基于偏微分方程图像处理的不对称差分数值方法
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摘要
图像处理的偏微分方程办法是一个新兴交叉学科分支,对于它的数值方法研究有重要的理论意义和实用价值。本学位论文针对图像处理中的几类经典的偏微分方程模型：中值曲率驱动方程(MCM方程)、仿射形态学尺度空间方程(AMSS方程)、非线性扩散(P-M)模型、全变差(TV)模型、测地线活动轮廓(GAC)模型,构造了基于上述模型的无条件稳定显式差分格式——不对称差分格式,将半隐式格式显式计算,采用线性化稳定性分析方法对该格式进行数值稳定性分析,给出格式的稳定性证明,讨论了格式的计算效率。
理论分析及数值实验表明,不对称差分格式在进行图像处理时,有效地平滑了噪声,提高了去噪效率,保持了边缘信息同时又加快了分割速度,与已有方法相比,是一种高效可行的数值方案。
Image processing based on the partial differential equation(PDE) is an emerging interdisciplinary branch, and its study of numerical methods has important theoretical significance and practical value. This paper constructs a new unconditional stability difference schemes——asymmetric difference scheme for several kinds of classical model:mean curvature motion(MCM equation)、affine morphological scale space(AMSS equation)、nonlinear diffusion (P-M)model、total variation (TV)model、geodesic active contour (GAC)model, analyzes the stability of these schemes using linear stability analysis and proposes the stability proof, discusses the computational efficiency of these schemes.
Theoretical analysis and numerical experiments show that, using the asymmetric difference scheme in image process, image can be smoothed better and improve the efficiency of denoising, maintain the edge information and enhances the segmentation speed. Compared with the existing methods, the asymmetric difference method is a feasible and efficient numerical scheme.
理论分析及数值实验表明,不对称差分格式在进行图像处理时,有效地平滑了噪声,提高了去噪效率,保持了边缘信息同时又加快了分割速度,与已有方法相比,是一种高效可行的数值方案。
Image processing based on the partial differential equation(PDE) is an emerging interdisciplinary branch, and its study of numerical methods has important theoretical significance and practical value. This paper constructs a new unconditional stability difference schemes——asymmetric difference scheme for several kinds of classical model:mean curvature motion(MCM equation)、affine morphological scale space(AMSS equation)、nonlinear diffusion (P-M)model、total variation (TV)model、geodesic active contour (GAC)model, analyzes the stability of these schemes using linear stability analysis and proposes the stability proof, discusses the computational efficiency of these schemes.
Theoretical analysis and numerical experiments show that, using the asymmetric difference scheme in image process, image can be smoothed better and improve the efficiency of denoising, maintain the edge information and enhances the segmentation speed. Compared with the existing methods, the asymmetric difference method is a feasible and efficient numerical scheme.
引文
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[9]朱春香,种兰祥,王大凯.平均曲率运动方程的三种实现方案的比较研究.计算机应用与软件,2008,25(6)：241～243
[10]P.Smereka.Semi—implicit level set methods for curvature and surface diffusion mo tion[J]. Journal of Scientific Computing,2003,19:1～3
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[18]MallatS著.陈刚,张亶,池哲儒译.应用数学和信号处理相遇.高校应用数学学报,1999,14(3)：350-364
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[21]Galvanin E A S,Dovale G M,DalPoz A P.The canny detector with edge region focusing using an anisotropic diffusion process[J].Pattern Recognition and Image Analysis,2006,16(4):614～621
[22]袁剑泽.PDE's在图像处理中的应用——2000年总结报告.西安交通大学人工智能与机器人研究所,2000
[23]Alvarez L,Guichard F,Lions P L,et al.Axioms and fundamental equation of image processing[J].Archive for rational Mechanics and Analusis,1993,16(9):200～257
[24]Rudin L,Osher S.Total variation based image restoration with free local constraints [A].in Proceedings IEEE Internat Conf Imag Proc,Austin:TX,IEEE Press,Piscataway,NJ,1994:31～35
[25]刘嘉,王宏琦.一种基于图割的交互式图像分割方法[J].电子与信息学报,2008,30(8)：1973-1976
[26]Chen Da-li,Xue Ding-yu,Pan Feng.An improved media filter based on automatic parameter tuning approach[C]//IEEE International Conference Proceedings on Mechatronics and Automation,2007:1305～1309
[27]李翠华,郑南宁.构造径向基函数的一般方法及其在图像处理中的应用.数 值计算与计算机应用,2000,(2)：81-87
[28]Kumar M,Stoll N,Kaber D,et al.Fuzzy filtering for an intelligent interpretation of medical data[C]//IEEE International Conference on Automation Science and Engineering,2007:225～230
[29]You Yu-Li,Kaveh D.Fourth-order partial differential equations for noise removal[J].IEEE Trans Image Processing 2000,9(10):1723-1730
[30]Galvanin E. A. S.,Do Vale G. M. and Dal Poz A. P.The canny detector with edge region focusing using an isotropic diffusion process[J].Pattren Recognition and Image Analysis,2006,16(4):614～621
[31]W.Joachim,M.bart.Viergever,et,al. An efficient and reliable scheme for nonlinear diffusion filtering[J].IEEE Transactions on image Processing,1998,7(3):398～ 410
[32]A.Ichigaya,M.Kurozumi,N.Hara,et.al.A method of estimating coding PSNR using quantized DCT Coefficients[J].IEEE Transactions on Circuits and Systems for Video Technology,2006,16(2):251～259
[33]Lysaker M,Stanley Osher,Tai Xue-cheng.Noise removal using smoothed normals and surface fitting[J].IEEE Transactions on Image Procedding,2004,13(10): 1345～1457
[34]Chen K.Adaptive smoothing via contextual and local discontinuities.IEEE Trans. Pattern Analysis and Machine Intelligence,2005,27(10):1552～1567
[35]J.K.Hawkins.Texture Properties for Pattern Recognition,in Picture Processing and Psychopiclorics,S. Kipkin and A.Rosenfeld Eds, Academic Press,New York,1980
[36]Paragios N,Mellina-Gottardo o,Ramesh V. Gradient vector flow fast geometric active contours[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,2004, 26(3):402～407
[37]Joachim Weicket,Coherence-enhancing diffusion filtering, International Journal of Computer Vision,April 1999,Vol.31,No.2/3:111～127
[38]Vese L,Osher S.Modeling textures with total variation minimization and oscillating patterns in image processing. Journal of Scientific Computing,2003,19:553～ 572
[39]Jia D Y,Huang F G,WEN X F.Fourth-order anisotropic diffusion equation for noise removal,ICSP,Beijing,China,2004,2:994-997
[40]Black M J,Sapro G,Marmont D H,et al.Robust anisotropic diffusion[J].IEEE Transactions on Image Processing 1998,7(3):421-432
[41]Catte F,Lions P L,Morel JM,et al.Image selective smoothing and edge detection by nonolinear diffusion[J].SIAM Journal on Numerical Analysis,1992,29(1): 82-193
[42]Galvanin E A S,Dovale G M,DalPoz A P.The canny detector with edge region focusing using an anisotropic diffusion process[J].Pattern Recognition and Image Analysis,2006,16(4):614～621
[2]王大凯,侯榆青,彭建业.图像处理的偏微分方程方法[M].北京：科学出版社,2008
[3]Perona P,Malik J.Scale-space and edge detection using anisotropic diffusion[J].IEEE Trans on Patt Anal and Mach Intell,1990,12(7):629-639
[4]冯象初,王卫卫.图像处理的变分和偏微分方程方法[M].科学出版社,2009
[5]吴斌,吴亚东.张红英.基于变分偏微分方程的图像复原技术[M].北京大学出版社,2008
[6]Rudin L,Osher S,Fatime E.Nonlinear total variation bases noise removal algorithm.Physisca D[J].1992,60:259～268
[7]Caselles V,Morel J M,Sapiro G.Geodesic active contours.Int.J.Comput. Vision, 1997,22:61～79
[8]张宜,陈刚.基于偏微分方程的图像处理[M].北京：高等教育出版社,2004
[9]朱春香,种兰祥,王大凯.平均曲率运动方程的三种实现方案的比较研究.计算机应用与软件,2008,25(6)：241～243
[10]P.Smereka.Semi—implicit level set methods for curvature and surface diffusion mo tion[J]. Journal of Scientific Computing,2003,19:1～3
[11]张文生.科学计算中的偏微分方程有限差分法[M].高等教育出版社,2006
[12]Peaceman D W,Rachford H H.The numerical solution of parabolic and elliptic differential equations.Journal Soc.Ind. Appl. Math,1995,3:28
[13]吴亚东,孙世新,张红英,等.一种基于图割的全变差图像去噪算法.电子学报,2007,35(2)：265-266
[14]熊保平,杜民.基于PDE图像去噪方法.计算机应用,2007,27(8)：2024-2029
[15]Marquina A,Osher S.Explicit algorithm for a new time dependent model based on level set motom for nonlinear deblurring and noise removel. SIAM.Sci.Comput.2000, 22:387-405
[16]Xu Jing, Shi Yuying and Wang Xiaoyu.Extrapolation Boundary Conditions for Total Variation-based Image Deblurring and Denoising.Chinese Journal of Computational Physics,2007,24(3):34～41
[17]Shi Yuying,Chang Qianshun.A Model for Image Deblurring and Denoising Based on the Rudin_Osher_Fatemi Model.Chinese Journal of Computational Physics,2006,23(5):551～558
[18]MallatS著.陈刚,张亶,池哲儒译.应用数学和信号处理相遇.高校应用数学学报,1999,14(3)：350-364
[19]Zhang Guangxin,Cai Hui,Cai Jinhui,Zhou Zekui. Study on Multi-Scale AMSS Operator for Fabric Flaw Detection. Chinese Journal of Sensors and Actuators,2006, 19(3):709～713
[20]祝轩,周明全等,曲率驱动与边缘停止相结合的非线性扩散及其在图像去噪中的应用,光子学报,2008,37(3)：609-612
[21]Galvanin E A S,Dovale G M,DalPoz A P.The canny detector with edge region focusing using an anisotropic diffusion process[J].Pattern Recognition and Image Analysis,2006,16(4):614～621
[22]袁剑泽.PDE's在图像处理中的应用——2000年总结报告.西安交通大学人工智能与机器人研究所,2000
[23]Alvarez L,Guichard F,Lions P L,et al.Axioms and fundamental equation of image processing[J].Archive for rational Mechanics and Analusis,1993,16(9):200～257
[24]Rudin L,Osher S.Total variation based image restoration with free local constraints [A].in Proceedings IEEE Internat Conf Imag Proc,Austin:TX,IEEE Press,Piscataway,NJ,1994:31～35
[25]刘嘉,王宏琦.一种基于图割的交互式图像分割方法[J].电子与信息学报,2008,30(8)：1973-1976
[26]Chen Da-li,Xue Ding-yu,Pan Feng.An improved media filter based on automatic parameter tuning approach[C]//IEEE International Conference Proceedings on Mechatronics and Automation,2007:1305～1309
[27]李翠华,郑南宁.构造径向基函数的一般方法及其在图像处理中的应用.数 值计算与计算机应用,2000,(2)：81-87
[28]Kumar M,Stoll N,Kaber D,et al.Fuzzy filtering for an intelligent interpretation of medical data[C]//IEEE International Conference on Automation Science and Engineering,2007:225～230
[29]You Yu-Li,Kaveh D.Fourth-order partial differential equations for noise removal[J].IEEE Trans Image Processing 2000,9(10):1723-1730
[30]Galvanin E. A. S.,Do Vale G. M. and Dal Poz A. P.The canny detector with edge region focusing using an isotropic diffusion process[J].Pattren Recognition and Image Analysis,2006,16(4):614～621
[31]W.Joachim,M.bart.Viergever,et,al. An efficient and reliable scheme for nonlinear diffusion filtering[J].IEEE Transactions on image Processing,1998,7(3):398～ 410
[32]A.Ichigaya,M.Kurozumi,N.Hara,et.al.A method of estimating coding PSNR using quantized DCT Coefficients[J].IEEE Transactions on Circuits and Systems for Video Technology,2006,16(2):251～259
[33]Lysaker M,Stanley Osher,Tai Xue-cheng.Noise removal using smoothed normals and surface fitting[J].IEEE Transactions on Image Procedding,2004,13(10): 1345～1457
[34]Chen K.Adaptive smoothing via contextual and local discontinuities.IEEE Trans. Pattern Analysis and Machine Intelligence,2005,27(10):1552～1567
[35]J.K.Hawkins.Texture Properties for Pattern Recognition,in Picture Processing and Psychopiclorics,S. Kipkin and A.Rosenfeld Eds, Academic Press,New York,1980
[36]Paragios N,Mellina-Gottardo o,Ramesh V. Gradient vector flow fast geometric active contours[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,2004, 26(3):402～407
[37]Joachim Weicket,Coherence-enhancing diffusion filtering, International Journal of Computer Vision,April 1999,Vol.31,No.2/3:111～127
[38]Vese L,Osher S.Modeling textures with total variation minimization and oscillating patterns in image processing. Journal of Scientific Computing,2003,19:553～ 572
[39]Jia D Y,Huang F G,WEN X F.Fourth-order anisotropic diffusion equation for noise removal,ICSP,Beijing,China,2004,2:994-997
[40]Black M J,Sapro G,Marmont D H,et al.Robust anisotropic diffusion[J].IEEE Transactions on Image Processing 1998,7(3):421-432
[41]Catte F,Lions P L,Morel JM,et al.Image selective smoothing and edge detection by nonolinear diffusion[J].SIAM Journal on Numerical Analysis,1992,29(1): 82-193
[42]Galvanin E A S,Dovale G M,DalPoz A P.The canny detector with edge region focusing using an anisotropic diffusion process[J].Pattern Recognition and Image Analysis,2006,16(4):614～621