基于逐段常数水平集方法的图像分割技术
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摘要
近年来基于水平集的图像分割方法受到了越来越多的重视,相比于传统的图像分割方法,该方法具有对初始轮廓线位置不敏感,拓扑适应性强等优点。
本文首先对图像分割的目的、意义进行了简单的概述。并对Mumford-Shah模型、Chan-Vese模型以及J.Lie、M.Lysaker和X-C Tai提出的一种逐段常数水平集方法进行了介绍和分析。
其次,在文献[10]基于Mumford-Shah模型的逐段常数水平集分割方法的基础上,为了提高图像分割效果和运算速度,分析并改进了惩罚项结构,提出了一种IMS(improved Mumford-Shah)分割模型。实验表明模型是可行有效的。
再次,为了消除纹理或噪声对图像分割的影响,在Mumford-Shah模型中引入了图像分解的思想,并结合逐段常数水平集方法,提出了MSWD(Mumford-Shah withDecomposition)分割模型,能够分离出图像中的纹理或噪声,避免了对分割结果带来的影响,且给出了详尽的运算过程,并通过仿真实验证明了模型的有效性和实用性。
最后针对脉冲噪声的特点给出了一种有效的迭代滤波算法。
Recently,a segmentation methods based on level set have attracted more and moreattentions. Compared to traditional image segmentation approaches, the method havemany prominent virtues: such as insensitivity to the initial curve position, the strongability to deal with the topological changes etc.
First of all, the paper summarizes the purpose and significance of image segmentation,introduce and analyze Mumford-Shah model,Chan-Vese model and piecewise constant levelset method.
Second, in the base of piecewise constant level set image segmentation method basedon Mumford-Shah model in [10], in order to improve the segmentation e?ect and increasethe algorithm speed, we analyze and improve structure of the term ,we propose a improvedMumford-shah model.The experiment shows the model is validity.
Third, in order to avoid the in?uence of texture, we adopt decomposition idea inMumford-Shah model, and combine with piecewise constant level set method. We proposea MSWD(Mumford-Shah with decomposition)model. the model can separate texture ornoise. We also show the algorithm methods, and testify the rationality and validity of themodel by experiment.
Finally, we propose an iterative filter algorithm based on impulse noise.
本文首先对图像分割的目的、意义进行了简单的概述。并对Mumford-Shah模型、Chan-Vese模型以及J.Lie、M.Lysaker和X-C Tai提出的一种逐段常数水平集方法进行了介绍和分析。
其次,在文献[10]基于Mumford-Shah模型的逐段常数水平集分割方法的基础上,为了提高图像分割效果和运算速度,分析并改进了惩罚项结构,提出了一种IMS(improved Mumford-Shah)分割模型。实验表明模型是可行有效的。
再次,为了消除纹理或噪声对图像分割的影响,在Mumford-Shah模型中引入了图像分解的思想,并结合逐段常数水平集方法,提出了MSWD(Mumford-Shah withDecomposition)分割模型,能够分离出图像中的纹理或噪声,避免了对分割结果带来的影响,且给出了详尽的运算过程,并通过仿真实验证明了模型的有效性和实用性。
最后针对脉冲噪声的特点给出了一种有效的迭代滤波算法。
Recently,a segmentation methods based on level set have attracted more and moreattentions. Compared to traditional image segmentation approaches, the method havemany prominent virtues: such as insensitivity to the initial curve position, the strongability to deal with the topological changes etc.
First of all, the paper summarizes the purpose and significance of image segmentation,introduce and analyze Mumford-Shah model,Chan-Vese model and piecewise constant levelset method.
Second, in the base of piecewise constant level set image segmentation method basedon Mumford-Shah model in [10], in order to improve the segmentation e?ect and increasethe algorithm speed, we analyze and improve structure of the term ,we propose a improvedMumford-shah model.The experiment shows the model is validity.
Third, in order to avoid the in?uence of texture, we adopt decomposition idea inMumford-Shah model, and combine with piecewise constant level set method. We proposea MSWD(Mumford-Shah with decomposition)model. the model can separate texture ornoise. We also show the algorithm methods, and testify the rationality and validity of themodel by experiment.
Finally, we propose an iterative filter algorithm based on impulse noise.
引文
[1] 章毓晋. 图像工程 (中册)[M]. 北京:清华大学出版社,2005.
[2] 王新成. 多媒体实用技术(图像分册)第一版 [M]. 成都电子科技大学出版社,1995.
[3] Caselles V,Morel J M,Sapiro G. Introduction to the special issue on partial di?erentialequation and geometry-driven di?usion in image processing and analysis. IEEE Trans onImage Processing, 1998,vol. 7:269-272..
[4] Chan T F,Shen J H, Vese L. Variational PDE models in image processing. Notice ofAMS.2003,50(1):14-26.
[5] L.I.Rudin, S.Osher,E.Fatemi.Nonlinear total cariation based noise removal algorithms. Phys-ica D,1992,vol.60,259-268.
[6] Marius Lysaker,Arvid Lundervold,Xue-Cheng Tai. Noise removel using fourth-order partialdi?erential equation with applications to medical magnetic resonance images in space andtime. IEEE trans on Image processing, 2003 vol.12:1579-1590.
[7] M. Bertalmio, L. Vese, G. Sapiro, and O. Osher. Simultaneous texture and structure imageinpainting. IEEE Trans. Image Process.2003, 10(8).
[8] T. F. Chan and J. Shen. Mathematical models for local nontexture inpaintings. SIAM J.Appl. Math.2002,62(3):1019-1043.
[9] T. Chan and L. A. Vese. Active contours without edges. IEEE Tran. image Proc.2001,10:266-277.
[10] J.Lie, M. Lysaker, and X.-C. Tai. A variant of the level set method and applications toimage segmentation. Math. Comp.2006.
[11] Johan Lie, Marius Lysaker, Xue-Cheng Tai: A binary level set model and some applicationsto Mumford-Shah image segmentation. IEEE Transactions on Image Processing 2006,15(5):1171-1181.
[12] 章毓晋. 图像分割 [M]. 北京:科学出版社,2001.
[13] M.Kass, A.Witkin, D.Terzopoulos. Snakes: Active contour models. In First InternationalConferenceon Computer Vision,1987.7,259-268.
[14] S.Kichenassamy, A.Kumar, P.Olver, A.Tannenbaum, A.Yezzi. Gradient ?ows and geometricactive contour models. In Proceedings of the 5th International Conference on ComputerVision,810-815, Boston,MA,June 1995.IEEE Computer Society Press.
[15] Caselles V, Kimmel R,Sapiro G. Geodesic active contours[J].International Journal of Com-puter Vision,1997,22(1):61-79.
[16] 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.
[17] 罗希平等. 图像分割方法综述 [J]. 模式识别与人工智能,1999,12(3):300-312.
[18] Aubert G, Kornprobst P. Mathematical problems in image processing, Applied MathematicsScience, Vol.147,Springer,2000.
[19] Sapiro G. Geometric Partial Di?erential Equation and Image Processing. London: Cam-bridge University Press,2001.
[20] Xu C, Pham D L, Prince J L. Image segmentation using deformable models. SPIE handbookson Medical imaging, Vol.3: Medical image analysis, May, 2000.
[21] T.F.Chan and L.A.Vese, Image segmentation using level sets and the piecewise constantMumford-Shan model, tech. rep, CAM Report 00-14, UCLA,Math, Depart, April 2000.revised December 2000.
[22] L.A.Vese and T.F.Chan, A new multiphase level set framework for image segmentation viathe Mumford-Shal model, International Journal of Computer Vision,50(2002), pp.271-293.
[23] Lars K. Nielsen, Xue-Cheng Tai, Sigurd Aannosen, and Magne Espedal. A binary levelset model for elliptic inverse problems with discontinuous coe?cients. CAM-report-05-51,UCLA, Applied Mathematics,2005.
[24] Xue-Cheng Tai and Hongwei Li. Piecewise constant level set methods (PCLSM) for ellipticinverse problems. CAM-report-05-59, UCLA, Applied Mathematics, 2005.
[25] Xue-Cheng Tai and Changhui Yao. Fast piecewise constant level set methods (PCLSM) withnewton updating. CAM-report-05-52, UCLA, Applied Mathematics, 2005.
[26] 曹远星, 董宇宁. 蛇模型综述 [J]. 信息技术,2006,3:113-116.
[27] Caselles V,Catte F, Coll T, and Dibos F. A geometric model for active contours[J]. NumerMath, 1993,66:1-31.
[28] Osher S, Sethian J A. Fronts propagating with curvature dependent speed: algorithms basedon Hamilton-Jacobi formulations[J]. Journal of Computational Physics, 1988, 79: 12-49.
[29] V. Caselles, F. Catt é, T. Coll, and F. Dibos, ”A geometric model for active contours inimage processing,” Numer. Math.1993, vol. 66, pp. 1-31.
[30] Mumford D, Shah J. Optimal approximation by piecewise smooth functions and associatedvariational problems[J]. Comm. PureAppl. Math, 1989, 42: 577-685.
[31] Song Gao, Tien D.Bui. Image segmentation and Selective smoothing by using Mumford-ShahModel [J]. IEEE Trans. on Image Processing. 2005,14(10): 1537-1549.
[32] L.Ambrosio and V.M.Tortorelli. Approximation of functionals depending on jumps by el-liptic functionals via -convergence. Communications on Pure and Applied Mathematics,1990,XLIII: 999-1036.
[33] A.Braides. Approximation of free-discontinuity problems, Vol.1694 of Lecture Notes in Math-ematics. Springer-Verlag,1998.
[34] K.Kunisch and X-C Tai, sequential and parallel splitting method for bilinear control prob-lems in Hilbert spaces, SIAM J. Numer. Anal,1997, 34: 91-118.
[35] Xue-Cheng Tai, Oddvar Christiansen, Ping Lin and Inge Skjaelaaen, Image segmentationusing some piecewise constant level set methods with MBO type of project. InternationalJournal of Computer Vision, JUN 2007,73 (1): 61-76 .
[36] Christiansen.O, Tai. X-C.,Fast Implementation of Piecewise Constant Level Set Methods,UCLA Applied Mathematics CAM-report 2006; 06(19).
[37] Aubert G, Vese L. A variational method in image recovery. SIAM journal of NumericalAnalysis,1997,34(5):1948-1979
[38] Kornprobst P, Deriche R, Aubert G. Nonlinear operators in image restoration. In Proceed-ings of the International conference on Computer Vision and Pattern Recognition, pages325-331, Puerto Rico, June 1997.IEEE computer Society, IEEE
[39] Y. Meyer, Oscillating Patterns in Image Processing and in Some Nonlinear Evolution Equa-tions, The Fifteenth Dean Jacquelines B. Lewis Memorial Lectures, 2001.
[40] L.Vese, S. Osher, Modeling Textures With Total Variation Minimization and OscillatingPatterns in Image processing, UCLA CAM Report 02-19, July 2002.
[41] S. Aliney. A property of the minimum vectors of a regularizing functional defined by meansof the absolute norm. IEEE Transactions on Signal Processing, 1997,45(4):913-917 .
[42] M. Nikolova. A varational approach to remove outliers and impulse noise. JMIV,2004,20(1-2):90-120.
[43] W. T. Yin, D. Goldfarb and S. Osher, Image cartoon-texture decomposition and featureselection using the total variation regularized functional, CAM report CAM05-47, 2005,UCLA, USA.
[44] T. Chan and S. Esedoglu. Aspects of Total Cariation Regularized Function Approximation,2004. CAM report 04-07, To Appear in SIAM Journal on Applied Mathematics.
[45] J-F. Aujol, G. Gilboa, T. Chan, S. Osher, ”Structure-texture image decomposition - model-ing, algorithms, and parameter selection”, International Journal of Computer Vision (IJCV),2006,67(1), pp. 111-136,
[46] Ko SJ,lee YH Centerweighted filters and their applications to image enhancement [J].IEEECircuits System,1991,38 (9):984-993
[47] Wang Zhou , Zhang D. Progressive switching median filter for the removal of impulse noisefrom highly corrupted images. IEEE Trans on Circuits and Systems Ⅱ,1999,46(1):78-80.
[48] Wang Junghua, L in L ianda. Improved median filter using minmax algorithm for imageproces-sing [ J ]. Electronics Letters, 1997, 33 (16): 1362 - 1363.
[49] Chen T, Ma K K, Chen L H. Tri - state median filter for image denoising [ J ]. IEEE Imageprocess,1999,8(12):1834-1838
[50] Florencio D A F, Schafer R W. Decision - based median filter using local signal statistics [J ]. Proc SP IE Symp Visual Comm. Image Process, 1994, 2038: 268 - 275.
[51] Lin Ho-ming, Wilson A N. Median filters with adaptive length [ J ].IEEE Transactions onCircuits and System, 1988, 35 ( 6 ) : 675 ～ 690.
[52] 曲延峰, 徐健. 有效去除图像中脉冲噪声的新型滤波算法 [ J ]. 计算机辅助设计与图形学学报, 2003, 15(4) : 397 - 401.
[53] 张宇, 王希勤, 彭应宁. 自适应中心加权的改进均值滤波算法 [J]. 清华大学学报 (自然科学版)1999, 39( 9) : 80.
[54] Wilkinson MHF. Optimizing edge detectors for robust automatic threshold selection: copy-ing with edge curvature and noise.GMIP,1998,60(4):385 201
[2] 王新成. 多媒体实用技术(图像分册)第一版 [M]. 成都电子科技大学出版社,1995.
[3] Caselles V,Morel J M,Sapiro G. Introduction to the special issue on partial di?erentialequation and geometry-driven di?usion in image processing and analysis. IEEE Trans onImage Processing, 1998,vol. 7:269-272..
[4] Chan T F,Shen J H, Vese L. Variational PDE models in image processing. Notice ofAMS.2003,50(1):14-26.
[5] L.I.Rudin, S.Osher,E.Fatemi.Nonlinear total cariation based noise removal algorithms. Phys-ica D,1992,vol.60,259-268.
[6] Marius Lysaker,Arvid Lundervold,Xue-Cheng Tai. Noise removel using fourth-order partialdi?erential equation with applications to medical magnetic resonance images in space andtime. IEEE trans on Image processing, 2003 vol.12:1579-1590.
[7] M. Bertalmio, L. Vese, G. Sapiro, and O. Osher. Simultaneous texture and structure imageinpainting. IEEE Trans. Image Process.2003, 10(8).
[8] T. F. Chan and J. Shen. Mathematical models for local nontexture inpaintings. SIAM J.Appl. Math.2002,62(3):1019-1043.
[9] T. Chan and L. A. Vese. Active contours without edges. IEEE Tran. image Proc.2001,10:266-277.
[10] J.Lie, M. Lysaker, and X.-C. Tai. A variant of the level set method and applications toimage segmentation. Math. Comp.2006.
[11] Johan Lie, Marius Lysaker, Xue-Cheng Tai: A binary level set model and some applicationsto Mumford-Shah image segmentation. IEEE Transactions on Image Processing 2006,15(5):1171-1181.
[12] 章毓晋. 图像分割 [M]. 北京:科学出版社,2001.
[13] M.Kass, A.Witkin, D.Terzopoulos. Snakes: Active contour models. In First InternationalConferenceon Computer Vision,1987.7,259-268.
[14] S.Kichenassamy, A.Kumar, P.Olver, A.Tannenbaum, A.Yezzi. Gradient ?ows and geometricactive contour models. In Proceedings of the 5th International Conference on ComputerVision,810-815, Boston,MA,June 1995.IEEE Computer Society Press.
[15] Caselles V, Kimmel R,Sapiro G. Geodesic active contours[J].International Journal of Com-puter Vision,1997,22(1):61-79.
[16] 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.
[17] 罗希平等. 图像分割方法综述 [J]. 模式识别与人工智能,1999,12(3):300-312.
[18] Aubert G, Kornprobst P. Mathematical problems in image processing, Applied MathematicsScience, Vol.147,Springer,2000.
[19] Sapiro G. Geometric Partial Di?erential Equation and Image Processing. London: Cam-bridge University Press,2001.
[20] Xu C, Pham D L, Prince J L. Image segmentation using deformable models. SPIE handbookson Medical imaging, Vol.3: Medical image analysis, May, 2000.
[21] T.F.Chan and L.A.Vese, Image segmentation using level sets and the piecewise constantMumford-Shan model, tech. rep, CAM Report 00-14, UCLA,Math, Depart, April 2000.revised December 2000.
[22] L.A.Vese and T.F.Chan, A new multiphase level set framework for image segmentation viathe Mumford-Shal model, International Journal of Computer Vision,50(2002), pp.271-293.
[23] Lars K. Nielsen, Xue-Cheng Tai, Sigurd Aannosen, and Magne Espedal. A binary levelset model for elliptic inverse problems with discontinuous coe?cients. CAM-report-05-51,UCLA, Applied Mathematics,2005.
[24] Xue-Cheng Tai and Hongwei Li. Piecewise constant level set methods (PCLSM) for ellipticinverse problems. CAM-report-05-59, UCLA, Applied Mathematics, 2005.
[25] Xue-Cheng Tai and Changhui Yao. Fast piecewise constant level set methods (PCLSM) withnewton updating. CAM-report-05-52, UCLA, Applied Mathematics, 2005.
[26] 曹远星, 董宇宁. 蛇模型综述 [J]. 信息技术,2006,3:113-116.
[27] Caselles V,Catte F, Coll T, and Dibos F. A geometric model for active contours[J]. NumerMath, 1993,66:1-31.
[28] Osher S, Sethian J A. Fronts propagating with curvature dependent speed: algorithms basedon Hamilton-Jacobi formulations[J]. Journal of Computational Physics, 1988, 79: 12-49.
[29] V. Caselles, F. Catt é, T. Coll, and F. Dibos, ”A geometric model for active contours inimage processing,” Numer. Math.1993, vol. 66, pp. 1-31.
[30] Mumford D, Shah J. Optimal approximation by piecewise smooth functions and associatedvariational problems[J]. Comm. PureAppl. Math, 1989, 42: 577-685.
[31] Song Gao, Tien D.Bui. Image segmentation and Selective smoothing by using Mumford-ShahModel [J]. IEEE Trans. on Image Processing. 2005,14(10): 1537-1549.
[32] L.Ambrosio and V.M.Tortorelli. Approximation of functionals depending on jumps by el-liptic functionals via -convergence. Communications on Pure and Applied Mathematics,1990,XLIII: 999-1036.
[33] A.Braides. Approximation of free-discontinuity problems, Vol.1694 of Lecture Notes in Math-ematics. Springer-Verlag,1998.
[34] K.Kunisch and X-C Tai, sequential and parallel splitting method for bilinear control prob-lems in Hilbert spaces, SIAM J. Numer. Anal,1997, 34: 91-118.
[35] Xue-Cheng Tai, Oddvar Christiansen, Ping Lin and Inge Skjaelaaen, Image segmentationusing some piecewise constant level set methods with MBO type of project. InternationalJournal of Computer Vision, JUN 2007,73 (1): 61-76 .
[36] Christiansen.O, Tai. X-C.,Fast Implementation of Piecewise Constant Level Set Methods,UCLA Applied Mathematics CAM-report 2006; 06(19).
[37] Aubert G, Vese L. A variational method in image recovery. SIAM journal of NumericalAnalysis,1997,34(5):1948-1979
[38] Kornprobst P, Deriche R, Aubert G. Nonlinear operators in image restoration. In Proceed-ings of the International conference on Computer Vision and Pattern Recognition, pages325-331, Puerto Rico, June 1997.IEEE computer Society, IEEE
[39] Y. Meyer, Oscillating Patterns in Image Processing and in Some Nonlinear Evolution Equa-tions, The Fifteenth Dean Jacquelines B. Lewis Memorial Lectures, 2001.
[40] L.Vese, S. Osher, Modeling Textures With Total Variation Minimization and OscillatingPatterns in Image processing, UCLA CAM Report 02-19, July 2002.
[41] S. Aliney. A property of the minimum vectors of a regularizing functional defined by meansof the absolute norm. IEEE Transactions on Signal Processing, 1997,45(4):913-917 .
[42] M. Nikolova. A varational approach to remove outliers and impulse noise. JMIV,2004,20(1-2):90-120.
[43] W. T. Yin, D. Goldfarb and S. Osher, Image cartoon-texture decomposition and featureselection using the total variation regularized functional, CAM report CAM05-47, 2005,UCLA, USA.
[44] T. Chan and S. Esedoglu. Aspects of Total Cariation Regularized Function Approximation,2004. CAM report 04-07, To Appear in SIAM Journal on Applied Mathematics.
[45] J-F. Aujol, G. Gilboa, T. Chan, S. Osher, ”Structure-texture image decomposition - model-ing, algorithms, and parameter selection”, International Journal of Computer Vision (IJCV),2006,67(1), pp. 111-136,
[46] Ko SJ,lee YH Centerweighted filters and their applications to image enhancement [J].IEEECircuits System,1991,38 (9):984-993
[47] Wang Zhou , Zhang D. Progressive switching median filter for the removal of impulse noisefrom highly corrupted images. IEEE Trans on Circuits and Systems Ⅱ,1999,46(1):78-80.
[48] Wang Junghua, L in L ianda. Improved median filter using minmax algorithm for imageproces-sing [ J ]. Electronics Letters, 1997, 33 (16): 1362 - 1363.
[49] Chen T, Ma K K, Chen L H. Tri - state median filter for image denoising [ J ]. IEEE Imageprocess,1999,8(12):1834-1838
[50] Florencio D A F, Schafer R W. Decision - based median filter using local signal statistics [J ]. Proc SP IE Symp Visual Comm. Image Process, 1994, 2038: 268 - 275.
[51] Lin Ho-ming, Wilson A N. Median filters with adaptive length [ J ].IEEE Transactions onCircuits and System, 1988, 35 ( 6 ) : 675 ～ 690.
[52] 曲延峰, 徐健. 有效去除图像中脉冲噪声的新型滤波算法 [ J ]. 计算机辅助设计与图形学学报, 2003, 15(4) : 397 - 401.
[53] 张宇, 王希勤, 彭应宁. 自适应中心加权的改进均值滤波算法 [J]. 清华大学学报 (自然科学版)1999, 39( 9) : 80.
[54] Wilkinson MHF. Optimizing edge detectors for robust automatic threshold selection: copy-ing with edge curvature and noise.GMIP,1998,60(4):385 201