小波域数字认证水印算法研究
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摘要
数字认证水印技术是数字水印技术的一个重要研究方向,它在帮助保持和验证数字图像等多媒体的内容方面具有重要的价值。
本文首先给出了数字水印的概念和应用领域。介绍了用于内容认证的数字水印技术的基本概念,基本特征,分析了它所面临的主要攻击手段,并着重总结归纳了认证水印目前的主要算法。同时介绍了小波分析理论的基础知识,以及它在数字图像处理和数字水印技术中的应用。在此基础上本文提出了三种小波变换域数字图像认证水印算法。第一种算法针对灰度图像设计,通过提取图像的特征信息作为水印信号,利用低维混沌系统,采用块平均量化的方法,在小波域实现水印的嵌入。第二种算法针对彩色图像设计,将有意义的水印图像嵌入到彩色图像的小波域中;利用时空混沌对算法进行保护,大大提高了水印系统的安全性;基于奇异值分解的水印嵌入策略则使算法获得了良好的图像质量和半脆弱性。在第二种算法的基础上,进一步提出了一种改进的基于奇异值分解的小波域彩色图像半脆弱水印算法,通过颜色模型的空间变换和对亮度信息的线性变换处理,解决了水印嵌入时存在的图像亮度发生变化的问题。所提出的认证水印算法,均提供了增强算法安全性的措施,并对此做了详细的分析论证。实验结果表明,上述提出的算法是有效可行的。文章最后,对认证水印的未来研究方向和发展前景做了分析和总结。
Digital watermarking technology for authentication is an important research branch of digital watermarking due to its significant effectiveness in the content verification of multimedia.
At first, this paper summaries the definition and application of digital watermarking and gives a general introduction of watermarking for authentication, including the definition, the main characteristics, the attacks threatening to it and the main algorithms designed at present. Wavelet theory and its application in digital image processing and digital watermarking are also introduced in detail. Based on the above, this paper proposes three new digital watermarking algorithms for image authentication in DWT domain. One of them is designed for gray images. The watermark is generated from the characteristics of the images and is then embedded into the wavelet domain applying the low-dimensional chaos and mean quantization method. Another is designed specially for color images where the meaningful watermark is embedded in the wavelet domain based on SVD. Spatiotemporal chaos is employed to protect the algorithm from attacks and the security of the scheme is highly improved. Good fidelity and semi-fragile ability is possessed for the use of SVD. Based on the second algorithm, an improved SVD-based wavelet domain watermarking algorithm is proposed. The spatial transform for color model and the linearly transform for the luminance information is performed so that the luminance-changing problem is well solved. All of the algorithms tell how to improve the security of the watermarking schemes with detailed analysis and demonstrations. The last part of the paper analyzes the research trend and usability of digital watermarking for multimedia authentication.
本文首先给出了数字水印的概念和应用领域。介绍了用于内容认证的数字水印技术的基本概念,基本特征,分析了它所面临的主要攻击手段,并着重总结归纳了认证水印目前的主要算法。同时介绍了小波分析理论的基础知识,以及它在数字图像处理和数字水印技术中的应用。在此基础上本文提出了三种小波变换域数字图像认证水印算法。第一种算法针对灰度图像设计,通过提取图像的特征信息作为水印信号,利用低维混沌系统,采用块平均量化的方法,在小波域实现水印的嵌入。第二种算法针对彩色图像设计,将有意义的水印图像嵌入到彩色图像的小波域中;利用时空混沌对算法进行保护,大大提高了水印系统的安全性;基于奇异值分解的水印嵌入策略则使算法获得了良好的图像质量和半脆弱性。在第二种算法的基础上,进一步提出了一种改进的基于奇异值分解的小波域彩色图像半脆弱水印算法,通过颜色模型的空间变换和对亮度信息的线性变换处理,解决了水印嵌入时存在的图像亮度发生变化的问题。所提出的认证水印算法,均提供了增强算法安全性的措施,并对此做了详细的分析论证。实验结果表明,上述提出的算法是有效可行的。文章最后,对认证水印的未来研究方向和发展前景做了分析和总结。
Digital watermarking technology for authentication is an important research branch of digital watermarking due to its significant effectiveness in the content verification of multimedia.
At first, this paper summaries the definition and application of digital watermarking and gives a general introduction of watermarking for authentication, including the definition, the main characteristics, the attacks threatening to it and the main algorithms designed at present. Wavelet theory and its application in digital image processing and digital watermarking are also introduced in detail. Based on the above, this paper proposes three new digital watermarking algorithms for image authentication in DWT domain. One of them is designed for gray images. The watermark is generated from the characteristics of the images and is then embedded into the wavelet domain applying the low-dimensional chaos and mean quantization method. Another is designed specially for color images where the meaningful watermark is embedded in the wavelet domain based on SVD. Spatiotemporal chaos is employed to protect the algorithm from attacks and the security of the scheme is highly improved. Good fidelity and semi-fragile ability is possessed for the use of SVD. Based on the second algorithm, an improved SVD-based wavelet domain watermarking algorithm is proposed. The spatial transform for color model and the linearly transform for the luminance information is performed so that the luminance-changing problem is well solved. All of the algorithms tell how to improve the security of the watermarking schemes with detailed analysis and demonstrations. The last part of the paper analyzes the research trend and usability of digital watermarking for multimedia authentication.
引文
[1]孙圣和,陆哲明,牛夏牧,数字水印技术与应用,北京,科学出版社,2004
[2]【美】Ingemar J.Cox 等著,王颖,黄志蓓译,数字水印,北京,电子工业出版社,2003
[3]王炳锡,陈琦,邓峰森,数字水印技术,西安,西安电子科技大学出版社,2003
[4]胡寿华,基于 MATLAB 的系统分析与设计:小波分析,西安,西安电子科技大学出版社,2000
[5]杨维明,时空混沌和藕合映象格子,上海,上海科技教育出版社,1994
[6]王晓丹,吴崇明,基于 MATLAB 的系统分析与设计:图像处理,西安,西安电子科技大学出版社,2000
[7]程正兴,小波分析算法与应用,西安,西安交通大学出版社,1998
[8]戴华,矩阵论,北京,科学出版社,2001
[9] 【 美 】 Ed. F. Deprettere , SVD and signal processing: algorithms, applications, and architectures,Amsterdam [Netherlands] ,North-Holland; 1988
[10] 程佩青,数字信号处理教程,北京,清华大学出版社,2001
[11] 闫敬文,数字图像处理技术与图像图形学基础教程,北京,科学出版社,2002
[12] Xiang Zhou, Xiaohui Duan, A Semi-Fragile Watermarking Scheme For Image Authentication, Proc.of the 10th International Multimedia Modelling Conference, 2004
[13] D. Kundur, D. Hatzinakos, Digital Watermarking for Telltale Tamper Proofing and Authentication, Proc. of the IEEE, 1999, 87:1167–1180
[14] N. Chotikakamthorn, W. Sangiamkun, Digital Watermarking Technique for Image Authentication by Neighboring Block Similarity Measure, Int. Conf.Electrical and Electronic Technology, 2001, pp. 743–747
[15] Jaejin Lee, Chee Sun Won, A Watermarking Sequence Using Parities of Error Control Coding For Image Authentication And Correction, Consumer Electronics, IEEE Transactions, 2000, 46 (2): 313 -317
[16]Shengsheng Yu, Yuping Hu, Content-based watermarking Scheme for Image Authentication, the 8th International Conference on Control, Autoamtion, Robostic and Vision, 2004
[17]Hongtao Lu, Ruiming Shen, Fragile Watermarking Scheme for Image Authentication, Electronics Letters, 2003, 39: 898-900
[18] A. H. Paquet, P. K. Ward, Wavelet-Based Digital Watermarking for Image Authentication, Int. Conf. Electrical and Computer Engineering, 2002, 2: 879–884
[19] M. Wu, B. Liu, Watermarking for Image Authentication, Int. Conf. Image Processing, 1998, 2: 437–441
[20] 贾云得,机器视觉,北京,科学出版社,2000
[21]丁科,何晨,蒋铃鸽,王宏霞,基于地址码的脆弱数字水印技术,上海交通大学学报,2004,38 (4)
[22]陈生潭,侯振华,王虹现,双重认证的变换域图像半脆弱数字水印算法,计算机辅助设计与图形学学报,2005, 17(5)
[23]谢荣生,秦红磊,郝燕玲等,混沌二维置换网络的设计及其在图像数字水印隐藏中的应用,信息隐藏全国学术研讨会(CIHW2000/2001)论文集,北京,2000:88-96
[24]Yuping Hu, Dezhi Han, Using Two Semi-Fragile Watermark for Image Authentication, Machine Learning and Cybernetics, Proceedings of 2005 International Conference on Volume 9, 18-21, 2005, pp: 5484 – 5489
[25]M. Yeung, F. Mintzer, An Invisible Watermarking Technique for Image Verification, Proc. of the International Conference on Image Processing, 1997, 1: 680-683
[26]P.W. Wong, A Public Key Watermarking for Image Verification and Authentication, Int. Conf. Image Processing, 1998, pp: 455–459
[27]M. Holliman, N. Memon, Counterfeiting Attacks on Oblivious Block-Wise Independent Invisible Watermarking Schemes, IEEE Trans. Image Processing, 2000, 9: 432–441
[28]Yuan-Liang Tang, Ching-Ting Chen, Image Authentication Using Relation Measures of Wavelet Coefficients, Proc. of the 2004 InternationalConference on e-Technology, e-Commerce and e-Service, 2004, pp: 541-545
[29]P. Bao, Xiaohu Ma, Image adaptive watermarking using wavelet domain singular value decomposition,IEEE Trans Circuits and Systems for Video Technology, 2005, 15(1): 96-102
[30]Dawei Zhao, Guanrong Chen, Wenbo Liu, A chaos-based robust wavelet-domain watermarking algorithm, Chaos, Solitons & Fractals, 2004, 22(1): 47-54
[31]Chuhong Fei, D. Kundur, R.H. Kwong, Analysis and design of secure watermark-based authentication systems, IEEE Trans Information Forensics and Security, 2006, 1(1): 43 -55
[32]Yongdong Wu, On the security of an SVD-based ownership watermarking. IEEE Trans Multimedia, 2005, 7(1): 624-627
[33]V.V. Astakhov, V.S. Anishchenko, A.V. Shabunin, Controlling spatiotemporal chaos in a chain of the coupled logistic maps, IEEE Trans Circuits and Systems I: Fundamental Theory and Applications, 1995, 42(6): 352 – 357
[34]Hai Wang, Jiandong Hu, Coupled-map spatiotemporal chaos synchronization and application in CDMA, Proc. Int. Conf. on Information, Communications and Signal Processing, 1997, 1: 69 – 73
[35]K. Maeno, Qibin Sun, Shih-Fu Chang, M. Suto, New semi-fragile image authentication watermarking techniques using random bias and nonuniform quantization, IEEE Trans Multimedia, 2006, 8(1): 32 – 45
[36]Guo-rui Feng, Ling-ge Jiang, Chen He, Yi Xue, Chaotic spread spectrum watermark of optimal space-filling curves, Chaos, Solitons & Fractals, 2006, 27(3): 580-587
[37]Qibin Sun, Shih-Fu Chang, A secure and robust digital signature scheme for JPEG2000 image authentication, IEEE Trans. Multimedia, 2005, 7(3):480 – 494
[38]Martinian E, Wornell G. W, Chen B, Authentication with distortion criteria, IEEE Trans. Information Theory, 2005, 51(7):2523 – 2542
[39]Siyue Chen, Leung H, Ergodic chaotic parameter modulation with application to digital image watermarking, IEEE Trans. Image Processing, 2005, 14(10):1590 – 1602
[40]M. Barni, F. Bartolini, A. Piva, Improved wavelet-based watermarking through pixel-wise masking, IEEE Trans. Image Processing, 2001, 10(5): 783–791
[41]Puate J,Jordan F,Using fractal compression scheme to embed a digital signature into an image,Proc. Of SPIE Photonics East’96 symposium, 1996:103-124
[42]Davis K J,Najarian K,Maximizing strength of digital watermarks using neural networks,Proc. of the International Joint Conference on Neural Network,2001,4:2893-2898
[43]Ching-Yung Lin, Shih-Fu Chang, A robust image authentication method distinguishing JPEG compression from malicious manipulation, Circuits and Systems for Video Technology, IEEE Transactions, 2001, 11(2):153 – 168
[44]Rey, C., Dugelay, J.-L, Blind detection of malicious alterations on still images using robust watermarks, Secure Images and Image Authentication (Ref. No. 2000/039), IEE Seminar on 10, 2000, pp:7/1 - 7/6
[45]Wenwu Zhu, Zixiang Xiong, Ya-Qin Zhang, Multiresolution watermarking for images and video: a unified approach, Image Processing, Proceedings. 1998 International Conference, 1998, 1:465 – 468
[46]Zhongkang Lu, Lin, W., Yang, X., EePing Ong; Susu Yao, Modeling visual attention's modulatory aftereffects on visual sensitivity and quality evaluation, Image Processing, IEEE Transactions, 2005, 14(11): 1928-1942
[47]Liehua Xie, Arce, G.R., Joint wavelet compression and authentication watermarking, Image Processing, Proceedings of International Conference, 1998, 2: 427 - 431
[48]武拴虎,谈正,基于混沌序列的视频图像数字水印隐藏与多分辨率检测,西安交通大学学报,2000,34(6):35-39
[49]王丽娜,于戈,王国仁,基于混沌特性改进的小波数字数字水印算法,电子学报,2001,29(10):1424-1426
[50]Silvesre G.C.M,et.al,Informed Audio Watermarking Scheme Using Digital Chaotic Signals,Proc. IEEE International Conference on Acoustics,Speech,and Signal Processing,2001,3:1361-1364
[51]刘瑞祯,谭铁牛, 基于奇异值分解的数字图像水印方法, 电子学报, 2001, 29 (2): 1- 4
[52]张维山,秀明,任勇,基于时空二维混沌的数字水印算法,微计算机信息,2005,21(11): 41-43
[53] J. Fridrich, R. Du, M. Long, Steganalysis of LSB Encoding in Color Images, IEEE International Conference on Multimedia and Expo, 2000, 3:1279-1283
[54]M. Kutter, F. A. P. Petitcoias, A Fair Benchmark for Image Watermarking Systems, Security and Watermarking of Multimedia Contents, 1999, SPIE-3657: 226-239
[55]Phen-Lan Lin, Po-Whei Huang, An-Wei Peng, A Fragile Watermarking Scheme for Image Authentication with Localization and Recovery, Proceedings of the IEEE Sixth International Symposium on Multimedia Software Engineering, 2004
[56] Phen-Lan Lin, Chung-Kai Hsieh, Po-Whei Huang, Hierarchical Watermarking Scheme for Image Authentication and Recovery, IEEE International Conference on Multimedia and Expo, 2004, pp: 963-966
[57]C.-T. Li, Reversible watermarking scheme with image-independent embedding capacity, IEE Proc.-Vis. Image Signal Process, 2005, 152(6): 779-786
[58]Zhenni Peng, Wenbo Liu, Color image authentication based on spatiotemporal chaos and SVD, Chaos, Solitons & Fractals, 2006(article in press,available online)
[2]【美】Ingemar J.Cox 等著,王颖,黄志蓓译,数字水印,北京,电子工业出版社,2003
[3]王炳锡,陈琦,邓峰森,数字水印技术,西安,西安电子科技大学出版社,2003
[4]胡寿华,基于 MATLAB 的系统分析与设计:小波分析,西安,西安电子科技大学出版社,2000
[5]杨维明,时空混沌和藕合映象格子,上海,上海科技教育出版社,1994
[6]王晓丹,吴崇明,基于 MATLAB 的系统分析与设计:图像处理,西安,西安电子科技大学出版社,2000
[7]程正兴,小波分析算法与应用,西安,西安交通大学出版社,1998
[8]戴华,矩阵论,北京,科学出版社,2001
[9] 【 美 】 Ed. F. Deprettere , SVD and signal processing: algorithms, applications, and architectures,Amsterdam [Netherlands] ,North-Holland; 1988
[10] 程佩青,数字信号处理教程,北京,清华大学出版社,2001
[11] 闫敬文,数字图像处理技术与图像图形学基础教程,北京,科学出版社,2002
[12] Xiang Zhou, Xiaohui Duan, A Semi-Fragile Watermarking Scheme For Image Authentication, Proc.of the 10th International Multimedia Modelling Conference, 2004
[13] D. Kundur, D. Hatzinakos, Digital Watermarking for Telltale Tamper Proofing and Authentication, Proc. of the IEEE, 1999, 87:1167–1180
[14] N. Chotikakamthorn, W. Sangiamkun, Digital Watermarking Technique for Image Authentication by Neighboring Block Similarity Measure, Int. Conf.Electrical and Electronic Technology, 2001, pp. 743–747
[15] Jaejin Lee, Chee Sun Won, A Watermarking Sequence Using Parities of Error Control Coding For Image Authentication And Correction, Consumer Electronics, IEEE Transactions, 2000, 46 (2): 313 -317
[16]Shengsheng Yu, Yuping Hu, Content-based watermarking Scheme for Image Authentication, the 8th International Conference on Control, Autoamtion, Robostic and Vision, 2004
[17]Hongtao Lu, Ruiming Shen, Fragile Watermarking Scheme for Image Authentication, Electronics Letters, 2003, 39: 898-900
[18] A. H. Paquet, P. K. Ward, Wavelet-Based Digital Watermarking for Image Authentication, Int. Conf. Electrical and Computer Engineering, 2002, 2: 879–884
[19] M. Wu, B. Liu, Watermarking for Image Authentication, Int. Conf. Image Processing, 1998, 2: 437–441
[20] 贾云得,机器视觉,北京,科学出版社,2000
[21]丁科,何晨,蒋铃鸽,王宏霞,基于地址码的脆弱数字水印技术,上海交通大学学报,2004,38 (4)
[22]陈生潭,侯振华,王虹现,双重认证的变换域图像半脆弱数字水印算法,计算机辅助设计与图形学学报,2005, 17(5)
[23]谢荣生,秦红磊,郝燕玲等,混沌二维置换网络的设计及其在图像数字水印隐藏中的应用,信息隐藏全国学术研讨会(CIHW2000/2001)论文集,北京,2000:88-96
[24]Yuping Hu, Dezhi Han, Using Two Semi-Fragile Watermark for Image Authentication, Machine Learning and Cybernetics, Proceedings of 2005 International Conference on Volume 9, 18-21, 2005, pp: 5484 – 5489
[25]M. Yeung, F. Mintzer, An Invisible Watermarking Technique for Image Verification, Proc. of the International Conference on Image Processing, 1997, 1: 680-683
[26]P.W. Wong, A Public Key Watermarking for Image Verification and Authentication, Int. Conf. Image Processing, 1998, pp: 455–459
[27]M. Holliman, N. Memon, Counterfeiting Attacks on Oblivious Block-Wise Independent Invisible Watermarking Schemes, IEEE Trans. Image Processing, 2000, 9: 432–441
[28]Yuan-Liang Tang, Ching-Ting Chen, Image Authentication Using Relation Measures of Wavelet Coefficients, Proc. of the 2004 InternationalConference on e-Technology, e-Commerce and e-Service, 2004, pp: 541-545
[29]P. Bao, Xiaohu Ma, Image adaptive watermarking using wavelet domain singular value decomposition,IEEE Trans Circuits and Systems for Video Technology, 2005, 15(1): 96-102
[30]Dawei Zhao, Guanrong Chen, Wenbo Liu, A chaos-based robust wavelet-domain watermarking algorithm, Chaos, Solitons & Fractals, 2004, 22(1): 47-54
[31]Chuhong Fei, D. Kundur, R.H. Kwong, Analysis and design of secure watermark-based authentication systems, IEEE Trans Information Forensics and Security, 2006, 1(1): 43 -55
[32]Yongdong Wu, On the security of an SVD-based ownership watermarking. IEEE Trans Multimedia, 2005, 7(1): 624-627
[33]V.V. Astakhov, V.S. Anishchenko, A.V. Shabunin, Controlling spatiotemporal chaos in a chain of the coupled logistic maps, IEEE Trans Circuits and Systems I: Fundamental Theory and Applications, 1995, 42(6): 352 – 357
[34]Hai Wang, Jiandong Hu, Coupled-map spatiotemporal chaos synchronization and application in CDMA, Proc. Int. Conf. on Information, Communications and Signal Processing, 1997, 1: 69 – 73
[35]K. Maeno, Qibin Sun, Shih-Fu Chang, M. Suto, New semi-fragile image authentication watermarking techniques using random bias and nonuniform quantization, IEEE Trans Multimedia, 2006, 8(1): 32 – 45
[36]Guo-rui Feng, Ling-ge Jiang, Chen He, Yi Xue, Chaotic spread spectrum watermark of optimal space-filling curves, Chaos, Solitons & Fractals, 2006, 27(3): 580-587
[37]Qibin Sun, Shih-Fu Chang, A secure and robust digital signature scheme for JPEG2000 image authentication, IEEE Trans. Multimedia, 2005, 7(3):480 – 494
[38]Martinian E, Wornell G. W, Chen B, Authentication with distortion criteria, IEEE Trans. Information Theory, 2005, 51(7):2523 – 2542
[39]Siyue Chen, Leung H, Ergodic chaotic parameter modulation with application to digital image watermarking, IEEE Trans. Image Processing, 2005, 14(10):1590 – 1602
[40]M. Barni, F. Bartolini, A. Piva, Improved wavelet-based watermarking through pixel-wise masking, IEEE Trans. Image Processing, 2001, 10(5): 783–791
[41]Puate J,Jordan F,Using fractal compression scheme to embed a digital signature into an image,Proc. Of SPIE Photonics East’96 symposium, 1996:103-124
[42]Davis K J,Najarian K,Maximizing strength of digital watermarks using neural networks,Proc. of the International Joint Conference on Neural Network,2001,4:2893-2898
[43]Ching-Yung Lin, Shih-Fu Chang, A robust image authentication method distinguishing JPEG compression from malicious manipulation, Circuits and Systems for Video Technology, IEEE Transactions, 2001, 11(2):153 – 168
[44]Rey, C., Dugelay, J.-L, Blind detection of malicious alterations on still images using robust watermarks, Secure Images and Image Authentication (Ref. No. 2000/039), IEE Seminar on 10, 2000, pp:7/1 - 7/6
[45]Wenwu Zhu, Zixiang Xiong, Ya-Qin Zhang, Multiresolution watermarking for images and video: a unified approach, Image Processing, Proceedings. 1998 International Conference, 1998, 1:465 – 468
[46]Zhongkang Lu, Lin, W., Yang, X., EePing Ong; Susu Yao, Modeling visual attention's modulatory aftereffects on visual sensitivity and quality evaluation, Image Processing, IEEE Transactions, 2005, 14(11): 1928-1942
[47]Liehua Xie, Arce, G.R., Joint wavelet compression and authentication watermarking, Image Processing, Proceedings of International Conference, 1998, 2: 427 - 431
[48]武拴虎,谈正,基于混沌序列的视频图像数字水印隐藏与多分辨率检测,西安交通大学学报,2000,34(6):35-39
[49]王丽娜,于戈,王国仁,基于混沌特性改进的小波数字数字水印算法,电子学报,2001,29(10):1424-1426
[50]Silvesre G.C.M,et.al,Informed Audio Watermarking Scheme Using Digital Chaotic Signals,Proc. IEEE International Conference on Acoustics,Speech,and Signal Processing,2001,3:1361-1364
[51]刘瑞祯,谭铁牛, 基于奇异值分解的数字图像水印方法, 电子学报, 2001, 29 (2): 1- 4
[52]张维山,秀明,任勇,基于时空二维混沌的数字水印算法,微计算机信息,2005,21(11): 41-43
[53] J. Fridrich, R. Du, M. Long, Steganalysis of LSB Encoding in Color Images, IEEE International Conference on Multimedia and Expo, 2000, 3:1279-1283
[54]M. Kutter, F. A. P. Petitcoias, A Fair Benchmark for Image Watermarking Systems, Security and Watermarking of Multimedia Contents, 1999, SPIE-3657: 226-239
[55]Phen-Lan Lin, Po-Whei Huang, An-Wei Peng, A Fragile Watermarking Scheme for Image Authentication with Localization and Recovery, Proceedings of the IEEE Sixth International Symposium on Multimedia Software Engineering, 2004
[56] Phen-Lan Lin, Chung-Kai Hsieh, Po-Whei Huang, Hierarchical Watermarking Scheme for Image Authentication and Recovery, IEEE International Conference on Multimedia and Expo, 2004, pp: 963-966
[57]C.-T. Li, Reversible watermarking scheme with image-independent embedding capacity, IEE Proc.-Vis. Image Signal Process, 2005, 152(6): 779-786
[58]Zhenni Peng, Wenbo Liu, Color image authentication based on spatiotemporal chaos and SVD, Chaos, Solitons & Fractals, 2006(article in press,available online)