基于PDE的新图像扩散模型及分类扩散
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摘要
偏微分方程是近三十年新兴的重要图像处理工具,为以数字图像处理为基础的众多领域注入了新的活力。图像复原是图像处理的重要组成部分,其目的是提高退化图像的质量,使其更加的清晰。图像的细节信息存在于图像的边缘部分,噪声可以分布于图像任何位置,噪声又与细节在频域内相混叠等等,使得滤除噪声和保护边缘成为一种矛盾。传统的方法较难处理这类问题,而新兴的偏微分工具提出的边缘保护思想及各向异性扩散等方法均为保护细节提供新的思路。
本文以图像抑噪为主要研究方向,分别从滤波模型、滤波方法及滤波思路三个方向出发,得到以下三个成果：
1.针对传统的滤波模型-PM模型存在逆扩散现象,提出一种适定的扩散方程。PM模型中的传导因子对边缘过于敏感,造成模型在边缘处出现逆扩散现象。通过更改模型的传导因子,降低其在边缘处的敏感性,进而得到一种适定的滤波方程。
2.提出三种图像特征扩散模型。三种模型均沿着图像特征方向进行扩散,这种扩散思路即使没有任何的边缘保护措施下依然会有效的避免破坏边缘信息,保护图像的细节部分。
3.提出分类扩散的思想。传统的图像扩散思路为平滑扩散。使用平滑抑噪思路使得边缘噪声较难滤除,影响图像的平滑性。分类扩散分为抑噪和平滑两部分,抑噪的主要是降低边缘噪声的噪声幅度,使其变为小噪声,而平滑滤波主要处理小噪声,平滑图像。从而既能有效的滤除图像的噪声,又能保证图像的光滑性。
Partial differential equation, as a new tool of image restoration in pasted thirty years, has injected new vitality to many image-processing-related fields. Image restoration is one of the most important parts of image processing, which is trying to turn the degraded images to clear high-quality pictures. As we know that, image information distribute in edge, noise can be anywhere, the two situations make denoising and details-protect to be a contradiction. However, traditional filters cannot deal with this problem. With the idea of edge protected and anisotropic diffusion introduce by PDEs, this contradiction can be well resolved.
This thesis is only focus on denoising problem, involved the restoration models, denoising methods and idea of noise removal three aspects. The main original contributions of this thesis are summarized as follows:
1. The classic model-PM model can cause reverse diffusion, which make the model instable. After analyzed the diffusion coefficient function of Perona&Malik (P-M) model, too sensitive at edge is the main reason for P-M model's ill-posed. Some amendments have done to the function, we got a posed diffusion model.
2. Proposed three discrete methods for a anisotropic diffusion operator. Anosotropic diffusion is based on the direction of image features. To avoid the damage, the smoothing has to be controlled one principle: minimal smoothing in the directions across the image features, and maximal smoothing in the directions along the image features.
3. Introduce a new diffusion way:classified diffusion. Smoothing is the only way that traditional diffusion to remove noise. However, marginal noise cannot be well smoothed, as a result, the smoothness of image cannot be improved a lot. Classified diffusion can be divided into two parts, noise-reduction and smoothing. The part of noise-reduction is to reduce the extent of marginal noise, that to be a minor noise. The other part is to remove minor noise, smoothing the image. Under two parts work, the image can be processed into a clear picture.
本文以图像抑噪为主要研究方向,分别从滤波模型、滤波方法及滤波思路三个方向出发,得到以下三个成果：
1.针对传统的滤波模型-PM模型存在逆扩散现象,提出一种适定的扩散方程。PM模型中的传导因子对边缘过于敏感,造成模型在边缘处出现逆扩散现象。通过更改模型的传导因子,降低其在边缘处的敏感性,进而得到一种适定的滤波方程。
2.提出三种图像特征扩散模型。三种模型均沿着图像特征方向进行扩散,这种扩散思路即使没有任何的边缘保护措施下依然会有效的避免破坏边缘信息,保护图像的细节部分。
3.提出分类扩散的思想。传统的图像扩散思路为平滑扩散。使用平滑抑噪思路使得边缘噪声较难滤除,影响图像的平滑性。分类扩散分为抑噪和平滑两部分,抑噪的主要是降低边缘噪声的噪声幅度,使其变为小噪声,而平滑滤波主要处理小噪声,平滑图像。从而既能有效的滤除图像的噪声,又能保证图像的光滑性。
Partial differential equation, as a new tool of image restoration in pasted thirty years, has injected new vitality to many image-processing-related fields. Image restoration is one of the most important parts of image processing, which is trying to turn the degraded images to clear high-quality pictures. As we know that, image information distribute in edge, noise can be anywhere, the two situations make denoising and details-protect to be a contradiction. However, traditional filters cannot deal with this problem. With the idea of edge protected and anisotropic diffusion introduce by PDEs, this contradiction can be well resolved.
This thesis is only focus on denoising problem, involved the restoration models, denoising methods and idea of noise removal three aspects. The main original contributions of this thesis are summarized as follows:
1. The classic model-PM model can cause reverse diffusion, which make the model instable. After analyzed the diffusion coefficient function of Perona&Malik (P-M) model, too sensitive at edge is the main reason for P-M model's ill-posed. Some amendments have done to the function, we got a posed diffusion model.
2. Proposed three discrete methods for a anisotropic diffusion operator. Anosotropic diffusion is based on the direction of image features. To avoid the damage, the smoothing has to be controlled one principle: minimal smoothing in the directions across the image features, and maximal smoothing in the directions along the image features.
3. Introduce a new diffusion way:classified diffusion. Smoothing is the only way that traditional diffusion to remove noise. However, marginal noise cannot be well smoothed, as a result, the smoothness of image cannot be improved a lot. Classified diffusion can be divided into two parts, noise-reduction and smoothing. The part of noise-reduction is to reduce the extent of marginal noise, that to be a minor noise. The other part is to remove minor noise, smoothing the image. Under two parts work, the image can be processed into a clear picture.
引文
[1]Aubert G, Kornprobst P. Mathematical Problems in Image Processing [M]. New York: Springer-Verlag.2002.
[2]邹谋炎。反卷积和信号复原[M].北京：国防工业出版社,2001.
[3]Sonka M, Hlavac V, Boyle R. Image Processing, Analysis and Machine Vison[M], PWS Publishing,1999.
[4]张亶,陈刚。基于偏微分方程的图象处理[M].北京：高等教育出版社.2004.
[5]Caselles V, Morel J M, Sapiro GS et al. Introduction to the special issue on partial differential equations and geometry-driven diffusion in image processing and analysis [J]. IEEE Transactions on Image Processing,1998,7(3):269-273.
[6]Witkin A P. Scale-space filtering, In Proceedings 8th International Joint Conference on Artificial Intelligence[J],1983,2:1019-1021.
[7]Koenderink J J. The structure of images[J]. Biological Cybernetics,1984,50:363-370.
[8]Hummel R A. Representations based on zero-crossings in scale-space[J]. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,1986,204-209.
[9]Perona P and Malik J. Scale-Space and edge detection using Anisotropic Diffusion [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,1990,12(07):629-639.
[10]Scherzer O, Weickert J. Relations between Regularization and Diffusion Filtering [J]. Journal of Mathematical Imaging and Vision,2000,12:43-63.
[11]Aubert G, Kornprobst P. Mathematical problems in Image Processing [M]. Berlin: Springer-Verlag,2001.
[12]Catte F, Lions P L, Morel J M. Image selective smoothing and edge detection by nonlinear diffusion [J]. SIAM J. Numer. Anal,1992,129:182-193.
[13]Alvarez L, Mazorra L. Signal and image restoration using shock filters and anisotropic diffusion [J]. SIAM Journal on Numerical Analysis,1994,31(02):590-605.
[14]Weickert J, Schnorr C. A Theoretical Framework for Convex Regularizers in PDE-Based Computation of Image Motion [J]. International Journal of Computer Vision,2001,45(3): 245-264.
[15]Gilboa G, Sochen N A, Zeevi Y Y. Forward-and- backward diffusion processes for adaptive image enhancement and denoising [J]. IEEE Transactions on Image Processing,2002,11(7): 689-703.
[16]Gilboa G, Zeevi Y Y, Sochen N A. Complex diffusion processes for image filtering [J]. Lecture Notes in Computer Science,2001,21(06):293-299.
[17]Cartoons Rene A, Zhong Siren. Adaptive smoothing respecting feature directions [J]. IEEE Transactions on Image Processing,1998,7(3):353-358.
[18]Rudin L, Osher S, Fatemi E. Nonlinear Total Variation Based Noise Removal Algorithms [J]. 1992,60:259-268.
[19]Engl H, Hanke M, Neubauer A. Regularization of Inverse Problems[M]. Dordrecht:Kluwer Academic Publishers,1996.
[20]Caselles V, Catte F, Coll B, et al. A geometric model for active contours in image Processing [J]. Numeric Mathematic,1993,66(1):1-31.
[21]Chen K, Member S. Adaptive Smoothing via Contextual and Local Discontinuities [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,2005,27(10):1552-1567.
[22]Jia D Y, Huang F G, WEN X F, Fourth-order anisotropic diffusion equations for noise removal[J], ICSP, Beijing, China,2004,2:994-997.
[23]Whitaker R T. Pizer S M. A multi-scale approach to nonuniform diffusion[J]. Computer Vision, Graphics, and Image Processing:Image Understanding,1993,57(1):99-110.
[24]You YuLi, Xu Wengyuan, Tannenbaum A, et al. Behavioral analysis of anisotropic diffusion in image processing[J], IEEE Transactions on Image Processing.1996,5(11):1539-1553.
[25]Chan T, Shen J, Vese L, Variational PDE models in image processing[J], Notice of Amer. Math. Soc.,2003,50(1),:14-26.
[26]D. Gabor. Information Therory in Electron Microscopy [J]. Lab, Invest.14,1965.801-80.
[27]Weickert J. Coherence-enhancing diffusion filtering[J]. International Journal of Computer Vision,1999,31(2-3):111-127.
[28]Chen Yunmei, Vemuri B C, Wang Li. Image denoising and segmentation via nonlinear diffusion[J]. Computers and Mathematics with Applications,2007,39(5/6):131-149.
[29]Wang Z, Bovik A C, Sheikh H R, et al. Image quality assessment:From error visibility to structural similarity [J]. IEEE Transaction on Image Processing,2004,13(4):600-612.
[30]Baldock R, Graham J. Image Processing and Analysis-A Practical Approach [M]. London: Oxford University Press.2000.
[31]Hansen P C, O'Leary D P. The use of the L-curve in the regularization of discrete ill-posed problems [J]. SIAM Journal on Scientific Computing.1993,14 (6):1487-1503.
[32]Marquina A, Osher S. Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal[J]. SIAM. Journal on Scientific Computing,2000,22:387-405.
[33]Osher S, Fedkiw R. Level set methods and dynamic implicit surfaces [M]. New York: Springer-Verlag.2003.
[34]Hansen P C, O'Leary D P. Analysis of discrete ill-posed problems by means of the L-curve [J]. SIAM Review.1992,34(4):561-580.
[35]Jain A K. Partial differential equations and finite-difference methods in image processing, part 1:Image representation [J], Journal of Optimization theory application.1977,23:65-91.
[36]Chan T F, Vese L A. Active contours without edges [J]. IEEE Transactions on Image Processing,2001,10(2):266-277,.
[37]Alvarez L, Guichard F, Lions P, et al. Axioms and fundamental equations of image processing[J]. Archive for Rational Mechanics and Analysis,1999,123(3),199-257
[38]Aubert G, Kornprobst P. Mathematical problems in image processing. Partial differential equations and the calculus of variations. Second edition[M]. New York:Springer,2006.
[39]Chan T, Vese L. Active contours without edges [J]. IEEE Transactions on Image Processing, 2001,10(2):266-277.
[40]Weickert J, Schnorr C. Variational optic flow computation with a spatio-temporal smoothness constraint[J]. Math. Image Vision,2001,14:245-255.
[41]Weber J, Malik J. Robust computation of optical flow in a multi-scale differential framework[M]. Computer Vision,1993.
[42]Adalsteinsson D, Sethian J A. The fast construction of extension velocities in level set methods[J]. Journal of Computational Physics,1999,148(1):2-22.
[43]Caselles V, Kimmel R, Sapiro G. Geodesic active contours. The International Journal of Computer Vision,1997,22(1):61-79.
[2]邹谋炎。反卷积和信号复原[M].北京：国防工业出版社,2001.
[3]Sonka M, Hlavac V, Boyle R. Image Processing, Analysis and Machine Vison[M], PWS Publishing,1999.
[4]张亶,陈刚。基于偏微分方程的图象处理[M].北京：高等教育出版社.2004.
[5]Caselles V, Morel J M, Sapiro GS et al. Introduction to the special issue on partial differential equations and geometry-driven diffusion in image processing and analysis [J]. IEEE Transactions on Image Processing,1998,7(3):269-273.
[6]Witkin A P. Scale-space filtering, In Proceedings 8th International Joint Conference on Artificial Intelligence[J],1983,2:1019-1021.
[7]Koenderink J J. The structure of images[J]. Biological Cybernetics,1984,50:363-370.
[8]Hummel R A. Representations based on zero-crossings in scale-space[J]. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,1986,204-209.
[9]Perona P and Malik J. Scale-Space and edge detection using Anisotropic Diffusion [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,1990,12(07):629-639.
[10]Scherzer O, Weickert J. Relations between Regularization and Diffusion Filtering [J]. Journal of Mathematical Imaging and Vision,2000,12:43-63.
[11]Aubert G, Kornprobst P. Mathematical problems in Image Processing [M]. Berlin: Springer-Verlag,2001.
[12]Catte F, Lions P L, Morel J M. Image selective smoothing and edge detection by nonlinear diffusion [J]. SIAM J. Numer. Anal,1992,129:182-193.
[13]Alvarez L, Mazorra L. Signal and image restoration using shock filters and anisotropic diffusion [J]. SIAM Journal on Numerical Analysis,1994,31(02):590-605.
[14]Weickert J, Schnorr C. A Theoretical Framework for Convex Regularizers in PDE-Based Computation of Image Motion [J]. International Journal of Computer Vision,2001,45(3): 245-264.
[15]Gilboa G, Sochen N A, Zeevi Y Y. Forward-and- backward diffusion processes for adaptive image enhancement and denoising [J]. IEEE Transactions on Image Processing,2002,11(7): 689-703.
[16]Gilboa G, Zeevi Y Y, Sochen N A. Complex diffusion processes for image filtering [J]. Lecture Notes in Computer Science,2001,21(06):293-299.
[17]Cartoons Rene A, Zhong Siren. Adaptive smoothing respecting feature directions [J]. IEEE Transactions on Image Processing,1998,7(3):353-358.
[18]Rudin L, Osher S, Fatemi E. Nonlinear Total Variation Based Noise Removal Algorithms [J]. 1992,60:259-268.
[19]Engl H, Hanke M, Neubauer A. Regularization of Inverse Problems[M]. Dordrecht:Kluwer Academic Publishers,1996.
[20]Caselles V, Catte F, Coll B, et al. A geometric model for active contours in image Processing [J]. Numeric Mathematic,1993,66(1):1-31.
[21]Chen K, Member S. Adaptive Smoothing via Contextual and Local Discontinuities [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,2005,27(10):1552-1567.
[22]Jia D Y, Huang F G, WEN X F, Fourth-order anisotropic diffusion equations for noise removal[J], ICSP, Beijing, China,2004,2:994-997.
[23]Whitaker R T. Pizer S M. A multi-scale approach to nonuniform diffusion[J]. Computer Vision, Graphics, and Image Processing:Image Understanding,1993,57(1):99-110.
[24]You YuLi, Xu Wengyuan, Tannenbaum A, et al. Behavioral analysis of anisotropic diffusion in image processing[J], IEEE Transactions on Image Processing.1996,5(11):1539-1553.
[25]Chan T, Shen J, Vese L, Variational PDE models in image processing[J], Notice of Amer. Math. Soc.,2003,50(1),:14-26.
[26]D. Gabor. Information Therory in Electron Microscopy [J]. Lab, Invest.14,1965.801-80.
[27]Weickert J. Coherence-enhancing diffusion filtering[J]. International Journal of Computer Vision,1999,31(2-3):111-127.
[28]Chen Yunmei, Vemuri B C, Wang Li. Image denoising and segmentation via nonlinear diffusion[J]. Computers and Mathematics with Applications,2007,39(5/6):131-149.
[29]Wang Z, Bovik A C, Sheikh H R, et al. Image quality assessment:From error visibility to structural similarity [J]. IEEE Transaction on Image Processing,2004,13(4):600-612.
[30]Baldock R, Graham J. Image Processing and Analysis-A Practical Approach [M]. London: Oxford University Press.2000.
[31]Hansen P C, O'Leary D P. The use of the L-curve in the regularization of discrete ill-posed problems [J]. SIAM Journal on Scientific Computing.1993,14 (6):1487-1503.
[32]Marquina A, Osher S. Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal[J]. SIAM. Journal on Scientific Computing,2000,22:387-405.
[33]Osher S, Fedkiw R. Level set methods and dynamic implicit surfaces [M]. New York: Springer-Verlag.2003.
[34]Hansen P C, O'Leary D P. Analysis of discrete ill-posed problems by means of the L-curve [J]. SIAM Review.1992,34(4):561-580.
[35]Jain A K. Partial differential equations and finite-difference methods in image processing, part 1:Image representation [J], Journal of Optimization theory application.1977,23:65-91.
[36]Chan T F, Vese L A. Active contours without edges [J]. IEEE Transactions on Image Processing,2001,10(2):266-277,.
[37]Alvarez L, Guichard F, Lions P, et al. Axioms and fundamental equations of image processing[J]. Archive for Rational Mechanics and Analysis,1999,123(3),199-257
[38]Aubert G, Kornprobst P. Mathematical problems in image processing. Partial differential equations and the calculus of variations. Second edition[M]. New York:Springer,2006.
[39]Chan T, Vese L. Active contours without edges [J]. IEEE Transactions on Image Processing, 2001,10(2):266-277.
[40]Weickert J, Schnorr C. Variational optic flow computation with a spatio-temporal smoothness constraint[J]. Math. Image Vision,2001,14:245-255.
[41]Weber J, Malik J. Robust computation of optical flow in a multi-scale differential framework[M]. Computer Vision,1993.
[42]Adalsteinsson D, Sethian J A. The fast construction of extension velocities in level set methods[J]. Journal of Computational Physics,1999,148(1):2-22.
[43]Caselles V, Kimmel R, Sapiro G. Geodesic active contours. The International Journal of Computer Vision,1997,22(1):61-79.