图像复原中正则化方法的研究及应用
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摘要
图像复原是图像处理中的一个重要问题,对于改善图像质量具有重要的意义。解决该问题的关键是对图像的退化过程建立相应的数学模型,然后通过求解该逆问题获得图像的复原模型并对原始图像进行合理估计。正则化方法是在求该逆过程中解决病态性问题的有效方法。本文针对图像复原中正则化方法的研究及应用进行了广泛深入的探讨与分析,主要工作和创新点如下:
第一,在正则化技术解决病态问题的基础上,从正则化方法数学理论入手,分析了图像的退化模型和图像复原的病态特征,重点讨论了正则化参数和正则项的选取,并总结了其求解算法及快速实现方法,完善了正则化方法图像复原的基本理论。
第二,在现有复原模型的完善上,重新构建正则化参数与正则化项,构造了新的具有空间自适应性质的正则化图像复原模型。运算中合理地设计算法,自适应的正则化参数可以自动修正到最优值,同时自适应加权的正则项使算法的性能更加完善。仿真计算表明,该方法可有效抑制图像边界的振铃效应并保护了图像的重要信息,并且比现有的方法具有更高的峰值信噪比。
第三,从传统的线性代数方法着手,提出了两种新的基于正则化思想的图像复原方法。一是从约束最小二乘出发,在加性噪声能量有界的前提下,采用正则化方法来克服病态问题,通过解一个单变量方程,并利用空域迭代运算实现了一种有效的图像复原;二是针对模糊图像的复原问题,从最小二乘算法出发,采用增量迭代的方法改善算法的收敛性,同时结合正则化技术克服问题的病态性质,并引入自适应的正则化参数,使其与图像复原的迭代运算同步进行并自动修正到最优值。计算结果表明,该方法可有效复原图像,在客观标准评价和主观视觉效果方面都有明显的改善。
Image restoration is one of the essential topics in image processing, which can greatly improve image quality. The key for this issue is to model the degradation processing in mathematical form, then solve this inverse problem to get a restoring model and an estimation of original image. Effective methods that can tackle the ill-posed problems are the so-called regularization methods during the inverse process. This thesis introduces and analyses in detail the research and application of regularization in image restoration, the main contributions and creativities are listed as follows:
Firstly, on the basis of the regularization technique for dealing with ill-posed problems, starting from the mathematical theories about regularization, analyzing the degraded model and the ill-posed character of image restoration, this paper mainly discusses the regularized parameter and regularized item, and the solution approaches and fast algorithms are also summarized, therefore perfects the essential theories about regularization in image restoration.
Secondly, to perfect the known restoring models, a new space-adaptive regularization model of image restoration is constructed by redesigning regularized parameter and regularized item. By contriving appropriate algorithms in the computations, the regularized parameter can automatically correct to the appropriate value due to its adaptive character, meanwhile, the adaptive effect is developed with a weight matrix included in the regularized item. Simulation results show that the ringing artifacts around the image are reduced, the main information is preserved, and the Peak Signal to Noise Ratio (PSNR) is higher than the known methods.
Thirdly, two new image restoration methods are proposed based on regularization idea from traditional linear algebra approach. On one hand, from the technique of constrained least squares and limited energy of additive noise, an effective restored approach by adopting regularization method to overcoming ill-posed problem, solving an equation with a single variable, and using space iterative algorithm is proposed; On the other hand, aiming at the restoration of blurred image, another effective restoration approach based on least-square algorithm is also proposed in this paper. This method firstly adopts increment iterative algorithm to improve convergence and meanwhile applies regularization technique to overcome ill-posed problem. In the computations, the regularized parameter has its adaptive character, which can be determined in terms of the restored image at each iteration step therefore automatically correct to the appropriate value. Numerical results show that these methods can effectively restore original image, and the objective standard evaluation and subjective visual effect are improved significantly.
第一,在正则化技术解决病态问题的基础上,从正则化方法数学理论入手,分析了图像的退化模型和图像复原的病态特征,重点讨论了正则化参数和正则项的选取,并总结了其求解算法及快速实现方法,完善了正则化方法图像复原的基本理论。
第二,在现有复原模型的完善上,重新构建正则化参数与正则化项,构造了新的具有空间自适应性质的正则化图像复原模型。运算中合理地设计算法,自适应的正则化参数可以自动修正到最优值,同时自适应加权的正则项使算法的性能更加完善。仿真计算表明,该方法可有效抑制图像边界的振铃效应并保护了图像的重要信息,并且比现有的方法具有更高的峰值信噪比。
第三,从传统的线性代数方法着手,提出了两种新的基于正则化思想的图像复原方法。一是从约束最小二乘出发,在加性噪声能量有界的前提下,采用正则化方法来克服病态问题,通过解一个单变量方程,并利用空域迭代运算实现了一种有效的图像复原;二是针对模糊图像的复原问题,从最小二乘算法出发,采用增量迭代的方法改善算法的收敛性,同时结合正则化技术克服问题的病态性质,并引入自适应的正则化参数,使其与图像复原的迭代运算同步进行并自动修正到最优值。计算结果表明,该方法可有效复原图像,在客观标准评价和主观视觉效果方面都有明显的改善。
Image restoration is one of the essential topics in image processing, which can greatly improve image quality. The key for this issue is to model the degradation processing in mathematical form, then solve this inverse problem to get a restoring model and an estimation of original image. Effective methods that can tackle the ill-posed problems are the so-called regularization methods during the inverse process. This thesis introduces and analyses in detail the research and application of regularization in image restoration, the main contributions and creativities are listed as follows:
Firstly, on the basis of the regularization technique for dealing with ill-posed problems, starting from the mathematical theories about regularization, analyzing the degraded model and the ill-posed character of image restoration, this paper mainly discusses the regularized parameter and regularized item, and the solution approaches and fast algorithms are also summarized, therefore perfects the essential theories about regularization in image restoration.
Secondly, to perfect the known restoring models, a new space-adaptive regularization model of image restoration is constructed by redesigning regularized parameter and regularized item. By contriving appropriate algorithms in the computations, the regularized parameter can automatically correct to the appropriate value due to its adaptive character, meanwhile, the adaptive effect is developed with a weight matrix included in the regularized item. Simulation results show that the ringing artifacts around the image are reduced, the main information is preserved, and the Peak Signal to Noise Ratio (PSNR) is higher than the known methods.
Thirdly, two new image restoration methods are proposed based on regularization idea from traditional linear algebra approach. On one hand, from the technique of constrained least squares and limited energy of additive noise, an effective restored approach by adopting regularization method to overcoming ill-posed problem, solving an equation with a single variable, and using space iterative algorithm is proposed; On the other hand, aiming at the restoration of blurred image, another effective restoration approach based on least-square algorithm is also proposed in this paper. This method firstly adopts increment iterative algorithm to improve convergence and meanwhile applies regularization technique to overcome ill-posed problem. In the computations, the regularized parameter has its adaptive character, which can be determined in terms of the restored image at each iteration step therefore automatically correct to the appropriate value. Numerical results show that these methods can effectively restore original image, and the objective standard evaluation and subjective visual effect are improved significantly.
引文
[1] Kenneth.R.Castleman(著),朱志刚等译,数字图像处理,北京:电子工业出版社,2002
[2] 阮秋奇,数字图像处理学,北京:电子工业出版社,2001
[3] 孙即祥,数字图像处理,石家庄:河北教育出版社,1993
[4] 章毓晋,图像处理和分析,北京:清华大学出版社,1999
[5] Digital Image Restoration, H.C.Andrews, B.R.Hunt, Englewood Cliffs, NJ: Prentice-Hall, 1977
[6] Katsaggelos A K, Digital Image Restoration, Berlin, Germany: Springer-Verlag, 1991
[7] 邹谋炎,反卷积和信号复原,北京:国防工业出版社,2001
[8] 肖庭延,于慎根,王延飞,反问题的数值解法,北京:科学出版社,2003
[9] J.L.Starck, E.Pantin, Deconvolution in astronomy: a review[C], PASP, 2002, October, 114: 1051~1069
[10] Vladimir Z. Mesarovic, Nikolas P. Galatsanos, Aggelos K. Katsaggelos, Regularized Constrained Total Least Squares Image Restoration, IEEE Transactions on Image Processing, Vol.4, No.8, August 1995: 1096~1108
[11] Chan T.F., Chui K wang Wong, Total variation blind deconvolution[J], IEEE Trans on Image Processing, 1998, 7(3): 370~375
[12] Georgios B. Giannakis, Robert W. Heath, Jr., Blind identification of multichannel FIR blurs and perfect image restoration[J], IEEE Trans on Image Processing, 2000, 9(11): 1877~1896
[13] 薛梅,邹采荣,杨娟,杨绿溪,一种空间自适应正则化图像盲复原算法,中国图象图形学报,2002,7(4):356～362
[14] 张航,罗大勇,图像盲复原算法研究现状及其展望,中国图象图形学报,2004,9(10):1145～1152
[15] Geman D, Yang C, Nonlinear image recovery with half-quadratic regularization, IEEE Trans on Image Processing, 1995, 4(7): 936~946
[16] Charbonnier P, Blance-Feraud L, Aubert G, Deterministic edge-preserving regularization in computed imaging, IEEE Trans on Image Processing, 1997, 6(2): 298~311
[17] Vogel CR, Oman ME. Fast, Robust total variation-based reconstruction of noisy, blurred images, IEEE Trans on Image Processing, 1998, 7(6): 813~824
[18] Mallat S, A wavelet Tour of Signal Processing, San Diego: Academic Press, 1998
[19] 蔡汉添,张军萍,一种基于小波变换的迭代正则化图像恢复算法,中国图象图形学报,1999,4(3):229～232
[20] 曾三友,丁立新,康立山,一种具有噪声估计能力的图像恢复正则化方法,武汉大学学报(理学版),2002,48(3):311～314
[21] Belge M, Kilmer ME, Miller EL, Wavelet domain image restoration with adaptive edge-preserving regularization. IEEE Trans on Image Processing, 2000, 9(4): 597~608
[22] 赵书斌,彭思龙,基于小波域 HMT 模型的图像超分辨重构,计算机辅助设计与图形学学报,2003,11(3):1347～1352
[23] 汪雪林,韩华,彭思龙,基于小波域局部高斯模型的图像复原,软件学报,2004,15(3):443～450
[24] Tikhonov A N, Arsenin V Y. Solutions of Ill-posed Problems, New York: Wiley, 1977
[25] Lagendijk R L, Biemond J, Boekee D E, Regularized iterative image restoration with ringing reduction, IEEE Trans on Acoust. Speech Signal Processing, 1988, 36: 1984~1888
[26] F.Malgouyres, Minimizing the Total Variation Under a General Convex Constraint for Image Restoration, IEEE Trans.on Image Processing, 2002, 11(12): 1450~1455
[27] P.Charbonnier, L.Blanc-Feraud, G.Aubert, M.Barlaud, Deterministic Edge-Preserving Regularization in Computed Imaging, IEEE Trans on Image Processing, 1997, 6(2): 298~311
[28] N.P.Galatsanos, A.K.Katsaggelos, Methods for Choosing the Regularization Parameter and Estimating the Noise Variance in Image Restoration and Their Relation, IEEE Trans on Image Processing, 1992, 1(3): 322~336
[29] M.G.Kang and A.K.Katsaggelos, General Choice of the Regularization Functional in Regularized Image Restoration, IEEE Trans. on Image Processing, 1995, 14(5): 594~602
[30] 余国华,田岩,一种新型的图像恢复算法,红外与激光工程,2004,33(1):34～37
[31] Plato R. Optimal Algorithms for Linear Ill-Posed Problems Yield Regularization Methods, Numer. Funct. Anal. Optim, 1990, 11: 111~118
[32] Brakhage H, On Ill-Posed Problems and the Method of Conjugate Gradients, In: Engl H W and Groetsch C W(eds): Inverse and Ill-Posed Problems, Academic Press, Boston, 1987,165~175
[33] Gfrerer H, An A-Posteriori Parameter Choice for Ordinary and Iterated Tikhonov Regularization of Ill-Posed Problems Leading to Optimal Convergence Rates, Mathematics of Computation, 1987, 49(180):507~522
[34] Moon Gi kang and Aggelos K, General Choice of the Regularization Functional in Regularized Image Restoration, IEEE Transactions on Image Processing, 1995, 14(5): 594~602
[35] Wufan Chen, Ming Chen and Jie Zhou, Adaptively Regularized Constrained Total Least- Squares Image Restoration, IEEE Transactions on Image Processing, 2000, 19(4): 588~596
[36] 周杰, 陈明, 陈武凡, RCTLS 图像恢复中局部线化的优化算法以及正则化参数的自适应选择, 计算机学报(增刊),1998,21: 341～346
[37] 陈武凡,李超,陈和晏,空域中退化图像恢复的有效算法,计算机学报,1999,22(12):1267～1271
[38] 王光新,王正明,段晓君,基于广义高斯噪声分布模型的迭代正则化图像复原,中国图象图形学报,2004,9(8):978～983
[39] Hong M C, Stathaki T, Katsaggelos A K, Iterative regularized least-mean mixed-norm image restoration[J], Optical Engineering, 2002, 41(10): 2515~2524
[40] 刘扬阳,金伟其,苏秉华,数字图像去模糊处理算法的对比研究,北京理工大学学报,2004,24(10):905～909
[41] 余昕,杨绿溪,邹采荣,基于确定性约束和局部空间自适应正则化的图像盲复原算法,数据采集与处理,2002,17(2):121～125
[42] You Y, Kaveh M, A regularization approach to joint blur identification and image restoration[J], IEEE Trans on Image Processing, 1996, 5(3): 416~428
[43] Lagendijk R L, Biemond J, Iterative Identification and Restoration of Images, Kluwer Academic Publishers, 1991
[44] Aghadasi F, Ward R K, Reduction of boundary artifacts in image restoration, IEEE Trans on Image Processing, 1996, 5(4): 611~618
[45] Charalambous C, Ghaddar F K, Kouris K, Tow iterative image restoration algorithms with application to nuclear medicine, IEEE Trans. on Medical Imaging, 1992, 11(1): 2~8
[46] Jeong H-S, Jung H-J, Joon K-P, Regularized iterative image interpolation and its application to spatially scalable coding[J], IEEE Trans on Consumer Electronics, 1998, 44(3): 1042~1047
[47] 周宏潮,基于稀疏参数模型及参数先验的图像分辨率增强方法研究,长沙:国防科技大学博士论文,2005
[48] 罗忠,朱重光,正则化数字图像盲恢复的一种新方法,测试技术学报,1999,13(2):67～75
[49] Teboul S, Blanc-Feraud L, Aubert G, Variational approach for edge-preserving regularization using coupled PDE’s, IEEE Trans on Image Processing, 1998, 7(3): 387~397
[50] 曾三友,康立山,丁立新,黄元江,一种基于正则化方法的准最佳图像复原技术,软件学报,2003,14(3):689～696
[51] Ramesh Neelamani, Hyeokho Choi, Richard Baraniuk, Fourier-Wavelet Regularized Deconvolution for Ill-Conditioned Systems, IEEE Trans on Signal Processing, 2004, 52(2): 418~433
[52] 谢美华,基于偏微分方程模型的图像去噪与分辨率增强技术研究,长沙:国防科技大学博士论文,2005
[53] 刘志军,丁明跃,周成平,刘买利,基于并行遗传算法的图像超分辨复原,中国图象图形学报,2004,9(1):62～68
[54] 王岩,梁甸农,郭汉伟,基于改进正则化方法的 SAR 图像增强技术,电子学报,2003,31(9):1307～1309
[2] 阮秋奇,数字图像处理学,北京:电子工业出版社,2001
[3] 孙即祥,数字图像处理,石家庄:河北教育出版社,1993
[4] 章毓晋,图像处理和分析,北京:清华大学出版社,1999
[5] Digital Image Restoration, H.C.Andrews, B.R.Hunt, Englewood Cliffs, NJ: Prentice-Hall, 1977
[6] Katsaggelos A K, Digital Image Restoration, Berlin, Germany: Springer-Verlag, 1991
[7] 邹谋炎,反卷积和信号复原,北京:国防工业出版社,2001
[8] 肖庭延,于慎根,王延飞,反问题的数值解法,北京:科学出版社,2003
[9] J.L.Starck, E.Pantin, Deconvolution in astronomy: a review[C], PASP, 2002, October, 114: 1051~1069
[10] Vladimir Z. Mesarovic, Nikolas P. Galatsanos, Aggelos K. Katsaggelos, Regularized Constrained Total Least Squares Image Restoration, IEEE Transactions on Image Processing, Vol.4, No.8, August 1995: 1096~1108
[11] Chan T.F., Chui K wang Wong, Total variation blind deconvolution[J], IEEE Trans on Image Processing, 1998, 7(3): 370~375
[12] Georgios B. Giannakis, Robert W. Heath, Jr., Blind identification of multichannel FIR blurs and perfect image restoration[J], IEEE Trans on Image Processing, 2000, 9(11): 1877~1896
[13] 薛梅,邹采荣,杨娟,杨绿溪,一种空间自适应正则化图像盲复原算法,中国图象图形学报,2002,7(4):356～362
[14] 张航,罗大勇,图像盲复原算法研究现状及其展望,中国图象图形学报,2004,9(10):1145～1152
[15] Geman D, Yang C, Nonlinear image recovery with half-quadratic regularization, IEEE Trans on Image Processing, 1995, 4(7): 936~946
[16] Charbonnier P, Blance-Feraud L, Aubert G, Deterministic edge-preserving regularization in computed imaging, IEEE Trans on Image Processing, 1997, 6(2): 298~311
[17] Vogel CR, Oman ME. Fast, Robust total variation-based reconstruction of noisy, blurred images, IEEE Trans on Image Processing, 1998, 7(6): 813~824
[18] Mallat S, A wavelet Tour of Signal Processing, San Diego: Academic Press, 1998
[19] 蔡汉添,张军萍,一种基于小波变换的迭代正则化图像恢复算法,中国图象图形学报,1999,4(3):229～232
[20] 曾三友,丁立新,康立山,一种具有噪声估计能力的图像恢复正则化方法,武汉大学学报(理学版),2002,48(3):311～314
[21] Belge M, Kilmer ME, Miller EL, Wavelet domain image restoration with adaptive edge-preserving regularization. IEEE Trans on Image Processing, 2000, 9(4): 597~608
[22] 赵书斌,彭思龙,基于小波域 HMT 模型的图像超分辨重构,计算机辅助设计与图形学学报,2003,11(3):1347～1352
[23] 汪雪林,韩华,彭思龙,基于小波域局部高斯模型的图像复原,软件学报,2004,15(3):443～450
[24] Tikhonov A N, Arsenin V Y. Solutions of Ill-posed Problems, New York: Wiley, 1977
[25] Lagendijk R L, Biemond J, Boekee D E, Regularized iterative image restoration with ringing reduction, IEEE Trans on Acoust. Speech Signal Processing, 1988, 36: 1984~1888
[26] F.Malgouyres, Minimizing the Total Variation Under a General Convex Constraint for Image Restoration, IEEE Trans.on Image Processing, 2002, 11(12): 1450~1455
[27] P.Charbonnier, L.Blanc-Feraud, G.Aubert, M.Barlaud, Deterministic Edge-Preserving Regularization in Computed Imaging, IEEE Trans on Image Processing, 1997, 6(2): 298~311
[28] N.P.Galatsanos, A.K.Katsaggelos, Methods for Choosing the Regularization Parameter and Estimating the Noise Variance in Image Restoration and Their Relation, IEEE Trans on Image Processing, 1992, 1(3): 322~336
[29] M.G.Kang and A.K.Katsaggelos, General Choice of the Regularization Functional in Regularized Image Restoration, IEEE Trans. on Image Processing, 1995, 14(5): 594~602
[30] 余国华,田岩,一种新型的图像恢复算法,红外与激光工程,2004,33(1):34～37
[31] Plato R. Optimal Algorithms for Linear Ill-Posed Problems Yield Regularization Methods, Numer. Funct. Anal. Optim, 1990, 11: 111~118
[32] Brakhage H, On Ill-Posed Problems and the Method of Conjugate Gradients, In: Engl H W and Groetsch C W(eds): Inverse and Ill-Posed Problems, Academic Press, Boston, 1987,165~175
[33] Gfrerer H, An A-Posteriori Parameter Choice for Ordinary and Iterated Tikhonov Regularization of Ill-Posed Problems Leading to Optimal Convergence Rates, Mathematics of Computation, 1987, 49(180):507~522
[34] Moon Gi kang and Aggelos K, General Choice of the Regularization Functional in Regularized Image Restoration, IEEE Transactions on Image Processing, 1995, 14(5): 594~602
[35] Wufan Chen, Ming Chen and Jie Zhou, Adaptively Regularized Constrained Total Least- Squares Image Restoration, IEEE Transactions on Image Processing, 2000, 19(4): 588~596
[36] 周杰, 陈明, 陈武凡, RCTLS 图像恢复中局部线化的优化算法以及正则化参数的自适应选择, 计算机学报(增刊),1998,21: 341～346
[37] 陈武凡,李超,陈和晏,空域中退化图像恢复的有效算法,计算机学报,1999,22(12):1267～1271
[38] 王光新,王正明,段晓君,基于广义高斯噪声分布模型的迭代正则化图像复原,中国图象图形学报,2004,9(8):978～983
[39] Hong M C, Stathaki T, Katsaggelos A K, Iterative regularized least-mean mixed-norm image restoration[J], Optical Engineering, 2002, 41(10): 2515~2524
[40] 刘扬阳,金伟其,苏秉华,数字图像去模糊处理算法的对比研究,北京理工大学学报,2004,24(10):905～909
[41] 余昕,杨绿溪,邹采荣,基于确定性约束和局部空间自适应正则化的图像盲复原算法,数据采集与处理,2002,17(2):121～125
[42] You Y, Kaveh M, A regularization approach to joint blur identification and image restoration[J], IEEE Trans on Image Processing, 1996, 5(3): 416~428
[43] Lagendijk R L, Biemond J, Iterative Identification and Restoration of Images, Kluwer Academic Publishers, 1991
[44] Aghadasi F, Ward R K, Reduction of boundary artifacts in image restoration, IEEE Trans on Image Processing, 1996, 5(4): 611~618
[45] Charalambous C, Ghaddar F K, Kouris K, Tow iterative image restoration algorithms with application to nuclear medicine, IEEE Trans. on Medical Imaging, 1992, 11(1): 2~8
[46] Jeong H-S, Jung H-J, Joon K-P, Regularized iterative image interpolation and its application to spatially scalable coding[J], IEEE Trans on Consumer Electronics, 1998, 44(3): 1042~1047
[47] 周宏潮,基于稀疏参数模型及参数先验的图像分辨率增强方法研究,长沙:国防科技大学博士论文,2005
[48] 罗忠,朱重光,正则化数字图像盲恢复的一种新方法,测试技术学报,1999,13(2):67～75
[49] Teboul S, Blanc-Feraud L, Aubert G, Variational approach for edge-preserving regularization using coupled PDE’s, IEEE Trans on Image Processing, 1998, 7(3): 387~397
[50] 曾三友,康立山,丁立新,黄元江,一种基于正则化方法的准最佳图像复原技术,软件学报,2003,14(3):689～696
[51] Ramesh Neelamani, Hyeokho Choi, Richard Baraniuk, Fourier-Wavelet Regularized Deconvolution for Ill-Conditioned Systems, IEEE Trans on Signal Processing, 2004, 52(2): 418~433
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