基于PDE和样本的图像修复
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摘要
图像修复是图像处理中的重要组成部分。2000年M.Bertalmio,G.Sapiro,V.Cadelles和C.Balle.ster联合发表的Image Inpainting,首次把图像的Inpainting,即图像的对残缺部分的填充修复引入到图像处理中来。由于图像inpainting本质上是一个病态问题,没有唯一解,所以人们从多个数学角度提出了多种方法。本文在第二章通过具体的模型,分别从偏微分方程、变分、小波、随机场、纹理等等图像处理领域,给出了一个较为详细的综述。
在上述的多种方法中,本文在第三章着重介绍了基于TV-Stokes方程的偏微分方程的方法和基于样本的纹理修复这两种相对较好的方法,而这两种方法在某些方面又存在不足之处。本文通过分析比较,对于一类图像,给出了一种综合这两种方法对图像进行修复的算法,使得对图像的修复同时具备了上述两种方法的优点,并且经过仿真实验,验证了这是一种有效的图像修复方法。
Image inpainting takes a crucial position in image processing.In 2000 M.Bertalmio,G.Sapiro,V.Cadelles and C.Ballester jointly published the Papers of Image Inpainting. It isthe first time that the image inpainting was introduced to image processing.Essentially theimage inpainting is an ill-posed problem,have not only one solution.So in recent years,a lotof image inpainting methods are developed from various mathematical points of view,suchas partial di?erential equations,texture, random field,wavelet and statistics,etc,In this pa-per,the basic idea and development of inpainting is discussed in detail by using manyinpainting models.
In the many methods above, two relatively better ways are introduced here:the TV-Stokes equation method which is based on partial di?erential equations and exemplar-basedimage inpainting method which is better for texture.Each of two methods has advantageand disadvantage.After comparison and analysis,this paper gives a combination of the twomethods above to inpaint a class of image,so the proposed method has the advantages ofabove two methods.Based on experiment and simulation,the proposed method is provedto be an e?ective image inpainting method.
在上述的多种方法中,本文在第三章着重介绍了基于TV-Stokes方程的偏微分方程的方法和基于样本的纹理修复这两种相对较好的方法,而这两种方法在某些方面又存在不足之处。本文通过分析比较,对于一类图像,给出了一种综合这两种方法对图像进行修复的算法,使得对图像的修复同时具备了上述两种方法的优点,并且经过仿真实验,验证了这是一种有效的图像修复方法。
Image inpainting takes a crucial position in image processing.In 2000 M.Bertalmio,G.Sapiro,V.Cadelles and C.Ballester jointly published the Papers of Image Inpainting. It isthe first time that the image inpainting was introduced to image processing.Essentially theimage inpainting is an ill-posed problem,have not only one solution.So in recent years,a lotof image inpainting methods are developed from various mathematical points of view,suchas partial di?erential equations,texture, random field,wavelet and statistics,etc,In this pa-per,the basic idea and development of inpainting is discussed in detail by using manyinpainting models.
In the many methods above, two relatively better ways are introduced here:the TV-Stokes equation method which is based on partial di?erential equations and exemplar-basedimage inpainting method which is better for texture.Each of two methods has advantageand disadvantage.After comparison and analysis,this paper gives a combination of the twomethods above to inpaint a class of image,so the proposed method has the advantages ofabove two methods.Based on experiment and simulation,the proposed method is provedto be an e?ective image inpainting method.
引文
[1] Xue-cheng Tai,Stanley Osher and Randi Holm.Image inpainting using a TV-Stokes equa-tion.in:Image Processing based on partial di?erential equations, Springer, Heidelberg,2006
[2] M.bertalmio, G.Sapiro, V.Cadelles and C.Ballester. Image inpainting.In KurtSkeley, com-puter Graphics proceedings. pages417-424, ACM Press,2000 SIGGRARPH Addison WesleyLongman, 2000
[3] M.bertalmio A.Bertozzi and G.Sapiro.Navier-Stokes,?uid dynamics,and image and video in-painting. IEEE computer Vision and Pattern Recognition,hawaii,1:I355-I362,December 2001;coe?cients.UCLA,Math.Depart,CAM-report 03-15,2003
[4] M.Bertalmio,L.Vese,G.Sapiro and S.Osher.Simultaneous strucure and texture image in-painting.In IEEE computer Society Conference on Computer Vision and Pattern Recogni-tion,volume2,June 2003
[5] C.Bsllester,M.Bertalmio,V.Cadelles,G.Sapiro and J.Verdera.Filling-in by joint interpolationof vector fields and grey levels.IEEE Trans.Image Process.10:1200-1211,2001
[6] T.Chan,J.Shen and L.Vese.Variation image inpainting.Commen Pure Appl Math. 58:579-619,2005
[7] T.Chan and J.Shen.Local inpainting models and TV inpainting. SIAM J.Appl.Math. 62:3,pp.1019 -1043, 2001
[8] T.Chan,J.Shen and H.M.Zhou.Total variation wavelet inpainting.Journal of MathematicalImaging and Vision,Volume 25, Number1,2006
[9] T.Chan and J.Shen. Nontexture inpainting by curvature driven di?usions. J.Visual Comm.Image Rep. 12:436-449, 2001
[10] T.Chan and J.Shen.Inpainting based on nonlinear transport and di?usion.In Inverse Prob-lems,Image Aanlysis,and Medical Imaging.Volume 313 of Contemp.Math.Z.Nashed andO.Scherzer,eds,AMS,Providence,RI,pages 53-65,2002
[11] T.Chan,S.H.Kang and J.Shen.Euler’s Elastica and Curvature Based Inpainting.SIAMJ.Appl.Math,2002
[12] J.Shen.Inpainting and the fundamental problem of image processing.SIAM News,363(5),2003
[13] TonyF.Chan.Jianhong(jackie)Shen:ImageProcessing and Snalysis,Society for Industrial andApplied Mathematics Phiadelphia,2001
[14] S.Esedoglu and R.March. Digital inpainting based on the Mumford-Shah-Euler image model.European J.Appl.Math.13:353-370,2002
[15] A. Criminisi,P. Pserez,and K.Toyama.Region filling and object removal by exemplarbasedimage inpainting.IEEE Transactions on Image Processing,2004.
[16] H.Grossauer and O.Scherzer.Using the Complex Ginzburg-Landau Equation for Digital In-painting in 2D and 3D.Scale Space Methods in Computer Vision,Lecture Notes in ComputerScience,2003.
[17] S.Geman and D.Geman.Stochastic relaxation,gibbs distributions,and the bayesian restorationof images.IEEE Transactions on Pattern Analysis and Machine Intelligence,1984.
[18] A.Levin,A.Zomet and Y.Weiss.Learing How to Inpainting from Global Image Statistics. ICCV,2003
[19] C.Balleste,S.Esedoglu,and A.Gillette.Inpainting of binary images using the Chan-Hilliardequation, IEEE Trans.Image Proc,2005
[20] Masnou.S.Disocclusion:A Variational Approach Using Level Lines.IEEE Trans.Image Process-ing,2002
[21] G.Aubert and P.Kornprobst.Mathematical Problems in Image Processing.Springer,2002
[22] 冈萨雷斯等.数字图像处理[M].第2版,北京:电子工业出版社,2003
[23] 章毓晋.图像处理和分析(图像工程上册)[M].北京:电子工业出版社,2002.
[24] 徐飞.MATLAB应用图像处理[M].第1版,西安:西安电子科技大学出版社2002
[25] 周廷芳等.基于径向基函数的图像修复技术 [J].中国图像图形学报,2004
[26] 程村.基于蒙特卡罗方法的图像修复快速算法 [J].工程地球物理学报,2006
[27] 霍星,檀结庆.连分式在图像修复中的应用 [J].量子电子学报,2006
[28] 刘昕,唐宝玲等.基于样本的图像修补[J].西安理工大学学报.Vol.21.No.1.2005
[29] 樊红丽,焦红和等.基于插值的残缺图像的修复算法[J].兵工学报, 2005
[30] 陈芳,叶正麟,王继红.基于统计特征的图像修复[J],计算机工程与应用,2007
[2] M.bertalmio, G.Sapiro, V.Cadelles and C.Ballester. Image inpainting.In KurtSkeley, com-puter Graphics proceedings. pages417-424, ACM Press,2000 SIGGRARPH Addison WesleyLongman, 2000
[3] M.bertalmio A.Bertozzi and G.Sapiro.Navier-Stokes,?uid dynamics,and image and video in-painting. IEEE computer Vision and Pattern Recognition,hawaii,1:I355-I362,December 2001;coe?cients.UCLA,Math.Depart,CAM-report 03-15,2003
[4] M.Bertalmio,L.Vese,G.Sapiro and S.Osher.Simultaneous strucure and texture image in-painting.In IEEE computer Society Conference on Computer Vision and Pattern Recogni-tion,volume2,June 2003
[5] C.Bsllester,M.Bertalmio,V.Cadelles,G.Sapiro and J.Verdera.Filling-in by joint interpolationof vector fields and grey levels.IEEE Trans.Image Process.10:1200-1211,2001
[6] T.Chan,J.Shen and L.Vese.Variation image inpainting.Commen Pure Appl Math. 58:579-619,2005
[7] T.Chan and J.Shen.Local inpainting models and TV inpainting. SIAM J.Appl.Math. 62:3,pp.1019 -1043, 2001
[8] T.Chan,J.Shen and H.M.Zhou.Total variation wavelet inpainting.Journal of MathematicalImaging and Vision,Volume 25, Number1,2006
[9] T.Chan and J.Shen. Nontexture inpainting by curvature driven di?usions. J.Visual Comm.Image Rep. 12:436-449, 2001
[10] T.Chan and J.Shen.Inpainting based on nonlinear transport and di?usion.In Inverse Prob-lems,Image Aanlysis,and Medical Imaging.Volume 313 of Contemp.Math.Z.Nashed andO.Scherzer,eds,AMS,Providence,RI,pages 53-65,2002
[11] T.Chan,S.H.Kang and J.Shen.Euler’s Elastica and Curvature Based Inpainting.SIAMJ.Appl.Math,2002
[12] J.Shen.Inpainting and the fundamental problem of image processing.SIAM News,363(5),2003
[13] TonyF.Chan.Jianhong(jackie)Shen:ImageProcessing and Snalysis,Society for Industrial andApplied Mathematics Phiadelphia,2001
[14] S.Esedoglu and R.March. Digital inpainting based on the Mumford-Shah-Euler image model.European J.Appl.Math.13:353-370,2002
[15] A. Criminisi,P. Pserez,and K.Toyama.Region filling and object removal by exemplarbasedimage inpainting.IEEE Transactions on Image Processing,2004.
[16] H.Grossauer and O.Scherzer.Using the Complex Ginzburg-Landau Equation for Digital In-painting in 2D and 3D.Scale Space Methods in Computer Vision,Lecture Notes in ComputerScience,2003.
[17] S.Geman and D.Geman.Stochastic relaxation,gibbs distributions,and the bayesian restorationof images.IEEE Transactions on Pattern Analysis and Machine Intelligence,1984.
[18] A.Levin,A.Zomet and Y.Weiss.Learing How to Inpainting from Global Image Statistics. ICCV,2003
[19] C.Balleste,S.Esedoglu,and A.Gillette.Inpainting of binary images using the Chan-Hilliardequation, IEEE Trans.Image Proc,2005
[20] Masnou.S.Disocclusion:A Variational Approach Using Level Lines.IEEE Trans.Image Process-ing,2002
[21] G.Aubert and P.Kornprobst.Mathematical Problems in Image Processing.Springer,2002
[22] 冈萨雷斯等.数字图像处理[M].第2版,北京:电子工业出版社,2003
[23] 章毓晋.图像处理和分析(图像工程上册)[M].北京:电子工业出版社,2002.
[24] 徐飞.MATLAB应用图像处理[M].第1版,西安:西安电子科技大学出版社2002
[25] 周廷芳等.基于径向基函数的图像修复技术 [J].中国图像图形学报,2004
[26] 程村.基于蒙特卡罗方法的图像修复快速算法 [J].工程地球物理学报,2006
[27] 霍星,檀结庆.连分式在图像修复中的应用 [J].量子电子学报,2006
[28] 刘昕,唐宝玲等.基于样本的图像修补[J].西安理工大学学报.Vol.21.No.1.2005
[29] 樊红丽,焦红和等.基于插值的残缺图像的修复算法[J].兵工学报, 2005
[30] 陈芳,叶正麟,王继红.基于统计特征的图像修复[J],计算机工程与应用,2007