基于小波包和全变差的图像去噪算法
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摘要
近年来,图像去噪问题一直都是国内外研究者所关注的研究课题,因为噪声去除的好坏,直接影响着图像的质量、视觉效果以及后续更高层次处理的需要.而图像去噪过程中,面临的关键问题是在抑制噪声的同时,如何更好地保护图像的重要细节特征.小波包分析技术和偏微分方程方法是图像去噪中两种比较有效的方法.本文对小波包分析方法和基于全变差的去噪方法进行了深入的研究和总结,根据它们各自的去噪特点,提出了一种新的去噪算法,即基于小波包和全变差的图像去噪算法.首先利用小波包变换的多分辨分析,对含噪图像进行四层分解；其次根据选定的信息代价函数来选取最优小波包基；再次对被选取的最优小波包基下的小波包系数进行阈值化处理,利用处理后的系数重构图像；最后采用全变差的方法,将初步处理的图像进行进一步去噪.数值实验表明本文的方法不仅提高了图像的峰值信噪比(PSNR)值而且获得了令人比较满意的视觉效果,因而证明了本文方法的可行性和有效性.
Image Denoising Algorithm Based On Wavelet Packet And Total Variation
Image is contaminated by different levels of noise produced in image acquisi-tion, transformation and storage,which makes image blurred.It not only undermines the image quality and applied results,but also affects the follow-up deep-level image processing.In order to improve image quality and the need for follow-up deep-level processing, image denoising has become an essential task in image pre-processing. During image denoising process, the key issue is how to suppress the noise while pre-serving image detailed features as much as possible(such as edge, texture, etc.).
Wavelet packet analysis and partial differential equation are the two more effi-cient image denoising methods and are developed in recent years.In this paper,we con-duct deep study and summary on wavelet packet analysis method and total variation method,in accordance with their respective denoising characteristics, propose a new denoising algorithm, which is denoising algorithm based on wavelet packet and total variation.
In the second chapter,we describe the basic theory of the wavelet transform and wavelet packet transform and their applications in image denoising.
In the third chapter,we describe the principle of denoising and introduce several classical image denoising models based on total variation method,and have done fur-ther analysis and research for them.If ROF Method is directly applied to the noisy image,it is easy to think noise as edge, making the ladder effect produced in the image flat areas,however, with the increase of iteration times, image will become blurred, and thus can not achieve a good denoising effect.2007 years, Tony F. Chan and Hao-Min Zhou,proposed TV wavelet thresholding denoising model based on wavelet and total variation method.Through the variational model to select and modify the wavelet co-efficients so that reconstructed image has fewer oscillations near edge while noise is smoothed.Although this method can to some extent reduce the oscillations, the noise has not been well suppressed.The reason may be that this method is not a more general approach, may gain a better denoising effect for a particular image.
The fourth chapter is the focus of this research work. Mainly combining advan-tages of wavelet packet analysis with advantages of total variation method, we propose a new image denoising method, which is image denoising algorithm based on wavelet packet and total variation. And comparing this method and present some classical denoising methods through numerical experiments, the experimental results show this method is feasible and effective.
In this paper, we exploit the advantages of wavelet packet, contain an initial noise filter.As wavelet packet well removes image noise in flat regions, thus avoiding the phenomenon noise is not well smoothed. And wavelet packet transform conducts a more elaborate division of image high frequency part,which makes certain detailed features can be protected in denoising processing.Thus we apply total variation method to the preliminary processed image, making image after denoising have a better visual effect.
With the analysis of both wavelet packet transform and denoising models based on total variation, this paper denoising algorithm can be described as follows:
STEP1:For noisy imageu0(x, y), we use wavelet packet method to conduct pre-liminary noise processing for it. Specific steps are as follows.
(1) We decompose imageuo(x,y) for four-layer.
(2) Select the appropriate information cost function Q, we select the best wavelet packet basis according to information cost function Q.
(3) To process wavelet packet coefficients under selected the best wavelet packet basis by the thresholding.
(4) we use processed wavelet packet coefficients to reconstruct image.
STEP2:Suppose new image after the initial treatment for u1(x, y), we use u1(x, y) to replace uo(x,y) of the following equation,and acquire final solution by solving the equation.
In numerical experiments, we compare this method with ROF method, soft thresh-olding denoising method and TV wavelet thresholding denoising method. Experimen-tal results show this method both in visual effect and in the peak signal to noise ratio respect, can achieve better results.
Image Denoising Algorithm Based On Wavelet Packet And Total Variation
Image is contaminated by different levels of noise produced in image acquisi-tion, transformation and storage,which makes image blurred.It not only undermines the image quality and applied results,but also affects the follow-up deep-level image processing.In order to improve image quality and the need for follow-up deep-level processing, image denoising has become an essential task in image pre-processing. During image denoising process, the key issue is how to suppress the noise while pre-serving image detailed features as much as possible(such as edge, texture, etc.).
Wavelet packet analysis and partial differential equation are the two more effi-cient image denoising methods and are developed in recent years.In this paper,we con-duct deep study and summary on wavelet packet analysis method and total variation method,in accordance with their respective denoising characteristics, propose a new denoising algorithm, which is denoising algorithm based on wavelet packet and total variation.
In the second chapter,we describe the basic theory of the wavelet transform and wavelet packet transform and their applications in image denoising.
In the third chapter,we describe the principle of denoising and introduce several classical image denoising models based on total variation method,and have done fur-ther analysis and research for them.If ROF Method is directly applied to the noisy image,it is easy to think noise as edge, making the ladder effect produced in the image flat areas,however, with the increase of iteration times, image will become blurred, and thus can not achieve a good denoising effect.2007 years, Tony F. Chan and Hao-Min Zhou,proposed TV wavelet thresholding denoising model based on wavelet and total variation method.Through the variational model to select and modify the wavelet co-efficients so that reconstructed image has fewer oscillations near edge while noise is smoothed.Although this method can to some extent reduce the oscillations, the noise has not been well suppressed.The reason may be that this method is not a more general approach, may gain a better denoising effect for a particular image.
The fourth chapter is the focus of this research work. Mainly combining advan-tages of wavelet packet analysis with advantages of total variation method, we propose a new image denoising method, which is image denoising algorithm based on wavelet packet and total variation. And comparing this method and present some classical denoising methods through numerical experiments, the experimental results show this method is feasible and effective.
In this paper, we exploit the advantages of wavelet packet, contain an initial noise filter.As wavelet packet well removes image noise in flat regions, thus avoiding the phenomenon noise is not well smoothed. And wavelet packet transform conducts a more elaborate division of image high frequency part,which makes certain detailed features can be protected in denoising processing.Thus we apply total variation method to the preliminary processed image, making image after denoising have a better visual effect.
With the analysis of both wavelet packet transform and denoising models based on total variation, this paper denoising algorithm can be described as follows:
STEP1:For noisy imageu0(x, y), we use wavelet packet method to conduct pre-liminary noise processing for it. Specific steps are as follows.
(1) We decompose imageuo(x,y) for four-layer.
(2) Select the appropriate information cost function Q, we select the best wavelet packet basis according to information cost function Q.
(3) To process wavelet packet coefficients under selected the best wavelet packet basis by the thresholding.
(4) we use processed wavelet packet coefficients to reconstruct image.
STEP2:Suppose new image after the initial treatment for u1(x, y), we use u1(x, y) to replace uo(x,y) of the following equation,and acquire final solution by solving the equation.
In numerical experiments, we compare this method with ROF method, soft thresh-olding denoising method and TV wavelet thresholding denoising method. Experimen-tal results show this method both in visual effect and in the peak signal to noise ratio respect, can achieve better results.
引文
[1]王大凯,候榆青,彭近业.图像处理的偏微分方程方法[M].北京：科学出版社,2008.
[2]KENNETH R.CASTLEMAN著,朱志刚,林学阖,石定机等译.数字图像处理[M].北京：电子工业出版社,2002.
[3]葛哲学,沙威.小波分析理论与MATLABR2007实现[M].北京：电子工业出版社,2007.
[4]成礼智,王红霞,罗永.小波的理论与应用[M].北京：科学出版社,2004.
[5]Henri Maitre著,孙洪译.现代数字图像处理[M].北京：电子工业出版社,2006.
[6]孙延奎.小波分析及其应用[M].北京：机械工业出版社,2005.
[7]C.SIDNEY BURRUS,RAMESH A.GOPINATH,HAITAO GUO等著,程正兴译.小波与小波变换导论[M].北京：机械工业出版社,2007.
[8]李世雄.小波变换及其应用[M].北京：高等教育出版社,1997.
[9]樊启斌.小波分析[M].武汉：武汉大学出版社,2008.
[10]胡昌华,李国华,周涛.基于MATLAB7.x的系统分析与设计—小波分析[M].西安：西安电子科技大学出版社,2008.
[11]高志,余啸海.MATLAB小波分析与应用[M].北京：国防工业出版社,2008.
[12]谢杰成,张大力.小波图像去噪综述[J].中国图像图形学报,2002,7(3)：209-217.
[13]赵瑞珍,宋国乡,王红.小波系数阈值估计的改进模型[J].西北工业大学学报2001年第19卷第4期.
[14]张黎,王立克,杨峰,李淑霞.小波阈值图像去噪研究与应用[J].中文核心期刊《微计算机信息》(管控一体化)2006年第22卷第10-3期.
[15]李敏,冯象初,杨文杰.基于全变差和小波硬阈值的图像去噪[J].中文核心期刊《微计算机信息》(管控一体化)2006年第22卷第6期.
[16]王正明,谢美华.偏微分方程在图像去噪中的应用[J].应用数学,2005,18(2)：219-224.
[17]Chan, T. F., and Zhou, H. M. (2000). Total variation improved wavelet thresh-olding in image compression. In Proceedings to the 2000 International Confer-ence on Image Processing, Vancouver, BC, Canada, Sept.10-13,2000, pp. 391-394.
[18]David L. Donoho.De-Noising by Soft-Thresholding.IEEE TRANSACTIONS ON INFORMATION THEORY, VOL.41, NO.3, MAY 1995.
[19]Tony F. Chan and Hao-Min Zhou. Total Variation Wavelet Thresholding. Journal of Scientific Computing, Vol.32, No.2, August 2007.
[20]Cand'es, E. J., and Guo, F. (2001). Edge-preserving image reconstruction from noisy radon data, in preparation (Invited Special Issue of the Journal of Signal Processing on Image and Video Coding Beyond Standards.).
[21]Rudin, L., Osher, S., and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D 60,259-268.
[22]Chan, T. F., and Zhou, H. M. (2000). Optimal constructions of wavelet coeffi-cients using total variation regularization in image compression. CAM Report, No.00-27, Dept. of Math., UCLA, July.
[23]Donoho, D. (1995). De-noising by soft thresholding. IEEE Trans. Inf. Th.41, 613-627.
[24]M. V. WICKERHAUSER (1993):Adapted Wavelet Analysis:Theory and Algo-rithms. New York:A. K. Peters.
[25]P. CHARBONNIER, L. BLANC-FERAUD, G. AUBERT, M. BARLAUD. De-terministic edge-preserving regularization in computed imaging[J]. IEEE Trans. Image Process.6 (2) (1997) 298-311.
[26]D. DONOHO, I. JOHNSTONE. Ideal spatial adaptation by wavelet shrinkage[J]. Biometrika 81 (1994) 425-455.
[27]S. MALLAT, S. ZHONG. Characterization of signals form multiscale edges[J]. IEEE Trans. Pattern Anal. Mach. Intelligence 14 (7) (1992) 710-732.
[28]P. PERONA, J. MALIK. Scale-space and edge detection using anisotropic diffu-sion[J]. IEEE Trans. Pattern Anal. Mach. Intelligence 12 (7) (1990) 629-639.
[29]KASHIF RAJPOOT, NASIR PAJPOOT, J.ALISON NOBEL. Discrete Wavelet Diffusion for Image Denoising[J]. Proceedings of the 3rd international conference on Image and Signal Processing.
[30]Donoho, D. (1999). Sparse Components of Images and Optimal Atomic Decom-positions. Tech. Report, Dept. of Stat., Stanford Univ.
[31]G. W. WEI. Generalized Perona-Malik equation for image restoration[J]. IEEE Signal Process. Lett., vol.6, no.1, pp.165-167, Jan.1999.
[32]J. WEICKERT. Anisotropic Diffusion in Image Processing[J]. Stuttgart, Ger-many:Teubner-Verlag,1998.
[33]S. MALLAT and W. L. HWANG. Singularity detection and processing with wavelets[J]. IEEE Trans. Inf. Theory, vol.38, no.2, pp.617-643, Mar.1992.
[34]J. H. VAN HATEREN, A. VAN DER SCHAAF (1998):Independent component filters of natural images compared with simple cells in the primary visual cortex. Proc. Roy. Soc. London Ser. B,265:359-366.
[35]PAVEL MRAZEK AND JOACHIM WEICKERT.Diffusion-Inspired Shrinkage Functions and Stability Results for Wavelet Denoising[J].International Journal of Computer Vision 64(2/3),171-186,2005 Springer Science+Business Media, Inc. Manufactured in The Netherlands.
[36]Perona, P., and Malik, J. (1990). Scale-space and edge detection using anisotropic diffusion. IEEE T Pattern Anal.12(7),629-639.
[37]D. Donoho, I. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika 81 (1994) 425-455.
[38]Feng Liu, Xiao E. Ruan, Wavelet-based diffusion approaches for signal denoising , Signal Processing 87 (2007) 1138-1146.
[39]Vogel, C., and Oman, M. (1998). Fast, robust total variation-based reconstruc-tion of noisy, blurred images. IEEE Trans. Image Process.7,813-824.
[2]KENNETH R.CASTLEMAN著,朱志刚,林学阖,石定机等译.数字图像处理[M].北京：电子工业出版社,2002.
[3]葛哲学,沙威.小波分析理论与MATLABR2007实现[M].北京：电子工业出版社,2007.
[4]成礼智,王红霞,罗永.小波的理论与应用[M].北京：科学出版社,2004.
[5]Henri Maitre著,孙洪译.现代数字图像处理[M].北京：电子工业出版社,2006.
[6]孙延奎.小波分析及其应用[M].北京：机械工业出版社,2005.
[7]C.SIDNEY BURRUS,RAMESH A.GOPINATH,HAITAO GUO等著,程正兴译.小波与小波变换导论[M].北京：机械工业出版社,2007.
[8]李世雄.小波变换及其应用[M].北京：高等教育出版社,1997.
[9]樊启斌.小波分析[M].武汉：武汉大学出版社,2008.
[10]胡昌华,李国华,周涛.基于MATLAB7.x的系统分析与设计—小波分析[M].西安：西安电子科技大学出版社,2008.
[11]高志,余啸海.MATLAB小波分析与应用[M].北京：国防工业出版社,2008.
[12]谢杰成,张大力.小波图像去噪综述[J].中国图像图形学报,2002,7(3)：209-217.
[13]赵瑞珍,宋国乡,王红.小波系数阈值估计的改进模型[J].西北工业大学学报2001年第19卷第4期.
[14]张黎,王立克,杨峰,李淑霞.小波阈值图像去噪研究与应用[J].中文核心期刊《微计算机信息》(管控一体化)2006年第22卷第10-3期.
[15]李敏,冯象初,杨文杰.基于全变差和小波硬阈值的图像去噪[J].中文核心期刊《微计算机信息》(管控一体化)2006年第22卷第6期.
[16]王正明,谢美华.偏微分方程在图像去噪中的应用[J].应用数学,2005,18(2)：219-224.
[17]Chan, T. F., and Zhou, H. M. (2000). Total variation improved wavelet thresh-olding in image compression. In Proceedings to the 2000 International Confer-ence on Image Processing, Vancouver, BC, Canada, Sept.10-13,2000, pp. 391-394.
[18]David L. Donoho.De-Noising by Soft-Thresholding.IEEE TRANSACTIONS ON INFORMATION THEORY, VOL.41, NO.3, MAY 1995.
[19]Tony F. Chan and Hao-Min Zhou. Total Variation Wavelet Thresholding. Journal of Scientific Computing, Vol.32, No.2, August 2007.
[20]Cand'es, E. J., and Guo, F. (2001). Edge-preserving image reconstruction from noisy radon data, in preparation (Invited Special Issue of the Journal of Signal Processing on Image and Video Coding Beyond Standards.).
[21]Rudin, L., Osher, S., and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D 60,259-268.
[22]Chan, T. F., and Zhou, H. M. (2000). Optimal constructions of wavelet coeffi-cients using total variation regularization in image compression. CAM Report, No.00-27, Dept. of Math., UCLA, July.
[23]Donoho, D. (1995). De-noising by soft thresholding. IEEE Trans. Inf. Th.41, 613-627.
[24]M. V. WICKERHAUSER (1993):Adapted Wavelet Analysis:Theory and Algo-rithms. New York:A. K. Peters.
[25]P. CHARBONNIER, L. BLANC-FERAUD, G. AUBERT, M. BARLAUD. De-terministic edge-preserving regularization in computed imaging[J]. IEEE Trans. Image Process.6 (2) (1997) 298-311.
[26]D. DONOHO, I. JOHNSTONE. Ideal spatial adaptation by wavelet shrinkage[J]. Biometrika 81 (1994) 425-455.
[27]S. MALLAT, S. ZHONG. Characterization of signals form multiscale edges[J]. IEEE Trans. Pattern Anal. Mach. Intelligence 14 (7) (1992) 710-732.
[28]P. PERONA, J. MALIK. Scale-space and edge detection using anisotropic diffu-sion[J]. IEEE Trans. Pattern Anal. Mach. Intelligence 12 (7) (1990) 629-639.
[29]KASHIF RAJPOOT, NASIR PAJPOOT, J.ALISON NOBEL. Discrete Wavelet Diffusion for Image Denoising[J]. Proceedings of the 3rd international conference on Image and Signal Processing.
[30]Donoho, D. (1999). Sparse Components of Images and Optimal Atomic Decom-positions. Tech. Report, Dept. of Stat., Stanford Univ.
[31]G. W. WEI. Generalized Perona-Malik equation for image restoration[J]. IEEE Signal Process. Lett., vol.6, no.1, pp.165-167, Jan.1999.
[32]J. WEICKERT. Anisotropic Diffusion in Image Processing[J]. Stuttgart, Ger-many:Teubner-Verlag,1998.
[33]S. MALLAT and W. L. HWANG. Singularity detection and processing with wavelets[J]. IEEE Trans. Inf. Theory, vol.38, no.2, pp.617-643, Mar.1992.
[34]J. H. VAN HATEREN, A. VAN DER SCHAAF (1998):Independent component filters of natural images compared with simple cells in the primary visual cortex. Proc. Roy. Soc. London Ser. B,265:359-366.
[35]PAVEL MRAZEK AND JOACHIM WEICKERT.Diffusion-Inspired Shrinkage Functions and Stability Results for Wavelet Denoising[J].International Journal of Computer Vision 64(2/3),171-186,2005 Springer Science+Business Media, Inc. Manufactured in The Netherlands.
[36]Perona, P., and Malik, J. (1990). Scale-space and edge detection using anisotropic diffusion. IEEE T Pattern Anal.12(7),629-639.
[37]D. Donoho, I. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika 81 (1994) 425-455.
[38]Feng Liu, Xiao E. Ruan, Wavelet-based diffusion approaches for signal denoising , Signal Processing 87 (2007) 1138-1146.
[39]Vogel, C., and Oman, M. (1998). Fast, robust total variation-based reconstruc-tion of noisy, blurred images. IEEE Trans. Image Process.7,813-824.