D.Gibson，Toby Berger著，朱山风，段上为 等译. Digital Compression for Multimedia Principles &Standards.电子工业出版社. 2000.
[2] 王汇源著.数字图像通信原理与技术.国防工业出版社. 2000.
[3] 李建平主编.小波分析与信号处理－理论、应用及软件实现.重庆出版社.1997.
[4] 吴乐南编著.数据压缩.电子工业出版社.2000.
[5] Daubechies I. Orthonormal bases of compactly supported wavelets. Comm. Pure Applied Math., vol. 41, no.7, pp.909-996, 1988.
[6] Mallat S. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. On Pattern analysis and machine intelligence, vol.11, no.7, pp.674-693, 1989.
[7] Cohen A, Daubechies I and Feauveau J.C. Bi-orthogonal bases of compactly supported wavelets. Comm. Pure Applied Math., vol. 4, pp.485-560, 1992.
[8] Sweldens W. The lifting scheme: A custom-design construction of biorthogonal wavelets. Appl Comut Harmon Analysis, vol.3, no.2, pp.186-200, April 1996.
[9] Daubechies I. and Sweldens. Factorizing the wavelet transform into lifting steps. Journal of Fourier Analysis and Applications, vol.4, no.3, pp.247-269, 1998.
[10] 程正兴. 小波分析算法与应用. 西安交通大学出版社.1997.
[11] 包革军，盖云英等.积分变换.哈尔滨工业大学出版社.1998.
[12] 胡昌华，张军波等. 基于MATLAB的系统分析与设计—小波分析. 西安电子科技大学出版社. 2000.
[13] 李水银，吴纪桃.分形与小波.科学出版社.2002.
[14] Bing-Bing Chai, Jozsef Vass etc. Significance-linked connected component analysis for wavelet image coding. IEEE Tran. Image processing, Vol.8, pp.774-783, June 1999.
[15] Stephane Mallat著，杨力华、戴道清等译. 信号处理的小波导引. 机械工业出版社.2002.
[16] 楼顺天，于卫.基于Matlab的系统分析与设计－系统工程. 西安电子科技大学出版社. 1999.
[17] 王欣，王德隽.离散信号的滤波.电子工业出版社.2002.
[18] Taubman,D. High performance scalable image compression with EBCOT. Image Processing, IEEE Transactions on , Volume: 9 Issue: 7,2000, pp: 1158 -1170.
[19] Taubman, D. High performance scalable image compression with EBCOT. ICIP Proceedings. 1999 International Conference on , Volume: 3, pp: 344 -348.
[20] Taubman, D.; Marcellin, M.W. JPEG2000: standard for interactive imaging. Proceedings of the IEEE, Volume: 90 Issue: 8, 2002. pp: 1336 -1357
[23] Taubman, D. Software architectures for jpeg2000. Digital Signal Processing, 2002. 14th International Conference on , Volume: 1 , 2002. pp: 197 -200
[24] Taubman, D.; Ordentlich, E.etc. Embedded block coding in JPEG2000. Image Processing, 2000. 2000 International Conference on, Volume: 2, pp: 33 -36
[25] JPEG 2000 Part I Final Committee Draft Version 1.0. ISO/IEC JTC 1/SC 29/WG 1 N1646R.
[26] W. Sweldens. The lifting scheme: A construction of second generation wavelets.
SIAM J.Math.Anal,1997,29(2),pp511-546.
[27] Calderbank A.R., Ingrid Daubechies.etc, Wavelet transforms that map integers to integers. Appl.comput.Harmon.Anal.,1998,5(3), pp332-369.
[28] Honggang Li, Qiao Wang,etc. A novel design of lifting scheme from general wavelet.Signal processing, IEEE Transactions on, 2001,49(3), pp1714-1716.
[29] W. Sweldens; P.Schroder. Building your own wavelets at home. In wavelets in Computer graphics. ACM SIGGRAPH Course notes,1996,pp15-87.
[30] Marco Grangetto, Enrico Magli.etc. Optimization and implementation of the integer wavelet transform for image coding. Image processing, IEEE Transactions on, 2002,11(6), pp596-603.
[31] Sonja Grgic, Mislav Grgic.etc. Performance analysis of Image compression using wavelets. Industrial electronics, IEEE Transactions on, 2001,48(3), pp682-694.
[32] R.coifman; V. Wickerhauser. Entropy-based algorithms for best basis selection. IEEE Trans. Inform. Theory,1992,38(3),pp 713-718.
[33] R.L.Claypoole; G.Davis.etc. Nonlinear wavelet transforms for image coding. Signal, system & computers, 1997,conf. Record of the 31st Asilomar. Vol1, pp662-667.
[34] Gemma Pilla; Henk J.A.M. Heijmans. Adaptive lifting schemes with perfect reconstruction. Signal processing, IEEE Transactions on, 2002,50(7), pp1620-1630.
[35] ?mer Nezih Gerek; A.Enis Cetin. Adaptive polyphase subband decomposition structures for image compression. Image processing, IEEE Transactions on, 2000,9(10), pp1649-1660.
[36] Nikolaos V.Boulgouris; Dimitrios Tzovaras.etc. Lossless image compression based on optimal prediction, adaptive lifting, and conditional Arithmetic coding.Image processing, IEEE Transactions on, 2001,10(1), pp1-14.
[37] Amir Said; William A.Pearlman. An image Multiresolution representation for lossless and lossy compression. Image processing, IEEE Transactions on, 1996,9(5), pp1303-1310.
[38] Julien Reichel; Gloria Menegaz.etc. Integer wavelet transform for embedded lossy to lossless image compression. Image processing, IEEE Transactions on, 2001,10(3), pp383-392.
[39] Detlev Marpe; Gabi Bl?ttermann.etc. A two-layered wavelet-based algorithm for efficient lossless and lossy image compression. Circuits and systems for video technology, 2000,10(7),pp 1094-1102.
[40] Amir Said; William A.Pearlman. A new, fast, and efficient image codec based on set partitioning in hierarchical trees. Circurts and systems for video technology, IEEE Transactions on, 1996,6(3), pp243-250.
[41] J.M. Shapior. Embedded image coding using zerotrees of wavelets doefficients. Signal processing, IEEE Trans. On, 1993,41(12),pp3445-3462.
[42] 王曜等. 结合小波预处理的图像编码方案.计算机学报.2003.vol.26 No.2,pp240-243.
[43]F.Sheng, A.Bilgin,etc. Lossy and lossless image compression using reversible integer wavelet transforms. ICIP 1998,vol3,pp876-880.
[44]M.Grangetto,etc. Lifting integer wavelets toward linearity. Proc.33rd ACSSC,1999,vol2,1647-1751.
[45]M.D.Adams,etc. Reversible integer-to-integer wavelet transforms for image compression: Performance evaluation and analysis. Image processing, IEEE Trans. On, 2000, vol9. no.6,pp1010-1024.
[46]M.Grangetto,etc. Minimaily non-linear integer wavelets for image coding. Proc.ICASSP 2000, vol.4,pp2039-2041.
[47]O.rioul and M.vetterli. Wavelets and signal processing. IEEE signal processing,Oct1991,pp14-18.
[48]M.D.Adams,etc. Performance evaluation of reversible integer-to-integer wavelet transforms for image compression. In proc.IEEE Data compression conf.Mar 1999,pp514-524.
[49] Jin Li,etc. An embedded still image coder with rate-distortion optimization. Image processing, IEEE Trans. On, 1999, vol8. no.7,pp913-923.
[50] Jin Li.Image compression-the Mechanics of the JPEG2000.WA98052.
[2] 王汇源著.数字图像通信原理与技术.国防工业出版社. 2000.
[3] 李建平主编.小波分析与信号处理－理论、应用及软件实现.重庆出版社.1997.
[4] 吴乐南编著.数据压缩.电子工业出版社.2000.
[5] Daubechies I. Orthonormal bases of compactly supported wavelets. Comm. Pure Applied Math., vol. 41, no.7, pp.909-996, 1988.
[6] Mallat S. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. On Pattern analysis and machine intelligence, vol.11, no.7, pp.674-693, 1989.
[7] Cohen A, Daubechies I and Feauveau J.C. Bi-orthogonal bases of compactly supported wavelets. Comm. Pure Applied Math., vol. 4, pp.485-560, 1992.
[8] Sweldens W. The lifting scheme: A custom-design construction of biorthogonal wavelets. Appl Comut Harmon Analysis, vol.3, no.2, pp.186-200, April 1996.
[9] Daubechies I. and Sweldens. Factorizing the wavelet transform into lifting steps. Journal of Fourier Analysis and Applications, vol.4, no.3, pp.247-269, 1998.
[10] 程正兴. 小波分析算法与应用. 西安交通大学出版社.1997.
[11] 包革军，盖云英等.积分变换.哈尔滨工业大学出版社.1998.
[12] 胡昌华，张军波等. 基于MATLAB的系统分析与设计—小波分析. 西安电子科技大学出版社. 2000.
[13] 李水银，吴纪桃.分形与小波.科学出版社.2002.
[14] Bing-Bing Chai, Jozsef Vass etc. Significance-linked connected component analysis for wavelet image coding. IEEE Tran. Image processing, Vol.8, pp.774-783, June 1999.
[15] Stephane Mallat著，杨力华、戴道清等译. 信号处理的小波导引. 机械工业出版社.2002.
[16] 楼顺天，于卫.基于Matlab的系统分析与设计－系统工程. 西安电子科技大学出版社. 1999.
[17] 王欣，王德隽.离散信号的滤波.电子工业出版社.2002.
[18] Taubman,D. High performance scalable image compression with EBCOT. Image Processing, IEEE Transactions on , Volume: 9 Issue: 7,2000, pp: 1158 -1170.
[19] Taubman, D. High performance scalable image compression with EBCOT. ICIP Proceedings. 1999 International Conference on , Volume: 3, pp: 344 -348.
[20] Taubman, D.; Marcellin, M.W. JPEG2000: standard for interactive imaging. Proceedings of the IEEE, Volume: 90 Issue: 8, 2002. pp: 1336 -1357
[23] Taubman, D. Software architectures for jpeg2000. Digital Signal Processing, 2002. 14th International Conference on , Volume: 1 , 2002. pp: 197 -200
[24] Taubman, D.; Ordentlich, E.etc. Embedded block coding in JPEG2000. Image Processing, 2000. 2000 International Conference on, Volume: 2, pp: 33 -36
[25] JPEG 2000 Part I Final Committee Draft Version 1.0. ISO/IEC JTC 1/SC 29/WG 1 N1646R.
[26] W. Sweldens. The lifting scheme: A construction of second generation wavelets.
SIAM J.Math.Anal,1997,29(2),pp511-546.
[27] Calderbank A.R., Ingrid Daubechies.etc, Wavelet transforms that map integers to integers. Appl.comput.Harmon.Anal.,1998,5(3), pp332-369.
[28] Honggang Li, Qiao Wang,etc. A novel design of lifting scheme from general wavelet.Signal processing, IEEE Transactions on, 2001,49(3), pp1714-1716.
[29] W. Sweldens; P.Schroder. Building your own wavelets at home. In wavelets in Computer graphics. ACM SIGGRAPH Course notes,1996,pp15-87.
[30] Marco Grangetto, Enrico Magli.etc. Optimization and implementation of the integer wavelet transform for image coding. Image processing, IEEE Transactions on, 2002,11(6), pp596-603.
[31] Sonja Grgic, Mislav Grgic.etc. Performance analysis of Image compression using wavelets. Industrial electronics, IEEE Transactions on, 2001,48(3), pp682-694.
[32] R.coifman; V. Wickerhauser. Entropy-based algorithms for best basis selection. IEEE Trans. Inform. Theory,1992,38(3),pp 713-718.
[33] R.L.Claypoole; G.Davis.etc. Nonlinear wavelet transforms for image coding. Signal, system & computers, 1997,conf. Record of the 31st Asilomar. Vol1, pp662-667.
[34] Gemma Pilla; Henk J.A.M. Heijmans. Adaptive lifting schemes with perfect reconstruction. Signal processing, IEEE Transactions on, 2002,50(7), pp1620-1630.
[35] ?mer Nezih Gerek; A.Enis Cetin. Adaptive polyphase subband decomposition structures for image compression. Image processing, IEEE Transactions on, 2000,9(10), pp1649-1660.
[36] Nikolaos V.Boulgouris; Dimitrios Tzovaras.etc. Lossless image compression based on optimal prediction, adaptive lifting, and conditional Arithmetic coding.Image processing, IEEE Transactions on, 2001,10(1), pp1-14.
[37] Amir Said; William A.Pearlman. An image Multiresolution representation for lossless and lossy compression. Image processing, IEEE Transactions on, 1996,9(5), pp1303-1310.
[38] Julien Reichel; Gloria Menegaz.etc. Integer wavelet transform for embedded lossy to lossless image compression. Image processing, IEEE Transactions on, 2001,10(3), pp383-392.
[39] Detlev Marpe; Gabi Bl?ttermann.etc. A two-layered wavelet-based algorithm for efficient lossless and lossy image compression. Circuits and systems for video technology, 2000,10(7),pp 1094-1102.
[40] Amir Said; William A.Pearlman. A new, fast, and efficient image codec based on set partitioning in hierarchical trees. Circurts and systems for video technology, IEEE Transactions on, 1996,6(3), pp243-250.
[41] J.M. Shapior. Embedded image coding using zerotrees of wavelets doefficients. Signal processing, IEEE Trans. On, 1993,41(12),pp3445-3462.
[42] 王曜等. 结合小波预处理的图像编码方案.计算机学报.2003.vol.26 No.2,pp240-243.
[43]F.Sheng, A.Bilgin,etc. Lossy and lossless image compression using reversible integer wavelet transforms. ICIP 1998,vol3,pp876-880.
[44]M.Grangetto,etc. Lifting integer wavelets toward linearity. Proc.33rd ACSSC,1999,vol2,1647-1751.
[45]M.D.Adams,etc. Reversible integer-to-integer wavelet transforms for image compression: Performance evaluation and analysis. Image processing, IEEE Trans. On, 2000, vol9. no.6,pp1010-1024.
[46]M.Grangetto,etc. Minimaily non-linear integer wavelets for image coding. Proc.ICASSP 2000, vol.4,pp2039-2041.
[47]O.rioul and M.vetterli. Wavelets and signal processing. IEEE signal processing,Oct1991,pp14-18.
[48]M.D.Adams,etc. Performance evaluation of reversible integer-to-integer wavelet transforms for image compression. In proc.IEEE Data compression conf.Mar 1999,pp514-524.
[49] Jin Li,etc. An embedded still image coder with rate-distortion optimization. Image processing, IEEE Trans. On, 1999, vol8. no.7,pp913-923.
[50] Jin Li.Image compression-the Mechanics of the JPEG2000.WA98052.
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