基于动态轮廓模型的图像分割算法研究
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摘要
图像分割已经成为一种重要的图像分析技术。在对图像的研究和应用中,人们往往仅对图像中的某些部分感兴趣,这些部分常称之为目标或者前景,对应图像中特定的、具有独特性质的区域。为了辨识和分析图像中的目标,需要将它们从图像中分离提取出来,进一步测量和利用。作为图像分析与图像理解的基础,图像分割是计算机识别领域中最基本、也是最困难的问题之一。现有的图像分割算法各有千秋,这些方法都是针对特定图像提出来的。由于图像的多样性和复杂性,目前还没有统一的分割实现方案。
通过分析国内外图像分割领域的研究现状和发展趋势,本文整理、归纳和总结了一般的图像分割理论、方法,对基于动态轮廓的分割算法进行了深入研究,重点研究适用于分割灰度分布不均匀的非同质特性图像的动态轮廓模型。
Chan-Vese(C-V)模型是最经典的动态轮廓模型,建立在边界演化理论和水平集方法基础上,通过动态轮廓不断演化来检测图像中的目标,轮廓能够在期望的边界处停止演化,演化的动力来自动态轮廓内部和外部的全局灰度统计信息,是一种基于区域信息的算法。该模型的显著特点是,待检测目标的边界不用通过梯度来定义,降低了模型对图像中噪声的敏感度,分割效果较好,但C-V模型在分割非同质特性图像时容易出现误分割。为改善C-V模型在分割非同质特性图像时出现的误分割问题,本文提出一种局部特性函数,用来限制轮廓演化所依据的统计信息参考范围。并将其引入到动态轮廓模型中,依据轮廓上各点的局部内部和局部外部统计信息来演化轮廓,构建由局部灰度统计信息驱动的动态轮廓模型。仿真实验表明,通过改变轮廓上各点演化趋势的参考信息,新模型在分割非同质特性图像时,轮廓能在期望的目标边界处停止演化,得到理想的分割结果。
Image segmentation has become an important image analysis technology. In the research and application of image, people often only interested in certain parts of the image, these sections often refer to objects or prospects.Objects and prospects correspond to the region with a particular and unique nature. In order to identify and analyze objects, we need to extract them from the image, on this basis, we could make further use of the image. As the basis of image analysis and understanding, image segmentation is one of the most basic but also difficult problems in the field of computer recognition. The existing image segmentation algorithms are special; these methods are aimed at specific images. Because of the diversity and complexity of image, there is no uniform segmentation implementation.
Through the analysis of the domestic and foreign image segmentation research situation and development trend, this paper arrangements and summarizes the general theory and method of image segmentation, and make further research on segmentation algorithm based on actice contours. Our key research is active contours model which is applicable to heterogeneous images.
Chan-Vese (CV) model is the most classical active contours model. C-V model bases on the theroy of boundary evolution and the method of level set, which can detect the object in the image by using an active contour. Active contour stop evoluting at the expected boundary. The force of evolution is decided by the global internal and external gray-scale mean of contours. It is a region-based algorithm. The outstanding feature of this model is that the object's boundaries to be detected are not defined by gradient. This feature reduces sensitivity of model to noise in image, but C-V model always generate wrong segmentation with heterogeneous image. In order to improve inaccurate segmentation when we use C-V model to segmente heterogeneous images, this paper propose a local property function to limit reference range of statictis information which is used by contour evolution, and import this function into active contours model to build a new model which is forced by local statistics information. When segmenting heterogeneous image, experiments show that according to the reference information of points which are on the active contour, the active contour can stops evoluting on the expected boundary and obtains the ideal segmentation results.
通过分析国内外图像分割领域的研究现状和发展趋势,本文整理、归纳和总结了一般的图像分割理论、方法,对基于动态轮廓的分割算法进行了深入研究,重点研究适用于分割灰度分布不均匀的非同质特性图像的动态轮廓模型。
Chan-Vese(C-V)模型是最经典的动态轮廓模型,建立在边界演化理论和水平集方法基础上,通过动态轮廓不断演化来检测图像中的目标,轮廓能够在期望的边界处停止演化,演化的动力来自动态轮廓内部和外部的全局灰度统计信息,是一种基于区域信息的算法。该模型的显著特点是,待检测目标的边界不用通过梯度来定义,降低了模型对图像中噪声的敏感度,分割效果较好,但C-V模型在分割非同质特性图像时容易出现误分割。为改善C-V模型在分割非同质特性图像时出现的误分割问题,本文提出一种局部特性函数,用来限制轮廓演化所依据的统计信息参考范围。并将其引入到动态轮廓模型中,依据轮廓上各点的局部内部和局部外部统计信息来演化轮廓,构建由局部灰度统计信息驱动的动态轮廓模型。仿真实验表明,通过改变轮廓上各点演化趋势的参考信息,新模型在分割非同质特性图像时,轮廓能在期望的目标边界处停止演化,得到理想的分割结果。
Image segmentation has become an important image analysis technology. In the research and application of image, people often only interested in certain parts of the image, these sections often refer to objects or prospects.Objects and prospects correspond to the region with a particular and unique nature. In order to identify and analyze objects, we need to extract them from the image, on this basis, we could make further use of the image. As the basis of image analysis and understanding, image segmentation is one of the most basic but also difficult problems in the field of computer recognition. The existing image segmentation algorithms are special; these methods are aimed at specific images. Because of the diversity and complexity of image, there is no uniform segmentation implementation.
Through the analysis of the domestic and foreign image segmentation research situation and development trend, this paper arrangements and summarizes the general theory and method of image segmentation, and make further research on segmentation algorithm based on actice contours. Our key research is active contours model which is applicable to heterogeneous images.
Chan-Vese (CV) model is the most classical active contours model. C-V model bases on the theroy of boundary evolution and the method of level set, which can detect the object in the image by using an active contour. Active contour stop evoluting at the expected boundary. The force of evolution is decided by the global internal and external gray-scale mean of contours. It is a region-based algorithm. The outstanding feature of this model is that the object's boundaries to be detected are not defined by gradient. This feature reduces sensitivity of model to noise in image, but C-V model always generate wrong segmentation with heterogeneous image. In order to improve inaccurate segmentation when we use C-V model to segmente heterogeneous images, this paper propose a local property function to limit reference range of statictis information which is used by contour evolution, and import this function into active contours model to build a new model which is forced by local statistics information. When segmenting heterogeneous image, experiments show that according to the reference information of points which are on the active contour, the active contour can stops evoluting on the expected boundary and obtains the ideal segmentation results.
引文
[1]章毓晋.中国图像工程:1995,中国图形学报,1996,1(1):78-83.
[2] Gonzalez.R.C,Woods.R.E.Digital Image Processing[M].Addison–Wesley,1st ed.1993.
[3]王新成.多媒体使用技术(图像分册)第一版[M].成都:电子科技大学出版社,1995.
[4]章毓晋.图像分割[M].科学出版社,2001.
[5] Pal N R,Pal S K.A review on image segmentation technique[sJ].Pattern Recognition,1993,26:1277-1294.
[6] Wilkinson M H F.Optimizing edge detectors for robust automatic threshold selection:coping with edge curvature and noise[J],Graphical Models and Image Processing,1998,60(4):385-401.
[7] Marr D,Hildreth E.Theory of edge detection[J].Proceedings of the Royal Society of London,Series B,Biological Sciences,1980,1167(207):187-217.
[8] Trucco E,Verri A.Introductory techniques for 3-D computer vision[M].Prentice Hall,1988.
[9] Canny J.A computational approach to edge detection[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,1986,8:679-698.
[10] Hong T H,Rosenfeld A.Compact region extraction using weighted pixel linking in a pyramid[J].IEEE-Transactions on Pattern Analysis and Machine Intelligence,1984,6(2):222237.
[11] Gross A D,Rosenfeld A.Multiresolution object detection and delineation[J].Computer Vision Graphics and Image Processing,1987,39:102-115.
[12] Horowitz S L,Pavlidis T.Picture segmentation by a tree traversal algorithm[J].J ACM,1976,23:368-388.
[13] Snyder W,Qi H.Machine Vision[M].Cambridge:Cambridge University,2004.
[14] Silverman J,Cooper D.Bayesian clustering for unsupervised estimation of surface andtexture models[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,1993,57:373-387.
[15] Lavalle M,Hutchinson S.A bayesian segmentation methodology for parametric image models[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,1995,17:211-217.
[16] Roerdink J,Meijster A.The watershed transform:Definitions,algorithms and parallel strategies[J].Fundamenta Informaticae,2000,41:187-228.
[17] Beucher S,Lantuejoul C.Use of watershed in contour detection[J].International Workshop in Image Processing:Real-time Edge and Motion Detection/Estimation,1979.
[18] Ederra G.Mathematical morphology techniques applied to anti-personnel mine detection[D].Belgium:Department of Electronics and Information Processing,Vrije University Brussel,1999.
[19] Beucher S.The watershed transformation applied to image segmentation[J].Proc.of Conference on Signal and Image Processing in Microscopy and Microanalysis,1991:299-314.
[20] Meyer F,Beucher S.Morphology segmentation[J].Journal of Visual Communication and Image Representation,1990,1(1):21-46.
[21] Beucher S.The watershed transform[EB/OL].http://cmm.ensmp.fr/~beucher/wtshed.html#watshed.
[22] Fukuda T,Morimoto Y,Morishita S,etc.Theory of communication[J].ACM Transactions on Database Systems,2001,2(26):179-213.
[23] Kass M,Witkin A,Terzopoulos D.Snakes,active contour model[J].International Journal of Computer Vision,1988:321-331.
[24] Osher S,Sethian J.Fronts propagating with curvature dependent speed:Algorithms based on Hamilton-jacobi formulations[J].Journal of Computational Physics,1988,79:12-49.
[25]王寿生等.数学物理方法[M].西安:西北工业大学,1992.
[26]徐建平,桂子鹏.变分方法[M].上海:同济大学出版社,1999.
[27]杨新.图像偏微分方程的原理与应用:上海交通大学出版社.2003.
[28] Sethian J A.Curvature and evolution of fronts[J].Communication of Mathematical Physics,1985,4(101):487-499.
[29] Sethian J A.A review of recent numerical algorithms for hypersurfaces moving with vurvature dependent speed[J].Differential Geometry,1989,31:131-161.
[30] Sethian J A.An analysis of flame propagation.Ph.D Thesis,Dept.of Mathematics,University of California,Berkeley,CA,1982.
[31] Adalsteinsson D,Sethian J A.The fast construction of extension velocities in level set methods[J].Journal of Computational Physics,1999,148:2-22.
[32] Chop D.Computing minimal surface via Level Set curvature flow[J].Journal of Computational Physics,1993,106:77-91.
[33] Kass M,Witkin A,Terzopoulos D.Snakes,active contour model.International Journal of Computer Vision[J],1988,1:321-331.
[34] Caselles V,Catte F,Coll T,etc.A geometric model for active contour[sJ].Numerische Mathematik,1993,66:1-31.
[35] Malladi R,Sethian J,Vemuri C.Shape modeling with front propagation:a level set approach[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,1995,17(2):158-175.
[36] Yezzi A,Kichenassamy S,Kumar A.A geometric snake model for segmentation of medical imagery[J].IEEE Transactions on Medical Imaging,1997,2:199-209.
[37] Cohen F,Cooper D.Maximum likelihood unsupervised textured image segmentation[J].Computer Vision,Graphics,and Image Process,1992,54:239-251.
[38] Caselles V,Kimmel R,Sapiro G.Geodesic active contour[J].International Journal of Computer Vision,1997,1:61-79.
[39] Caselles V,Kimmel R,Sapiro G.Geodesic active contour[J].Proc.of IEEE Internationl Conference on Computer Vision,1995:694-699.
[40] Kichenassamy S,Kumar A,Olver P,etc.Gradient flows and geometric active contour models[J].Proc.of IEEE Internationl Conference on Computer Vision,1995:810-815.
[41] Sapiro G.Color snakes[R].America:Hewlett-Packard Labs,1995.
[42] Aubert G,Blanc-Feraud L[J].Some remarks on the equivalence between 2d and 3dclassical snakes and geodesic active contours.International Journal of Computer Vision,1999,1:19-28.
[43]Aubert G,Blanc-Feraud L.An elementary proof of the equivalence between 2d and 3d classical snakes and geodesic active contours[R].France:INRIA,1998.
[44] Xu C,Yezzi A,Prince J.A summary of geometric level set analogues for a general calss of parametric active contour and surface models[J].Proc.of IEEE Workshop on Variational,Geometric and Level Set Method in Computer Vision,2001:104-111.
[45] Xu C,Yezzi A,Prince J.on the relationship between parametric and geometric active contours[J].Proc.of IEEE Conference on Signals,Systems and Computers,2000:483-489.
[46] Sapiro G.Color snakes[J].Computer Vision and Image Understanding,1997,2:247-253.
[47] Sapiro G.Vector (self) snakes:a geometric framework for color,texture,and multiscale image segmentation[J].Proc.of IEEE International Conference on Image Processing,1996:817-820.
[48] Sapiro G.Vector-valued acative contours[J].Proc.of IEEE Conference on Computer Vision and Pattern Recognition,1996:680-685.
[49] Paragios N,Mellina-Gottardo O,Ramesh V.Gradient vector flow fast geometric active contours[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,2004,3(26):402-407.
[50] Paragios N,Mellina-Gottardo O,Ramesh V.Gradient vector flow fast geodesic active contours[J].Proc.of IEEE International Conference on Computer Vision,2001,1:7-14.
[51] Xu C,Prince J.Handbook of Medical Imaging:Processing and Analysis Management,ch.Gradient Vector Flow Deformable Models[M].United States:Academic Press,2000.
[52] Xu C,Pham D,Prince J.Handbook of Medical Imaging:Processing and Analysis Management,ch.Image Segmentation Using Deformable Model[sM].United States:Academic Press,2000:129-174.
[53] Xu C,Prince J.Generalized gradient vector flow external forces for active contours[J].Signal Processing,1998,3:131-139.
[54] Xu C,Prince J.Snakes,shapes,and gradient vector flow[J].IEEE Transactions on Image Processing,1998,3:359-369.
[55]李俊,杨新,施鹏飞.基于Mumford-Shah模型的快速水平集图像分割方法[J].计算机学报,1997,25(11):1175-1183.
[56] Chan T,Vese L.Active contours without edges[J].IEEE Transactions on Image Processing,2001,2:266-277.
[57] Chan T,Vese L.Active contours without edges[R].America:UCLA,Computational Applied Math Group,1998.
[58] Mumford D,Shah J.Optimal approximation by piecewise smooth functions and associated variational problems[J].Communication Pure and Applied Mathematics,1989,42:577-685.
[59] Mumford D,Shah J.Boundary detection by minimizing functionals.Proc.of IEEE Conference on Computer Vision and Pattern Recognition[J].1985:19-23.
[60] Morel J,Solimini S.Variational Methods in Image Segmentation[R].Progress in nonlinear differential equations and their applications,Bostion:Birkhauser,1995.
[61] Zhao H,Chan T,Merriman B,etc.A variational level set approach to multiphase motion[J].Journal of Computational Physics,1996,127:179-195.
[62] Li C,Xu C,Gui C,etc.Level set evolution without re-ininialization:A new variational formulation[J].IEEE International Conference on Computer Vision and Pattern Recogniton,2005:430-436.
[63] Oshar S,Fedkiw R.Level Set Methods and Dynamic Implicit Surfaces.New York:Cambridge University Press,2003.
[64] Osher S,Tsai R.Level set methods and their applications in image science.Communication Mathematics Science,2003,4(1):1-20.
[2] Gonzalez.R.C,Woods.R.E.Digital Image Processing[M].Addison–Wesley,1st ed.1993.
[3]王新成.多媒体使用技术(图像分册)第一版[M].成都:电子科技大学出版社,1995.
[4]章毓晋.图像分割[M].科学出版社,2001.
[5] Pal N R,Pal S K.A review on image segmentation technique[sJ].Pattern Recognition,1993,26:1277-1294.
[6] Wilkinson M H F.Optimizing edge detectors for robust automatic threshold selection:coping with edge curvature and noise[J],Graphical Models and Image Processing,1998,60(4):385-401.
[7] Marr D,Hildreth E.Theory of edge detection[J].Proceedings of the Royal Society of London,Series B,Biological Sciences,1980,1167(207):187-217.
[8] Trucco E,Verri A.Introductory techniques for 3-D computer vision[M].Prentice Hall,1988.
[9] Canny J.A computational approach to edge detection[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,1986,8:679-698.
[10] Hong T H,Rosenfeld A.Compact region extraction using weighted pixel linking in a pyramid[J].IEEE-Transactions on Pattern Analysis and Machine Intelligence,1984,6(2):222237.
[11] Gross A D,Rosenfeld A.Multiresolution object detection and delineation[J].Computer Vision Graphics and Image Processing,1987,39:102-115.
[12] Horowitz S L,Pavlidis T.Picture segmentation by a tree traversal algorithm[J].J ACM,1976,23:368-388.
[13] Snyder W,Qi H.Machine Vision[M].Cambridge:Cambridge University,2004.
[14] Silverman J,Cooper D.Bayesian clustering for unsupervised estimation of surface andtexture models[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,1993,57:373-387.
[15] Lavalle M,Hutchinson S.A bayesian segmentation methodology for parametric image models[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,1995,17:211-217.
[16] Roerdink J,Meijster A.The watershed transform:Definitions,algorithms and parallel strategies[J].Fundamenta Informaticae,2000,41:187-228.
[17] Beucher S,Lantuejoul C.Use of watershed in contour detection[J].International Workshop in Image Processing:Real-time Edge and Motion Detection/Estimation,1979.
[18] Ederra G.Mathematical morphology techniques applied to anti-personnel mine detection[D].Belgium:Department of Electronics and Information Processing,Vrije University Brussel,1999.
[19] Beucher S.The watershed transformation applied to image segmentation[J].Proc.of Conference on Signal and Image Processing in Microscopy and Microanalysis,1991:299-314.
[20] Meyer F,Beucher S.Morphology segmentation[J].Journal of Visual Communication and Image Representation,1990,1(1):21-46.
[21] Beucher S.The watershed transform[EB/OL].http://cmm.ensmp.fr/~beucher/wtshed.html#watshed.
[22] Fukuda T,Morimoto Y,Morishita S,etc.Theory of communication[J].ACM Transactions on Database Systems,2001,2(26):179-213.
[23] Kass M,Witkin A,Terzopoulos D.Snakes,active contour model[J].International Journal of Computer Vision,1988:321-331.
[24] Osher S,Sethian J.Fronts propagating with curvature dependent speed:Algorithms based on Hamilton-jacobi formulations[J].Journal of Computational Physics,1988,79:12-49.
[25]王寿生等.数学物理方法[M].西安:西北工业大学,1992.
[26]徐建平,桂子鹏.变分方法[M].上海:同济大学出版社,1999.
[27]杨新.图像偏微分方程的原理与应用:上海交通大学出版社.2003.
[28] Sethian J A.Curvature and evolution of fronts[J].Communication of Mathematical Physics,1985,4(101):487-499.
[29] Sethian J A.A review of recent numerical algorithms for hypersurfaces moving with vurvature dependent speed[J].Differential Geometry,1989,31:131-161.
[30] Sethian J A.An analysis of flame propagation.Ph.D Thesis,Dept.of Mathematics,University of California,Berkeley,CA,1982.
[31] Adalsteinsson D,Sethian J A.The fast construction of extension velocities in level set methods[J].Journal of Computational Physics,1999,148:2-22.
[32] Chop D.Computing minimal surface via Level Set curvature flow[J].Journal of Computational Physics,1993,106:77-91.
[33] Kass M,Witkin A,Terzopoulos D.Snakes,active contour model.International Journal of Computer Vision[J],1988,1:321-331.
[34] Caselles V,Catte F,Coll T,etc.A geometric model for active contour[sJ].Numerische Mathematik,1993,66:1-31.
[35] Malladi R,Sethian J,Vemuri C.Shape modeling with front propagation:a level set approach[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,1995,17(2):158-175.
[36] Yezzi A,Kichenassamy S,Kumar A.A geometric snake model for segmentation of medical imagery[J].IEEE Transactions on Medical Imaging,1997,2:199-209.
[37] Cohen F,Cooper D.Maximum likelihood unsupervised textured image segmentation[J].Computer Vision,Graphics,and Image Process,1992,54:239-251.
[38] Caselles V,Kimmel R,Sapiro G.Geodesic active contour[J].International Journal of Computer Vision,1997,1:61-79.
[39] Caselles V,Kimmel R,Sapiro G.Geodesic active contour[J].Proc.of IEEE Internationl Conference on Computer Vision,1995:694-699.
[40] Kichenassamy S,Kumar A,Olver P,etc.Gradient flows and geometric active contour models[J].Proc.of IEEE Internationl Conference on Computer Vision,1995:810-815.
[41] Sapiro G.Color snakes[R].America:Hewlett-Packard Labs,1995.
[42] Aubert G,Blanc-Feraud L[J].Some remarks on the equivalence between 2d and 3dclassical snakes and geodesic active contours.International Journal of Computer Vision,1999,1:19-28.
[43]Aubert G,Blanc-Feraud L.An elementary proof of the equivalence between 2d and 3d classical snakes and geodesic active contours[R].France:INRIA,1998.
[44] Xu C,Yezzi A,Prince J.A summary of geometric level set analogues for a general calss of parametric active contour and surface models[J].Proc.of IEEE Workshop on Variational,Geometric and Level Set Method in Computer Vision,2001:104-111.
[45] Xu C,Yezzi A,Prince J.on the relationship between parametric and geometric active contours[J].Proc.of IEEE Conference on Signals,Systems and Computers,2000:483-489.
[46] Sapiro G.Color snakes[J].Computer Vision and Image Understanding,1997,2:247-253.
[47] Sapiro G.Vector (self) snakes:a geometric framework for color,texture,and multiscale image segmentation[J].Proc.of IEEE International Conference on Image Processing,1996:817-820.
[48] Sapiro G.Vector-valued acative contours[J].Proc.of IEEE Conference on Computer Vision and Pattern Recognition,1996:680-685.
[49] Paragios N,Mellina-Gottardo O,Ramesh V.Gradient vector flow fast geometric active contours[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,2004,3(26):402-407.
[50] Paragios N,Mellina-Gottardo O,Ramesh V.Gradient vector flow fast geodesic active contours[J].Proc.of IEEE International Conference on Computer Vision,2001,1:7-14.
[51] Xu C,Prince J.Handbook of Medical Imaging:Processing and Analysis Management,ch.Gradient Vector Flow Deformable Models[M].United States:Academic Press,2000.
[52] Xu C,Pham D,Prince J.Handbook of Medical Imaging:Processing and Analysis Management,ch.Image Segmentation Using Deformable Model[sM].United States:Academic Press,2000:129-174.
[53] Xu C,Prince J.Generalized gradient vector flow external forces for active contours[J].Signal Processing,1998,3:131-139.
[54] Xu C,Prince J.Snakes,shapes,and gradient vector flow[J].IEEE Transactions on Image Processing,1998,3:359-369.
[55]李俊,杨新,施鹏飞.基于Mumford-Shah模型的快速水平集图像分割方法[J].计算机学报,1997,25(11):1175-1183.
[56] Chan T,Vese L.Active contours without edges[J].IEEE Transactions on Image Processing,2001,2:266-277.
[57] Chan T,Vese L.Active contours without edges[R].America:UCLA,Computational Applied Math Group,1998.
[58] Mumford D,Shah J.Optimal approximation by piecewise smooth functions and associated variational problems[J].Communication Pure and Applied Mathematics,1989,42:577-685.
[59] Mumford D,Shah J.Boundary detection by minimizing functionals.Proc.of IEEE Conference on Computer Vision and Pattern Recognition[J].1985:19-23.
[60] Morel J,Solimini S.Variational Methods in Image Segmentation[R].Progress in nonlinear differential equations and their applications,Bostion:Birkhauser,1995.
[61] Zhao H,Chan T,Merriman B,etc.A variational level set approach to multiphase motion[J].Journal of Computational Physics,1996,127:179-195.
[62] Li C,Xu C,Gui C,etc.Level set evolution without re-ininialization:A new variational formulation[J].IEEE International Conference on Computer Vision and Pattern Recogniton,2005:430-436.
[63] Oshar S,Fedkiw R.Level Set Methods and Dynamic Implicit Surfaces.New York:Cambridge University Press,2003.
[64] Osher S,Tsai R.Level set methods and their applications in image science.Communication Mathematics Science,2003,4(1):1-20.