摘要
多相图像分割是目前国际上图像处理、计算机视觉等领域研究的热点问题之一,在医学诊断、计算机辅助手术、机器视觉、基于遥感图像的资源分类等领域具有重要应用。
由于问题的复杂性,具有拓扑自适应能力的水平集方法成为多相图像分割的主流方法,而图像分割的变分方法能方便地集成图像的边缘、区域、形状、纹理及运动场信息,易于建立图像分割的集成化模型,具有很好的通用性和可扩展性。本文对基于变分水平集的多相图像分割的数学模型、数值方法及应用进行了系统的研究,所开展的主要工作包括:
1)对国内、国际近年发展的图像分割的变分水平集方法及相关数值差分方法进行了较全面的综述,并分析了不同模型所存在的主要问题。
2)在对基于边缘和区域的两相图像分割的变分水平集方法研究基础上,提出了传统方法无需水平集函数重新初始化的改进模型,并将所得到的结果推广到多相图像分割的变分水平集模型。
3)建立了基于区域的图像分割的通用模型,该模型能方便地应用于不同噪声分布的图像分割。本文还将上述模型由分段常值图像分割延伸到分段光滑的图像分割。
4)基于多水平集函数建立了多相图像分割的通用模型,可适应于任意相数的图像分割。提出了基于通过变分水平集方法以模块化的形式集成多种图像分割模型成分,易于实现复杂的多相图像分割,具有较好的拓展能力。
5)对含噪声和纹理分布区域的多相图像分割,本文将多相图像分割和图像扩散模型结合,建立了含噪声和纹理图像分割的多相图像分割模型。采用图像扩散的方法去除噪声,并对纹理区域进行扩散,从而可将不同区域看作近似分段常值和分段光滑区域处理。
6)针对分段光滑图像多相图像分割中水平集函数演化方程的不规则边界条件,本文采用了图像修复技术,将差分计算的区域自动扩展到规则的网格空间,并得到理想结果。
此外,本文利用人工图像对所提出的方法进行了验证,还将上述结果对遥感图像处理领域的海岸带分类、海岸线及河道提取、海岛植被分布信息提取及海洋溢油区域提取等进行了初步实验。
Multiphase image segmentation is one of the hot issues in the fields of image processing, computer vision etc. It has important applications in medical diagnoses, computer-aided surgery, machine vision, resource classification based on remote sensing and so on.
Due to complexity of the problem, level set method with topologically self-adaptive capability becomes the mainstream method of multiphase image segmentation. On the other hand, variational method can conveniently combine different types of image information, such as edge, region, shape, texture and motion field, making it a better choice for building integrated image segmentation models with good universality and expansibility. In this paper, the mathematical models, numerical methods and applications of multiphase image segmentation based on variational level set methods are studied. The main content of the research work is:
1) Previous research works in recent years on variational level set methods for image segmentation and related numerical difference methods are studied extensively and described in detail. Major problems for each method are analyzed.
2) An improved model of traditional variational level set method for two phase image segmentation based on edge and region is proposed. In this new method, re-initialization of level set function is no longer needed. This method is also extended to accommodate multiphase image segmentation.
3) A general region-based image segmentation model suitable for image with different noise distribution is established. The application domain for this method is later extended from piecewise constant image to piecewise smooth image.
4) A general image segmentation model based on multiple level set functions that can be applied to images with an arbitrary number of phases is established. A new implementation strategy that can integrate aspects of different segmentation models through modularization is proposed. This new strategy has good extensibility and can be used to implement complex multiphase image segmentation operations.
5) A new model for segmentation of multiphase image with noise and texture distribution is designed by combining multiphase image segmentation methods established above with image diffusion method. Image diffusion can reduce noise and transform textured regions into piecewise constant and piecewise smooth regions.
6) For segmentation of piecewise smooth multiphase image, techniques for image inpainting are used to extend difference equation solution space into regular grid space. By doing so, the irregular boundary conditions of level set function evolution equation are eliminated.
Finally, the proposed methods are tested and verified on some artificial images and a number of remote sensing image processing tasks, including coastal zone classification, coastline and watercourse extraction, acquisition of island vegetation distribution information and extraction of oil spill region.
引文
[1]章毓晋.图象分割.北京:科学出版社,2001.1-7
[2] Aubert G., Kornprobst P. Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences, Springer-Verlag, 2002. 1-4
[3] Chan T. F., Shen J., Vese L. Variational PDE models in image processing. Notices Amer. Math. Soc. , 2003, 50(1):14–26
[4] Paragios N., Chen Y., Faugeras O. Handbook of Mathematical Models in Computer Vision, Springer-Verlag, New York, 2006. 63-174
[5] Suri J. S., Liu K., Singh S., et al. Shape Recovery Algorithms Using Level Sets In 2-D/3-D Medical Imagery: A State-Of-The-Art Review. IEEE Transactions on information Technology in Biomedicine, 2002, 6 (1): 8-28
[6] Cremers D., Sochen N., Schn?rr C. A Multiphase Dynamic Labeling for Variational Recognition-driven Image Segmentation. International Journal of Computer Vision, 2006, 66(1): 1573-1405
[7] Paragios N. A Level Set Approach for Shape-driven Segmentation and Tracking of the Left Ventricle. Journal of LATEX Class Files, 2002, 8(1): 1-4
[8] Terzopoulos D. Deformable Models: Classic, Topology-Adaptive and Generalized Formulations. In: Osher, S, and Paragios, N. Geometric Level Set Methods in Imaging, Vision, and Graphics, Springer, 2003, 21-40
[9] Osher S, Sethian J. Fronts Propagating with Curvature Dependent Speed: Algorithms Based on the Hamilton-Jacobi Formulation. Journal of Computational Physics, 1988, 79(1): 12-49
[10] Sethian J. A. Level Set Methods and Fast Marching Methods, 2nd ed. Cambridge Monographs on Applied and Computational Mathematics. Cambridge: Cambridge University Press, 1999
[11] Osher S., Fedkiw R. Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag, New York, 2002
[12] Zhao H. K., Chan T., Merriman B., Osher S. A variational level set approach to multiphase motion. J.Comput. Phys. 1996, 127(1): 179–195,
[13] Kass M., Witkin A., Terzopoulos D. Snakes: Active Contour Models. International Journal of Computer Visions, 1987, 1(4): 321–331
[14] Cohen L. D., Cohen I. Finite-Element Methods For Active Contour Models and Balloons for 2-D and 3-D Images. IEEE Trans. on Pattern Anal. Machine Intell. , 1993, 15(11): 1131-1147
[15] Brigger P., Hoeg J., Unsser M. B-spline snakes: a flexible tool for parametric contour detection. IEEE Trans Image Process , 2000, 9(9): 884–896
[16] Cohen L. D. On active contour models and balloons. CVGIP: Imag. Under., 1991, 53(2): 211-218
[17] Ivins J. Statistical snakes: active region models: [PhD Thesis]. University of Sheffield, 1996
[18] Xu C. Deformable Models with Application to Human Cerebral Cortex Reconstruction from Magnetic Resonance Images [PhD Thesis]. The Johns Hopkins University, 2000
[19] Wei M., Zhou Y., Wan M. A Fast Snake Model Based on Non-Linear Diffusion for Medical Image Segmentation. Computerized Medical Imaging and Graphics ,2004, 28(3): 109–117
[20] Park H. W., Todd S., Yongmin K. Active contour model with gradient directional information: directional snake, IEEE Trans Circuits Syst Video Technol, 2001, 11(2): 252–256
[21] Hou Z., Han C. Force Field Analysis Snake: an Improved Parametric Active Contour Model. Pattern Recognition Letters, 2005, 26 (5): 513-526
[22] Gunn S. R., Nixon M. S. A robust snake implementation: a dual active contour. IEEE Trans Pattern Anal Mach Intell, 1997, 19(1): 63–68
[23] Neuenschwander W. M., Fun P., Iverson L., et al. Ziplock snakes. Int J Comput Vision, 1997, 25(3): 191–201
[24] Ip H. H. S., Shen D. An affine-invariant active contour model (ai-snake) for model-based segmentation. Image Vision Comput , 1998, 16(4): 75–86
[25] McInerney T., Terzopoulos D. T-snakes: Topology Adaptive Snakes. Medical Image Analysis, 2000, 4 (2): 73-91
[26] Caselles V., Catte F., Coll T., Dibos F. A Geometric Model For Active Contours In Image Processing. Numerische Mathematik, 1993, 66 (1): 1-31
[27] Malladi R., Sethian J. A., Vemuri B. C. Shape Modeling With Front Propagation - A Level Set Approach. IEEE Transactions On Pattern Analysis And Machine Intelligence, 1995, 17(2): 158-175
[28] Kichenassamy S., Kumar A., Olver P., Tannenbaum A., Yezzi A. Gradient Flows and Geometric Active Contour Models. In the Proceedings of IEEE International Conference in Computer Vision, Cambridge, MA, USA, 1995. 810-815
[29] Caselles V., Kimmel R., and Sapiro G. Geodesic Active Contours. In the Proceedings of International Conference on Computer Vision, Cambridge, MA, USA, 1995. 694-699
[30] Sapiro G. Color Snakes, Technique Report, Hewlett-Packard Labs, 1995
[31] Kimmel R. Fast Edge Integration. In Osher S., Paragios N. Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer, 2003. 59-78
[32] Mumford D., Shah J. Optimal approximation by piecewise smooth functions and assosiated variational problems. Comm. Pure Appl. Math, 1989, 42(5): 577-685
[33] Ambrosio L., Tortorelli V. Approximation of functionals depending on jump by elliptic functionals via G-Convergence. Comm.. Pure Appl. Math, 1990, 43(8): 999-1036
[34] Braides A. Approximation of Free-discontinuity Problems. Vol. 1694 of Lecture Notes in Mathematics, Springer-Verlag, 1998
[35] Braides A., Dal Maso G. Non-local Approximation of the Mumford-Shah Functional. Cal. Var. Partial Differ. Equ., 1997, 5(4): 293-322
[36] Chambolle A. Image Segmentation by Variational Methods: Mumford and Shah Functional and Discrete Approximations. Siam J. Appl. Math., 1995, 55(3): 827-863
[37] Bourdin B., Chambolle A. Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. NUMERISCHE MATHEMATIK, 2000,85 (4): 609-646
[38] Chan T. F. , Vese L. A. Active contours without edges. IEEE Transactions on Image Processing, 2001,10(2): 266-277
[39] Yezzi A., Tsai A., Willsky A. A Statistical Approach to Snakes for Bimodal and Trimodal Imagery. Seventh IEEE International Conference on Computer Vision, October 1999, 898-903
[40] Zhu S., Yuille A. L. Region competition: Unifying snakes, region growing, energy/ Bayes/MDL for multi-band image segmentation. IEEE Trans. Pattern Anal. Mach. Intel. PAMI, 1996, 18 (9): 884-900
[41] Paragios N. Geodesic Active Regions and Level Set Methods: Contributions and Applications in Artificial Vision, PhD Thesis, Ecole Superiore en Science Informatique, Universite de Nice-Sophia Antipolis/I.N.R.I.A., 2000
[42] Sifakis E., Garcia C., Tziritas G. Bayesian level sets for image segmentation. Journal of Visual Communication and Image Representation, 2002, 13(1-2): 44-64
[43] Chesnaud C., Refregier P., and Boulet V. Statistical region snake-based segmentation adapted to different physical noise models. IEEE Trans. Pattern Anal. Machine Intel., 1999, 21(11): 1145-1157
[44] Galland F., Bertaux N., Refregier P. Multi-component image segmentation in homogeneous regions based on description length minimization: Application to speckle, Poisson and Bernoulli noise. Pattern Recognition, 2005, 38 (11): 1926-1936
[45] Sarti A. Corsi C., Mazzini E., et al. Maximum likelihood segmentation of ultrasound images with Rayleigh distribution. IEEE Transactions on Ultrasonics Ferroelectronics and Frequency Control, 2005,52 (6): 947-960
[46] Jehan-Besson S., Barlaud M., Aubert G. DREAM(2)S: Deformable Regions driven by an Eulerian Accurate Minimization Method for image and video Segmentation. International Journal of Computer Vision, 2003, 53 (1): 45-70
[47] Ronfard R. Region-based strategies for active contour models. International Journal of Computer Vision, 1994, 13(2): 229-251
[48] Tsai A., Yezzi A., Willsky A. Curve Evolution Implementation of the Mumford-Shah Functional for Image Segmentation, Denoising, Interpolation, and Magnification. IEEE Transactions on Image Processing, 2001,10(8): 1169-1186
[49] Leventon M., Grimson E., and Faugeras O. Statistical Shape Influence in Geodesic Active Contours. In IEEE Conference on Computer Vision and Pattern Recognition, 2000. 316-322
[50] Chen Y., Thiruvenkadam H., Tagare H., et al. On the Incorporation of Shape Priors in Geometric Active Contours. In IEEE Workshop in Variational and Level Set Methods, 2001. 145-152
[51] Chen Y., Thiruvenkadam H., Gopinath K., Brigg R. FMR Image Registration Using the Mumford-Shah Functional and Shape Information, In World Multiconference on Systems, Cybernetics and Informatics, 2002. 580-583
[52] Vemuri B., Chen Y. Joint Image Registration and Segmentation. In: Osher S, Paragios N. Geometric Level Set Methods in Imaging, Vision, and Graphics, Springer, 2003. 251-270
[53] Chan T. F., and Zhu W. Level Set Based Shape Prior Segmentation. UCLA CAM 03-66, November 2003
[54] Cremers D., Tischhauser F., Weickert J., Schnorr C. Diffusion Snakes: Introducing statistical shape knowledge into the Mumford–Shah functional. Int. J. of Computer Vision, 2002, 50(3): 295-313
[55] Paragios N. A level set approach for shape-driven segmentation and tracking of the left ventricle. IEEE Transactions on Medical Imaging, 2003, 22 (6): 773-776
[56] Gout C., Le Guyader C., Vese L. Segmentation under geometrical conditions using geodesic active contours and interpolation using level set methods,.Numerical Algorithms, 2005, 39 (1-3): 155-173
[57] Bergtholdt M., Schnorr C. Shape priors and online appearance learning for variational segmentation and object recognition in static scenes. Lecture Notes in Computer Science, 2005, 3663: 342-350
[58] Bergtholdt M., Cremers D., Schnorr C. Variational Segmentation with Shape Priors. In: Paragios N., Chen Y., Faugeras O. Handbook of Mathematical Models in Computer Vision. Springer Verlag, New York, 2006. 131-144
[59] Tsai A., Yezzi A., Wells W., et al. A shape-based approach to the segmentation of medical imagery using level sets. IEEE Trans. on Medical Imaging, 2003, 22 (2): 137-154
[60] Yang J., Duncan J. S. 3D Image Segmentation of Deformable Objects with Shape- Appearance Joint Prior Models. Lecture Notes in Computer Science, 2003, 2878: 573–580
[61] Cheng L., Yang J., Fan X., et al. A generalized level set formulation of the Mumford-Shah functional for brain MR image segmentation. Lecture Notes in Computer Science, 2005, 3565: 418-430
[62] Shang Y., Yang X., Zhu M., et al. Region and Shape Prior Based Geodesic Active Contour and Application in Cardiac Valve Segmentation. O. Gervasi et al. (Eds.): ICCSA 2005, LNCS 3483. 1102– 1110
[63] Paragios N. Curve Propagation, level Set Methods and Grouping. In: Paragios N., Chen Y., Faugeras O. Handbook of Mathematical Models in Computer Vision. Springer-Verlag, New York, 2006. 145-160
[64] Pluempitiwiriyawej C., Moura J. M. F., et al. STACS: New active contour scheme for cardiac MR image segmentation. IEEE Transactions on Medical Imaging, 2005, 24 (5): 593-603
[65] Samson C., Blanc-Feraud L., Aubert G., et al. A level set model for image classification. International Journal of Computer Vision, 2000, 40 (3): 187-197
[66] Paragios N., Deriche R. Geodesic Active Regions and Level Set Methods for SupervisedTexture Segmentation. International Journal of Computer Vision, 2002, 46(3): 223-247
[67] Brox T., Weickert J. Level Set Based Image Segmentation with Multiple Regions. Lecture Notes in Computer Science, 2004, 3175: 415-423
[68] Vese L. A., Chan T. F. A multiphase level set framework for image segmentation using the Mumford and Shah model. International Journal of Computer Vision, 2002, 50 (3): 271-293
[69] Vese L. A. Multiphase Object Detection and Image Segmentation. In: Osher S., Paragios N. Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer, 2003. 175-194
[70] Angelini E. D., Song T., Mensh B. D., Laine A. Multi-phase Three-Dimensional Level Set Segmentation of Brain MRI. C. Barillot, D.R. Haynor, and P. Hellier (Eds.): MICCAI 2004, LNCS 3216. 318–326
[71] Lie J., Lysaker M., Tai X.-C. A variant of the level set method and applications to image segmentation. UCLA, CAM-03-50, 2003
[72] Lie J., Lysaker M., Tai X.-C. A Binary Level Set Model and some Applications to Mumford-Shah Image Segmentation. UCLA, CAM-04-31, 2004
[73] Chung G., Vese L. A. Energy Minimization Based Segmentation and Denoising using a Multilayer Level Set Approach. Lecture Notes in Computer Science, 2005, 3757: 439-455
[74] Mansouri A-R. Mitiche A., Vázquez C. Multiregion competition: A Level Set Extension of Region Competition to Multiple Region Image Partitioning, Computer Vision and Image Understanding, 2006, 101 (3): 137-150
[75] Horn B., Schunck B. Determinating Optical Flow. Artificial Intelligence, 1981, 17(2): 185-203
[76] Lucas B., Kanade T. An Iterative Image Registration Technique with an Application to Stereo Vision. In Proc. 7th International Joint Conference on Artificial Intelligence, Vancouver, British Columbia , 1981. 674-679
[77] Aubert G., Deriche R., Kornprobst P. Computing Optical Flow via Variational Techniques. SIAM Journal on Applied Mathematics, 1999, 60(1):156–182
[78] Weickert J., Bruhn A., Brox T., Papenberg N. A survey on variational optic flow methods for small displacements. Preprint No. 152, Department of Mathematics, Saarland University, Saarbrücken, Germany, September 2005
[79] Dydenko I., Jamal F., Bernard O., D’hooge J. Magnin I. E., Friboulet D. A level set framework with a shape and motion prior for segmentation and region tracking in echocardiography, Medical Image Analysis . Preprint, www.elsevier.com/locate/media, 2005
[80] Cremers D., Soatto S. Motion Competition: A Variational Approach to Piecewise Parametric Motion Segmentation. International Journal of Computer Vision, 2005, 62(3): 249–265
[81] Paragios N., Deriche R. Variational Principles in Optical Flow Estimation and Tracking. In: Osher S., Paragios N. Geometric Level Set Methods in Imaging Vision and Graphics , Springer Verlag, 2003. 299-317
[82] Brox T., Rousson M., Deriche R., Weickert J. Unsupervised Segmentation IncorporatingColour, Texture, and Motion. LNCS 2756: 353-360, August 2003
[83] Chan T. F., Esedoglu S., Nikolova M., Algorithms for Finding Global Minimizers of Denoising and Segmentation Models. UCLA CAM Report 04-54, September 2004
[84] Rudin L. I., Osher S., Fatemi E. Nonlinear Total Variation Based Noise Removal Algorithms. Physica D, 1992, 60(1-4):259-268
[85] Bresson X., Esedoglu S., Vandergheynst P., Thiran J. P., Osher S. Global Minimizers of the Active Contour/Snake Model. UCLA CAM Report 05-04, January 2005
[86] Leung H., Osher S. Global Minimization of the Active Contour Model with TV-Inpainting and Two-Phase Denoising. N. Paragios et al. (Eds.): VLSM 2005, LNCS 3752:149–160, 2005
[87] Xu C., Prince J. L. Global Optimality of Gradient Vector Flow. Conference on Information Sciences and Systems, Princeton University, March 15-17, 2000
[88] Appleton B., Talbot H. Globally Optimal Geodesic Active Contours. Journal of Mathematical Imaging and Vision, 2005, 23(1): 67–86
[89] Kervrann C., Trubuil A. Optimal Level Curves and Global Minimizers of Cost Functionals in Image Segmentation. Journal of Mathematical Imaging and Vision, 2002,17(2): 153–174,
[90]李海云,李光颖,王筝.一种基于水平集的脊柱MRI分割算法的研究.北京生物医学工程,2004,23(3):178-204
[91]张继武,张道兵,史舒娟,等.基于水平集方法的数字胸片分割.中国图像图形学报,2004,9(12):1459-1465
[92]曹国,杨新.基于水平集方法的航拍图片人工区域的分割检测.模式识别与人工智能,2006,19(4):526-530
[93]郑罡,王惠南,李远禄,等.基于Chan-Vese模型的多目标层次分割算法.中国图像图形学报,2006,11(6):804-810
[94]李俊韬,范跃祖,张海.基于Mumford-Shah模型的运动目标检测.电子技术应用,2005,31(9):4-6
[95]陈强,周则明,屈颖歌,等.左心室核磁共振的自动分割.计算机学报,2005,28(6):991-999
[96]蔡超,周成平,丁明跃,等.基于C-V方法改进的红外自动分割.华中科技大学学报,2006,34(3):62-64
[97]张治国,周越,谢凯.一种基于Mumford-Shah模型的脑肿瘤水平集分割方法.上海交通大学学报,2005,39(12):1955-1962
[98]朱昀,杨杰,于志强.基于混合水平集模型的CT颅面眼眶分割.生物医学工程学杂志,2005,22(5):875-879
[99]陆宏炯.处理碎片的多个水平集方法.上海交通大学学报,2004,38(6):1031-1034
[100] Zheng Gang, Wang Huinan. Multi-Phase Active Contour Model for Image Segmentation Based on Level Sets. Transactions of Nanjing University of Aeronautics & Astronautics, 2006,23(2):132-137
[101]杨丽,杨新.基于区域划分的曲线演化多目标分割.计算机学报,2004,27(3):420-425
[102]刘国才,王耀男.基于水平集逐层迭代算法的多层Mumford-Shah分割、去噪与重建模型.自动化学报,2006,32(4):534-540
[103] Huang Fuzhen, Su Jianbo, Xi Yugeng. Incorporating Prior Shape Into Geometric Active Contours for Face Contour Detection. Chinese Journal of Electronics, 2004, 13(3):552-555
[104]周则明,王元全,王平安,等.基于水平集的3D左心室表面重建.计算机研究与发展,2005,42(7):1173-1178
[105]金大年,杨丰,陈武凡.基于贝叶斯分类的水平集MR分割方法.中国医学物理学杂志,2005,22(4):568-571
[106]朱付平,田捷,林瑶,等.基于Level Set方法的医学分割.软件学报,2002,13(9):1866-1872
[107]李俊,杨新,施鹏飞.基于Mumford-Shah模型的快速水平集分割方法.计算机学报,2002,25(11):1175-1183
[108]肖亮,吴慧中,韦志辉,等.分割中分段光滑Mumford-Shah模型的水平集算法.计算机研究与发展,2004,41(1):129-135
[109] Xu C., Pham D. L. ,Prince J. L. Image Segmentation Using Deformable Models, SPIE Press, May 2000: 129-174
[110] Terzopoulos D., Witkin A., Kass M. Constraints on deformable models: recovering 3D shape and nonrigid motion, Artificial Intelligence, 1988, 36(1): 91–123
[111] Danielsson P. E. Euclidean distance mapping. Comp. Graph. Imag. Proc., 1980, 14:. 227–248
[112] Borgefors G. Distance transformations in arbitrary dimensions. Comp. Vis. Graph Imag. Proc., 1984, 27: 321–345
[113] Amini A. A., Weymouth T. E., Jain R. C. Using dynamic programming for solving variational problems in vision. IEEE Trans. PAMI , 1990, 12(9): 855–867
[114] McInerney T. , Terzopoulos D. A dynamic finite element surface model for segmentation and tracking in multidimensional medical images with application to cardiac 4D image analysis. Comp. Med. Imag. Graph., 1995, 19(1): 69–83
[115] Kimia B. B. Conservation Laws and a Theory of Shape. Ph.D. thesis, McGill Centre for Intelligent Machines, McGill University, Montreal, Canada, 1990.
[116] Adalsteinsson D., Sethian J. A. The fast construction of extension velocities in level set methods. J. Computational Physics, 1999, 148(1): 2–22
[117] Yezzi J. A., Kichenassamy S., Kumar A., et al. A Geometric Snake Model for Segmentation of Medical Imagery. IEEE Transactions on Medical Imaging, 1997, 16(2):199-209
[118] Siddiqi K., Lauziere Y. B., Tannenbaum A., Zucker S. W. Area and length minimizing flows for shape segmentation. IEEE Trans. on Imag. Proc., 1998, 7(3): 433–443
[119] Chan T. F. , Shen J. Variational Image Inpainting , Communications on Pure and Applied Mathematics, 2005, 58(5): 579-619
[120] Tikhonov A. N. , Arsenin V. Y. Solutions of Ill-Posed Problems. Mathematics of Computation,1978, 32(144):1320-1322
[121] Aubert G. , Vese L. A Variational Method in Image Recovery. SIAM Journal on Numerical Analysis, 1997, 34(5): 1948-1979
[122] Perona P. , Malik J. Scale-space and Edge Detection Using Anisotropic Diffusion. IEEE Tran. On Pattern Anal. and Math. Intel. , 1990, 12(7): 629-639
[123] Nordstr?m N. Biased Anisotropic Diffusion– A Unified Regularization and Diffusion Approach to Edge Detection, Image Vision Comput. , 1990, 8(4):318-327
[124] Osher S., Shu C. High-order Essentially Nonoscillatory Schemes for Hamilton-Jacobi Equations. SIAM J. Numer. Anal., 1991, 28(4): 907-922
[125] Weickert J. Applications of Nonlinear Diffusion in Image Processing and Computer Vision. Acta Mathematica Universitatics Comenianae, 2001, LXX (1): 33-50
[126] Weickert J., Fast Methods for Implicit Active Contour Models. In: Osher S., Paragios N. Geometric Level Set Methods in Imaging, Vision, and Graphics, Springer-Verlag, New York, 2003. 43-58
[127]陈秋晓,骆剑承,周成虎,等.基于多特征的遥感影像分类方法.遥感学报,2004,8(3):239-245
[128]刘永学,李满春,毛亮.基于边缘的多光谱遥感图像分割方法.遥感学报,2006,10(3):350-356
[129]王小龙,张杰,初佳兰.基于光学遥感的海岛潮间带和湿地信息提取.海洋科学进展,2005,23(4):477-481
[130]初佳兰,张杰,王小龙,等.SPOT、QuickBird卫星遥感数据提取东沙岛植被信息的比较.海洋学研究,2006,24(2):79-85
[131]翟继双,王超,王正志.一种基于多闭值的形态学提取遥感图象海岸线特征方法.中国图象图形学报,2003,8(7)
[132]冯兰娣,孙效功,胥可辉.利用海岸带遥感图像提取岸线的小波变换方法.青岛海洋大学学报,2002,32 (5):777-781
[133]陆立明,王润生,李武皋.基于合成孔径雷达回波数据的海岸线提取方法.软件学报,2004,15(4):531-536
[134]马小峰,赵冬至,邢小罡,等.海岸线卫星遥感提取方法研究.海洋环境科学,2007,26(2):185-189
[135]李林茹,高双喜,曹淑服.基于小波变换和梯度矢量流Snake模型的ERS-1 SAR图像的海岸线探测.河北工业科技,2004,21(4):24-26
[136]梁小祎,张杰,孟俊敏.溢油SAR图像分类中的纹理特征选择.海洋科学进展,2007,25(3):346-354