非线性扩散和变分模型在图像去噪中的应用
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摘要
图像是人们获取信息的重要渠道。但在图像的获取、传输、存储等过程中往往因为各种原因而掺杂入噪声。因此,在进一步使用图像前去除噪声,提高图像质量成为数字图像处理中的重要研究部分。本文对基于非线性扩散和变分方法的图像降噪技术进行了研究,主要包括以下主要内容。对非线性扩散技术和基于变分方法的图像降噪技术的发展与现状进行了阐述。在非线性扩散技术中,介绍了P-M图像扩散模型和几种在此基础上的改进方法,分析了这几种模型的去噪和边缘保持特点。对基于变分方法的ROF模型及其改进模型进行了分析,讨论了这些模型的边缘保持作用。在以上研究的基础上,本文主要工作如下:
介绍了TV流扩散和正、逆向扩散技术和图像耦合技术,证明了TV流模型的边缘扩散性质,在此基础上提出了基于TV流的彩色图像耦合扩散模型,将TV流和正、逆向扩散技术运用到彩色图像扩散中;通过实验证明正、逆向扩散技术在矢量图像扩散工作中的作用。
阶梯现象是ROF模型的主要不足之一,本文分析了ROF模型产生阶梯化现象的原因,研究了Bing Song自适应去噪模型的去噪和边缘保持性质,证明了其在不同参数p下的边缘扩散性质;同时对Blomgren等人提出的梯度自适应改进模型进行了介绍,分析了改进模型的优点和不足,提出了基于梯度自适应改进模型的改进函数,提升了Blomgren模型的边缘保持能力,使之能够适应各种对比度的图像。
通过对以上方法分别进行仿真实验,结果表明:在彩色图像中,耦合的TV流扩散模型较未耦合模型有更好的去噪及边缘保持效果,且正逆向扩散在耦合模型中仍能保持作用;利用改进的梯度自适应函数,模型较好地适应了各种对比度下的图像,在消除图像阶梯化现象的同时体现出较好地去噪和边缘保持效果。
Image is an important source of information. Due to noise mixed with process of information acquisition, transmission, storage and so forth, which degrades the quality of image. Image denoising becomes an important task of digital image processing. A comprehensive research on edge preservation, channel coupling, adaptive gradient function of image denoising using nonlinear diffusion and total variation method are made in this dissertation, which mainly includes the following contents:
At first, the recent developments and application fields of nonlinear diffusion and total variation method were introduced; some improved models which based on P-M and ROF model are listed too. The principles of these methods are analyzed. Based on these analyses, the main work can be listed as follow:
Both TV flow and edge enhancing flow are analyzed; a basic coupled method for color image is mentioned. In order to remove noise effectively and preserve edges and key details in color image, considering the information of each channel of color image and the advantages of denoising and edges preservation of TV flow, a channel couple-d diffusion model which based on TV flow is proposed, the function of the forward and backward diffusion are also included.
The staircase which is the main side effects of ROF model is analyzed, and the characteristics of Bing Song adaptive method are also studied and its property of edge diffusion with respect to different region of parameter p are proved. Besides, Bomgren adaptive gradient function method is also introduced, some of its characteristics of edge preservation and denoising effect are listed. Due to the defect of the adaptive gradient function which limited the method's denoising and edge preservation abilities, an improved adaptive gradient function is proposed, new function allow researcher choosing parameter beta according to the magnitude of objects in the images to rescale the transition region, through this method, the model can prevent some low-intensity edges being smoothed.
At last, the methods for image denoising proposed in this paper are tested respectively. Experimental results show that the channel coupled TV diffusion model is better preserving geometric information such as edges in addition to its effectiveness for image denoising. Besides, the properties of forward and backward diffusion based on the new model are not changed in color image denoising; At the mean time, the results show that Bomgren model which adopted improved adaptive gradient function not only prevent the staircase, but well capture edge and remove noise with different intensity of images.
介绍了TV流扩散和正、逆向扩散技术和图像耦合技术,证明了TV流模型的边缘扩散性质,在此基础上提出了基于TV流的彩色图像耦合扩散模型,将TV流和正、逆向扩散技术运用到彩色图像扩散中;通过实验证明正、逆向扩散技术在矢量图像扩散工作中的作用。
阶梯现象是ROF模型的主要不足之一,本文分析了ROF模型产生阶梯化现象的原因,研究了Bing Song自适应去噪模型的去噪和边缘保持性质,证明了其在不同参数p下的边缘扩散性质;同时对Blomgren等人提出的梯度自适应改进模型进行了介绍,分析了改进模型的优点和不足,提出了基于梯度自适应改进模型的改进函数,提升了Blomgren模型的边缘保持能力,使之能够适应各种对比度的图像。
通过对以上方法分别进行仿真实验,结果表明:在彩色图像中,耦合的TV流扩散模型较未耦合模型有更好的去噪及边缘保持效果,且正逆向扩散在耦合模型中仍能保持作用;利用改进的梯度自适应函数,模型较好地适应了各种对比度下的图像,在消除图像阶梯化现象的同时体现出较好地去噪和边缘保持效果。
Image is an important source of information. Due to noise mixed with process of information acquisition, transmission, storage and so forth, which degrades the quality of image. Image denoising becomes an important task of digital image processing. A comprehensive research on edge preservation, channel coupling, adaptive gradient function of image denoising using nonlinear diffusion and total variation method are made in this dissertation, which mainly includes the following contents:
At first, the recent developments and application fields of nonlinear diffusion and total variation method were introduced; some improved models which based on P-M and ROF model are listed too. The principles of these methods are analyzed. Based on these analyses, the main work can be listed as follow:
Both TV flow and edge enhancing flow are analyzed; a basic coupled method for color image is mentioned. In order to remove noise effectively and preserve edges and key details in color image, considering the information of each channel of color image and the advantages of denoising and edges preservation of TV flow, a channel couple-d diffusion model which based on TV flow is proposed, the function of the forward and backward diffusion are also included.
The staircase which is the main side effects of ROF model is analyzed, and the characteristics of Bing Song adaptive method are also studied and its property of edge diffusion with respect to different region of parameter p are proved. Besides, Bomgren adaptive gradient function method is also introduced, some of its characteristics of edge preservation and denoising effect are listed. Due to the defect of the adaptive gradient function which limited the method's denoising and edge preservation abilities, an improved adaptive gradient function is proposed, new function allow researcher choosing parameter beta according to the magnitude of objects in the images to rescale the transition region, through this method, the model can prevent some low-intensity edges being smoothed.
At last, the methods for image denoising proposed in this paper are tested respectively. Experimental results show that the channel coupled TV diffusion model is better preserving geometric information such as edges in addition to its effectiveness for image denoising. Besides, the properties of forward and backward diffusion based on the new model are not changed in color image denoising; At the mean time, the results show that Bomgren model which adopted improved adaptive gradient function not only prevent the staircase, but well capture edge and remove noise with different intensity of images.
引文
1D.Marr,Vision[M].W.H.Freeman and Company,1982.
2Witkin,Scale-space Filtering[J].Int.Joint conf.Artif.Intell.,pp.19831019-1021.
3J.J.Koenderink,The Structure of Image[J].BiD1Cybern,1984,50:363-370.
4R.A.Hummel,Representations Based on Zero-crossing in Scale-space[C].Proc.IEEE Computer Vision and Pattern Recog.Conf.1986,204-209.
5P.Perona,J.Malik,Scale-Space and Edge Detection Using Anisotropic Diffusion[J].IEEE Trans.Pattern Analysis and Machine Intelligence,1990,12(7):629-639.
6王毅,张良培,李平湘.各向异性扩散平滑滤波的改进算法[J].中国图象图形学报,2006,11(2):210-216.
7Weickert J.,Coherence-Enhancing Diffusion Filtering[J].Int.J Compute Vision,1999,31(2/3):111-127.
8F.Andreu,C.Ballester,V.Caselles,and J.M.Maz'on.Minimizing Total Variation Flow[J].Differential and Integral Equations,2001,14(3):321-360,March.
9F.Oibos and G.Koepfler.Global Total Variation Minimization[J].SIAM Journal on Numerical Analysis,2000,37(2):646-664.
10V.I.Tsurkov.An Analytical Model of Edge Protection Under Noise Suppression by Anisotropic Diffusion[J].Journal of Computer and Systems Sciences International,2000,39(3):437-440.
11G.Gerig,O.K"ubler,R.Kikinis,and F.A.Jolesz.Nonlinear Anisotropic Filtering of MRI Data[J].IEEE Transactions on Medical Imaging,1992,11:221-232.
12A.N.Tikhonov and V.Y.Arsenin,Solutions of Ill-Posed Problems[M].John Wiley,New York,1977
13L.Rudin,S.Osher,and E.Fatemi.Nonlinear Total Variation based Noise Removal Algorithms[J].Physical D,1992,60:259-268.
14P.Blomgren and T.ChaR.Color TV:Total Variation Methods for Restoration of Vector-Valued Images[J].IEEE Trans.Image Proc.,1998,7:304-309.
15S.Osher,M.Burger,D.Goldfarb,J.Xu,and W.Yin.An Iterative Segularization Method for Total Variation Based Image Restoration[J].UCLA CAM Report,2004,04-13.
16T.Chan and S.Esedoglu.Aspects of Total Variation Regularized L1 Function Approximation[J].UCLA CAM Report,2004,04-07.
17S.Esedoglu and S.Osher.Decomposition of Images by the Anisotropic Rudin-Osher-Fatemi Model[J].Comm.Pure Appl.Math.,2004,57:1609-1626.
18P.Blomgren,P.Mulet,T.Chan,and C.Wong.Total Variation Image Restoration:Numerical Methods and Extensions.In ICIP,pages 384-387,Santa Barbara,1997.
19A.Chambolle and P.Lions.Image Recovery via Total Variation Minimization and Related Problems[J].Numer.Math.,1997,76:167-188.
20T.Chan and C.Wong.Total Variation Blind Deconvolution[J].IEEE Trans.Image Process.1998,7:370-375.
21T.ChanandJ.Shen.Mathematical Models of Local Non-texture Inpaintings[J].SIAM J.Appl.Math.,2001,62:1019-1043.
22Y.Meyer.Oscillating Patterns in Image Processing and in some Nonlinear Evolution Equations[M].AMS,2001.
23T.Brox.From Pixels to Regions:Partial Differential Equations in Image Analysis [M].Ph.D.Dissertation,Mathematical Image Analysis Group,Department of Mathematics and Computer Science,Saarland University,Germany,April 2005.
24仵冀颖,阮秋琦.偏微分方程在图像去噪中的应用[J].计算机工程与应用,2006,22:0069-0071.
25贾迪野,黄凤岗,苏菡.一种新的基丁高阶非线性扩散的图像平滑方法[J].计算机学报,2005,28(5):882-891.
26F.Catte,P.L.Lions,J.M.Morel and T.Coil,Image Selective Smoothing and Edge Detection by Nonlinear Diffusion[J].SIAM J.Numer.Anal.1992,29:182-193.
27L.Alvarez,P.L.Lions,J.M.Morel.Image Selective Smoothing and Edge Detection by Nonlinear Diffusion[J].SIAM J Number Anal.,1992,29(3):845-866.
28A.Chambolle and P.Lions.Image Recovery via Total Variation Minimization and Related Problems[J].Numer.Math.,1997,76:167-188.
29R.Acar and C.R.Vogel.Analysis of total variation penalty methods for ill-posed problems[J].Inverse Problems,1994,10:1217-1229.
30D.Strong and T.F.Chan,Spatialy and Scale Adaptive Total Variation Based Regularization and Anisotropic Diffusion in Image Processing[J].UCLA Math Department CAM Report,1996,11:96-46,.
31D.Strong.Adaptive Total Variation Minimization Image Restoration[M].PhD thesis,1997.
32Y.Chen and W.Wunderli.Adaptive Total Variation for Image Restoration in BY Space [J].J.Math.Anal.Appl,2002,272:117-137.
33C.A.Z.Barcelos,Y.Chen,Heat Flows and Related Minimization Problem in Image Restoration[J].Computers and Mathematics with Applications,2000,89:81-97.
34L.Rudin,P.L.Lions,S.Osher,Multiplicative Denoising and Deblurring Theory and Algorithms in Geometric Level Sets Methods in Imaging Vision and Graphics [M].Springer-Verlag,2003
35S.Osher,N.Paragios,Geometric Level Sets Methods in Imaging Vision Graphics [M].Springer-Verlag Press,2003
36L.Rudin,S.J.Osher,C.Fu,Total Variation Based Restoration of Noisy Blurred Images[J].SIAM Journal of Numerical Analysis,accepted for publication,1993.
37G.Aubert,and L.Vese,A Variational Method in Image Recovery[J].SIAM J.Numer. Anal.,1991,34(5):1948-1979.
38P.Charbonnier,Blanc-Feraud,G.Aubert,and M.Barlaud,Deterministic Edge-Preserving Regularization in Computed Imaging[J].IEEE Trans.Image Processing,1997,Febr.Vol.6,No.2.
39S.Teboul,L.Blanc-Feraud,G.Aubert,and M.Barlaud,Variational Approach for Edge-Preserving Regularization Ising Coupled PDE' s[J].IEEE Trans.Image Processing,1998,March Vol.7,No.3.
40F.Dibos and G.Koepfler.Global Total Variation Minimization[J].SIAM Journal on Numerical Analysis,2000,37(2):646-664.
41F.Andreu,C.Ballester,V.Caselles,and J.M.Maz'on.Minimizing Total Variation Flow.Differential and Integral Equations[J].2001,14(3):321-360.
42T.Brox,M.Welk,G.Steidl,and J.Weickert.Equivalence Results for TV Diffusion and TV Regularisation[A].In L.D.Griffin and M.Lillholm,Editors,Scale-Space Methods in Computer Vision[J].volume 2695 of Lecture Notes in Computer Science,Springer,Berlin,2003,pp 86-100.
43G.Steidl,J.Weickert,T.Brox,P.Mr' azek,and M.Welk.On the equivalence of soft wavelet shrinkage,total variation diffusion,total variation regularization,and SIDEs[J].SIAM Journal on Numerical Analysis,2004,42(2):686-713.
44V.I.Tsurkov.An Analytical Model of Edge Protection Under Noise Suppression by Anisotropic Diffusion[J].Journal of Computer and Systems Sciences International,2000,39(3):437-440.
45S.L.Keeling and R.Stollberger.Nonlinear Anisotropic Diffusion Filters for Wide Range Edge Sharpening[J].Inverse Problems,2002,18:175-190.
46朱立新,王平安,夏德深.引入耦合梯度保真项的非线性扩散图像去噪方法[J].计算机研究与发展,2007年44(8):1390-1398
47Bing Song,Topics in Variational PDE Image Segmentation,Inpainting and Denoising [M].Ph.D.Thesis,2003.
48P.Blomgren,P.Mulet,T.Chan,and C.Wong.Total Variation Image Restoration:Numerical Methods and Extensions[J].In ICIP,1997,384-387.
49A.Chambolle and P.Lions.Image Recovery via Total Variation Minimization and Related Problems[J].Numer.Math.,1997,76:167-188.
50T.Chan,M.Marquina,and-P.Mulet.High-order Total Variation-based Image Restoration[J].SIAM J.Sci.Comput.,2000,22:503-516.
51T.F.Chan,S.Esedoglu,and F.Park.A Fourth Order Dual Method for Staircase.Reduction in Texture Extraction and Image Restoration Problems[J].Technical Report.UCLA,CA,CAM 2005,05-28.
2Witkin,Scale-space Filtering[J].Int.Joint conf.Artif.Intell.,pp.19831019-1021.
3J.J.Koenderink,The Structure of Image[J].BiD1Cybern,1984,50:363-370.
4R.A.Hummel,Representations Based on Zero-crossing in Scale-space[C].Proc.IEEE Computer Vision and Pattern Recog.Conf.1986,204-209.
5P.Perona,J.Malik,Scale-Space and Edge Detection Using Anisotropic Diffusion[J].IEEE Trans.Pattern Analysis and Machine Intelligence,1990,12(7):629-639.
6王毅,张良培,李平湘.各向异性扩散平滑滤波的改进算法[J].中国图象图形学报,2006,11(2):210-216.
7Weickert J.,Coherence-Enhancing Diffusion Filtering[J].Int.J Compute Vision,1999,31(2/3):111-127.
8F.Andreu,C.Ballester,V.Caselles,and J.M.Maz'on.Minimizing Total Variation Flow[J].Differential and Integral Equations,2001,14(3):321-360,March.
9F.Oibos and G.Koepfler.Global Total Variation Minimization[J].SIAM Journal on Numerical Analysis,2000,37(2):646-664.
10V.I.Tsurkov.An Analytical Model of Edge Protection Under Noise Suppression by Anisotropic Diffusion[J].Journal of Computer and Systems Sciences International,2000,39(3):437-440.
11G.Gerig,O.K"ubler,R.Kikinis,and F.A.Jolesz.Nonlinear Anisotropic Filtering of MRI Data[J].IEEE Transactions on Medical Imaging,1992,11:221-232.
12A.N.Tikhonov and V.Y.Arsenin,Solutions of Ill-Posed Problems[M].John Wiley,New York,1977
13L.Rudin,S.Osher,and E.Fatemi.Nonlinear Total Variation based Noise Removal Algorithms[J].Physical D,1992,60:259-268.
14P.Blomgren and T.ChaR.Color TV:Total Variation Methods for Restoration of Vector-Valued Images[J].IEEE Trans.Image Proc.,1998,7:304-309.
15S.Osher,M.Burger,D.Goldfarb,J.Xu,and W.Yin.An Iterative Segularization Method for Total Variation Based Image Restoration[J].UCLA CAM Report,2004,04-13.
16T.Chan and S.Esedoglu.Aspects of Total Variation Regularized L1 Function Approximation[J].UCLA CAM Report,2004,04-07.
17S.Esedoglu and S.Osher.Decomposition of Images by the Anisotropic Rudin-Osher-Fatemi Model[J].Comm.Pure Appl.Math.,2004,57:1609-1626.
18P.Blomgren,P.Mulet,T.Chan,and C.Wong.Total Variation Image Restoration:Numerical Methods and Extensions.In ICIP,pages 384-387,Santa Barbara,1997.
19A.Chambolle and P.Lions.Image Recovery via Total Variation Minimization and Related Problems[J].Numer.Math.,1997,76:167-188.
20T.Chan and C.Wong.Total Variation Blind Deconvolution[J].IEEE Trans.Image Process.1998,7:370-375.
21T.ChanandJ.Shen.Mathematical Models of Local Non-texture Inpaintings[J].SIAM J.Appl.Math.,2001,62:1019-1043.
22Y.Meyer.Oscillating Patterns in Image Processing and in some Nonlinear Evolution Equations[M].AMS,2001.
23T.Brox.From Pixels to Regions:Partial Differential Equations in Image Analysis [M].Ph.D.Dissertation,Mathematical Image Analysis Group,Department of Mathematics and Computer Science,Saarland University,Germany,April 2005.
24仵冀颖,阮秋琦.偏微分方程在图像去噪中的应用[J].计算机工程与应用,2006,22:0069-0071.
25贾迪野,黄凤岗,苏菡.一种新的基丁高阶非线性扩散的图像平滑方法[J].计算机学报,2005,28(5):882-891.
26F.Catte,P.L.Lions,J.M.Morel and T.Coil,Image Selective Smoothing and Edge Detection by Nonlinear Diffusion[J].SIAM J.Numer.Anal.1992,29:182-193.
27L.Alvarez,P.L.Lions,J.M.Morel.Image Selective Smoothing and Edge Detection by Nonlinear Diffusion[J].SIAM J Number Anal.,1992,29(3):845-866.
28A.Chambolle and P.Lions.Image Recovery via Total Variation Minimization and Related Problems[J].Numer.Math.,1997,76:167-188.
29R.Acar and C.R.Vogel.Analysis of total variation penalty methods for ill-posed problems[J].Inverse Problems,1994,10:1217-1229.
30D.Strong and T.F.Chan,Spatialy and Scale Adaptive Total Variation Based Regularization and Anisotropic Diffusion in Image Processing[J].UCLA Math Department CAM Report,1996,11:96-46,.
31D.Strong.Adaptive Total Variation Minimization Image Restoration[M].PhD thesis,1997.
32Y.Chen and W.Wunderli.Adaptive Total Variation for Image Restoration in BY Space [J].J.Math.Anal.Appl,2002,272:117-137.
33C.A.Z.Barcelos,Y.Chen,Heat Flows and Related Minimization Problem in Image Restoration[J].Computers and Mathematics with Applications,2000,89:81-97.
34L.Rudin,P.L.Lions,S.Osher,Multiplicative Denoising and Deblurring Theory and Algorithms in Geometric Level Sets Methods in Imaging Vision and Graphics [M].Springer-Verlag,2003
35S.Osher,N.Paragios,Geometric Level Sets Methods in Imaging Vision Graphics [M].Springer-Verlag Press,2003
36L.Rudin,S.J.Osher,C.Fu,Total Variation Based Restoration of Noisy Blurred Images[J].SIAM Journal of Numerical Analysis,accepted for publication,1993.
37G.Aubert,and L.Vese,A Variational Method in Image Recovery[J].SIAM J.Numer. Anal.,1991,34(5):1948-1979.
38P.Charbonnier,Blanc-Feraud,G.Aubert,and M.Barlaud,Deterministic Edge-Preserving Regularization in Computed Imaging[J].IEEE Trans.Image Processing,1997,Febr.Vol.6,No.2.
39S.Teboul,L.Blanc-Feraud,G.Aubert,and M.Barlaud,Variational Approach for Edge-Preserving Regularization Ising Coupled PDE' s[J].IEEE Trans.Image Processing,1998,March Vol.7,No.3.
40F.Dibos and G.Koepfler.Global Total Variation Minimization[J].SIAM Journal on Numerical Analysis,2000,37(2):646-664.
41F.Andreu,C.Ballester,V.Caselles,and J.M.Maz'on.Minimizing Total Variation Flow.Differential and Integral Equations[J].2001,14(3):321-360.
42T.Brox,M.Welk,G.Steidl,and J.Weickert.Equivalence Results for TV Diffusion and TV Regularisation[A].In L.D.Griffin and M.Lillholm,Editors,Scale-Space Methods in Computer Vision[J].volume 2695 of Lecture Notes in Computer Science,Springer,Berlin,2003,pp 86-100.
43G.Steidl,J.Weickert,T.Brox,P.Mr' azek,and M.Welk.On the equivalence of soft wavelet shrinkage,total variation diffusion,total variation regularization,and SIDEs[J].SIAM Journal on Numerical Analysis,2004,42(2):686-713.
44V.I.Tsurkov.An Analytical Model of Edge Protection Under Noise Suppression by Anisotropic Diffusion[J].Journal of Computer and Systems Sciences International,2000,39(3):437-440.
45S.L.Keeling and R.Stollberger.Nonlinear Anisotropic Diffusion Filters for Wide Range Edge Sharpening[J].Inverse Problems,2002,18:175-190.
46朱立新,王平安,夏德深.引入耦合梯度保真项的非线性扩散图像去噪方法[J].计算机研究与发展,2007年44(8):1390-1398
47Bing Song,Topics in Variational PDE Image Segmentation,Inpainting and Denoising [M].Ph.D.Thesis,2003.
48P.Blomgren,P.Mulet,T.Chan,and C.Wong.Total Variation Image Restoration:Numerical Methods and Extensions[J].In ICIP,1997,384-387.
49A.Chambolle and P.Lions.Image Recovery via Total Variation Minimization and Related Problems[J].Numer.Math.,1997,76:167-188.
50T.Chan,M.Marquina,and-P.Mulet.High-order Total Variation-based Image Restoration[J].SIAM J.Sci.Comput.,2000,22:503-516.
51T.F.Chan,S.Esedoglu,and F.Park.A Fourth Order Dual Method for Staircase.Reduction in Texture Extraction and Image Restoration Problems[J].Technical Report.UCLA,CA,CAM 2005,05-28.