摘要
均值移动算法是一种迭代算法,在图像处理和计算机视觉中得到广泛的应用,但其理论分析仍然存在一定的不足。本课题完善了均值移动算法的理论证明,研究其在图像平滑、分割和目标跟踪中的应用。本论文完成的主要工作如下:
针对以往文献对均值移动算法密度函数分析的不足,讨论了轮廓函数k(x)与以x为自变量的核函数k(?)之间的凹凸性关系,以及核函数k(?)与密度函数f(x)之间的凹凸性关系。比较分析了均值移动算法两种运动方式的迭代时间长短和步数多少。根据柯西收敛定理更好的证明了均值移动算法的收敛性。提出并证明了同窗内均值移动矢量间的夹角不大于90度等。总结了均值移动步长为牛顿步长、高斯-牛顿步长或其它非线性步长的条件;讨论了均值移动点在运动方向上取密度极大值的位置;分析了均值移动函数在其收敛点处密度一阶导数为零的充要条件等;
研究了均值移动算法在图像平滑、分割和运动目标跟踪中的应用,针对图像平滑,提出了基于柱状HSV空间的均值移动图像平滑算法;分析了基于柱状HSV空间的多幅同场景图像的最优组合方法;通过实验比较分析了在不同颜色步长和空间步长情况下的图像平滑效果。针对图像分割,提出了根据当前帧图像边缘(由均值移动算法求出)和差分优化模板相结合的当前帧运动目标分割算法。针对目标跟踪,提出了双窗特征提取算法和基于像素分量的目标特征更新算法,以及跟踪窗自适应旋转和缩放算法。
The mean shift algorithm is an iteration algorithm, which has been widely used in the image processing and the computer vision, still has some theoretical proficiency. Therefore, the theoretical proven of the mean shift algorithm is perfected as well as its application in the image smoothing, image segmentation and object tracking is studied in the paper. The main finished tasks in this paper are as fellows:
In view of the deficiency in the analysis of the density function of the mean shift algorithm, the relation of the concavity and convexity between the profile function k(x) and the kernel function k(||x-c||~2), the kernel k(||x - c||~2) and the density function f(x) are discussed. Besides, the iterative step and time of two motion rules of the mean shift algorithm are compared. Some theorems are analyzed and proposed, such as the convergence of the mean shift algorithm is proved using the Cauchy theorem; the one that the angle of the mean shift vectors in the same window is less than 90 degree,etc. some conditions are summarized while the condition of the mean-shift step is Newton step, quasi-newton step or other step; the local maximum value of density in the moving direction is discussed; and the necessary and sufficient condition is finally analyzed that the first derivative of the density function is equal to zero on the convergence point, etc.
At last, the application of the mean shift algorithm in image smoothing, image segmentation and object tracking is tested; the image smoothing algorithm based on the cylindrical HSV space is studied in virtue of image smoothing problems; the optimal combination methods based on the multiple same scene images in the cylindrical HSV space is proved; the effect of the image smoothing is discussed with different color and space steps. Considering the image segmentation problems, the object segmentation algorithm based on the image edges of the current frame (the edge is gotten based on the mean shift algorithm) and the optimal difference template is proposed. With a view to object tracking problems, the two-window feature extracting algorithm, the object feature updated algorithm based on the pixel, and the linear combination method of the tracking windows are proposed separately.
引文
[1] McFarlane, N.J.B. and C. P Schofield, Segmentation and tracking of piglets in images. Machine Vision and Applications.1995:187-193P
[2] K. Fukunaga and L. D. Hostetler. The estimation of the gradient of a density function, with applications in pattern recognition. IEEE Trans. Information Theory. 1975 (21) :32-40P
[3] Y.Cheng, Mean shift. Mode seeking, and clustering. IEEE Trans.Pattern Analysis and Machine Intelligence. 1995, 17(8):790-799P
[4] C. Yang, R. Duraiswami, D. DeMenthon and L. Davis. Mean-Shift Analysis Using Quasi-Newton Methods. In IEEE International Conference on Image Processing. 2003(3):447-450P
[5] Comaniciu D, Ramesh V, Meer P. Real-Time tracking of non-rigid objects using mean shift. In: Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition (CVPR).2000: 142-149P
[6] M. Fashing and C. Tomasi. Mean shift is a bound optimization. IEEE Trans. Pattern Anal. Machine Intell. 2005, 25(3):471-474P
[7] 李乡儒,吴福朝,吴战义.均值漂移算法的收敛性.软件学报.2005,16(3) : 365-374页
[8] Comaniciu D, Ramesh V. Mean shift and optimal prediction for efficient object tracking. In: Mojsilovic A, Hu J, eds. Proc. of the IEEE Int'1 Conf. on Image Processing (ICIP). 2000:70-73P
[9] Comaniciu D, Ramesh V, Meer P. Kernel-Based object tracking. IEEE Trans. on Pattern Analysis and Machine Intelligence. 2003,25(5):564-575P
[10] Zhou XS, Comaniciu D, Krishnan S. An information fusion framework for robust shape tracking. Workshop on Statistical and Computational Theories of Vision SCTV. 2003:1-24P
[11] Comaniciu D, Meer P. Mean shift: A robust approach toward feature space analysis. IEEE Trans. on Pattern Analysis and Machine Intelligence.2002,24(5):603-619P
[12] Comaniciu D, Meer P. Mean shift analysis and applications. In:Proc. of the IEEE Int'1 Conf. on Computer Vision (ICCV). 1999:1197-1203P
[13] Barash D, Comaniciu D. A common framework for nonlinear diffusion, adaptive smoothing,bilateral filtering and mean shift. Image and Video Computing. 2003,22(1):73-81P
[14] Comaniciu D,Ramesh V,Meer P. The variable band width mean shift and data-driven scale selection. In: Proc.of the IEEE Int' 1 Conf.on Computer Vision (ICCV).2001:438-445P
[15] Comaniciu D.An algorithm for data-driven bandwidth selection.IEEE Trans,on Pattern Analysis and Machine Intelligence. 2003,25(2):281-288P
[16] Comaniciu D,Ramesh V,Bue AD. Multivariate saddle point detection for statistical clustering. In:Proc. of the European Conf. Computer Vision (ECCV).2002:561-576P
[17] Bradski GR.Computer vision face tracking for use in a perceptual user interface. In: Regina Spencer Sipple. IEEE Workshop on Applications of Computer Vision. Stoughton: Printing House. 1998:214-219P
[18] Collins RT.Mean-Shift blob tracking through scale space. In: Proc.of the Conf.on Computer Vision and Pattern Recognition(CVPR).2003:18-20P
[19] C.Yang,R. Duraiswami,and L. Davis. Similarity Measure for Nonparametric Kernel Density Based Object Tracking. Eighteenth Annual Conference Neural Information Processing Systems.2005:1561-1568P
[20] C.Yang,R. Duraiswami,A. Elgammal and L.Davis. Real-Time Kernel-Based Tracking in Joint Feature-Spatial Spaces. Tech. Rep.CS-TR-4567. University of Maryland. 2003
[21] 朱胜利，朱善安，李旭超．快速运动目标的Mean shift跟踪算法．光电工程．2006，33(5)：66-70页
[22] 贾静平，张艳宁，柴艳妹，赵荣椿．目标多自由度Mean Shift序列图像跟踪算法．西北工业大学学报．2005，23(5)：618-622页
[23] 李培华．一种改进的mean shift跟踪算法．自动化学报．2007，33(4)P：347-354页
[24] 程伟，杨杰．一种基于均值移位的红外目标跟踪新方法．红外与毫米波学报．2005，24(3)：231-235P
[25] K.Deguchi,O.Kawanaka and T.Okatani.Object tracking by the mean-shift of regional colour distribution combined with the particle- filter algorithm. Proc. ICPR. 2004:506-509P
[26] A. Bandera,J.M.Perez-Lorenzo et al.Mean shift based clustering of Hough domain for fast line segment detection. Pattern Recognition Letters. 2006 (27):578-586P
[27] Yuzhong Wang etal.Unsupervised color-texture segmentation based on soft criterionwith adaptive mean-shift clustering.PRL.2006,27(5): 386-392P
[28] Wang J,Thiesson B,Xu YQ,Cohen M. Image and video segmentation by anisotropic kernel mean shift.In:Proc.of the European Conf.on Computer Vision(ECCV).2004
[29] M.(?).Carreira-Perpi(?)án. Fast Nonparametric Clustering with Gaussian Blurring Mean-Shift,Proc. 23rd Int'l Conf. Machine Learning,2006
[30] Georgescu,B.Shimshoni,Meer P. Mean shift based clustering in high dimensions: a texture classification example. In: IEEE Int.Conf. on Computer Vision. 2003(2):456-463P
[31] H.Tek,D.Comaniciu,J. Williams. Vessel Detection by Mean-Shift Based Ray Propagation,IEEE Workshop on Mathematical Methods in Biomedical Image Analysis,Hawaii,2001
[32] 刘蓉，段福庆等．基于均值漂移的自适应滤波及其在光谱信号处理中的应用．电子与信息学报．2006，28(2)：312-316页
[33] 朱秀昌，刘峰，胡栋．数字图像处理与图像通信．北京：北京邮电大学出版社．2002：67-84页
[34] 吕知辛，黄尊灵．图像平滑的高斯加权平均算法．计算机系统应用．1998(7)：41-42页
[35] 崔鹏，张立昂，吉利久，高立．两种平滑优化算法在一维情况下的比较研究．北京大学学报．2003，39(5)：681-689页
[36] 贾永红．计算机图像处理与分析．武昌：武汉大学出版社．2001：51-75页
[37] 张霞，葛芦生．自适应中值滤波算法的研究与应用．微计算机信息.2007(6)：217-218，237页
[38] 胡旺，李志蜀，黄奇．基于双窗口和极值压缩的自适应中值滤波．中国图象图形学报．2007，12(1)：43-50页
[39] 张根耀，赵国际，赵宗涛等．运动目标识别中的噪声滤除技术．西北大学学报．2003，33(2)：147-150页
[40] Donoho D L.Denoising by soft thresholding. IEEE Trans Information Theory,1995,41(3):613-627P
[41] Donoho D L,Johnstone I M. Ideal spatial adaptation by wavelet shrinkage.Biometrika. 1994,81(3):425-455P
[42] Donoho D L,Johnstone I M,Kerkyacharian G etal,Wavelet shrinkage:symptopia. Journal of the Royal Statistical Society. 1995,57(2):301-369P
[43] Donoho D L.De-noising via soft thresholding. Technical report. California: Stanford University. 1992
[44] Guichard F,Morel J M. Partial differential equations and image iterative filtering. Proc Tutorial International Conference Image Processing. Washington,DC. 1995:131-150P
[45] Witkin A. Scale space filtering. In: Proc 8th International Joint Conference Artificial Intelligence. 1983: 1019-1021P
[46] Charbonnier P,Feraud L B,Aubert G et al.Two deterministic half-quadratic regularization algorithms for computed imageing. Proceedings of IEEE ICIP.1994(2):168-172P
[47] 章毓晋．图像分析．北京：清华大学出版社．2005：73页
[48] 韩思奇，王蕾．图像分割的阈值法综述．系统工程与电子技术．2002，124(16)：91-102页
[49] Doyle W. Operation Useful for Similarity-Invariant Pattern Recognition. J. Asssoc.Comput. Mach. 1962 (9) :259-267P
[50] Lee S,Chung S. A Comparative Performance Study of Several Global Thresholding Techniques for Segmentation. Computer Vision,Graphics and Image Processing. 1990(52):171-190P
[51] Ostu N A. Threshold Selection Method from Gray-Level Histograms.IEEE Trans,on System,Man and Cybernetics. 1979,9(1):62-66P
[52] Tsai W H. Moment-Preserving Thresholding: A New Approach. Computer Vision,Graphics and Image Processing.1985(29):377-393P
[53] Kittler J,Illingworth J. Minimum Error Thresholding. Pattern Recognition. 1986(19):41-47P
[54] Cho S,Haralick R,Yi S. Improvement of Kittler and Illingworth's Minimum Error Thresholding. Pattern Recognition. 1989(22):609-617P
[55] [55]Rosenfeld A,De La Torre P. Histgram Concavity Analysis as an Aid in Threshold Selection. IEEE Tranactions on Systems,Man and Cybernetics. 1983,13(3):231-235P
[56] Pun T. A New Method for Gray-Level Picture Thresholding Using the Entroy of the Histgran. Signal Processing. 1980(2):223-237P
[57] Kapur J,Sahop P,Wong A. A New Method for Grey-Level Picture Thresholding Using the Entropy of the Histogram. Computer Vision Graphics and Image Processing. 1985(29):210-239P
[58] Kapur J. Maximum Entropy Models in Science and Engineering. Wiley Eastern,New Delhi. 1989
[59] Yen J C,Chang F J,Chang S. A New Criterion for Automatic Multilevel Thresholding. IEEE Trans,on Image Processing.1995(4):370-378P
[60] J. Hartigan and M. Wong. A k-means clustering algorithm. Applied Statistics,1979 (28) :100-108P
[61] J. C. Dunn. A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters,Journal of Cybernetics. 1973(3):32-57P
[62] Dempster A. P.,Laird N. M.,Rubin D. B. Maximum likeli-hood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. 1977,39(1):1-38P
[63] S. C. Johnson. Hierarchical Clustering Schemes,Psychometrika. 1967(2):241-254P
[64] Mangin J F,Frouin V,Bloch I,Regis J,Lopez-Krahe J.From 3D Magnetic Resonance Images to Structural Representations of the Cortex Topography Using Topology Preserving Deformations.Journal of Mathematical Imaging and Vision. 1995,5(3):297-318P
[65] Canny J. A Computational Approach to Edge Detection. IEEE Trtans on Pattern Analysis and Machine Intelligence. 1986,8(6) :679-698P
[66] 章毓晋．图像处理和分析．北京：清华大学出版社，1999年
[67] Nalwa V S,Binford T O. On Detecting Edges. IEEE Trans on Pattern Analysis and Machine Intelligence. 1986,8(6):699-711P
[68] Goshtasby A. Design and Recovery of 2-D and 3-D Shapes Using Rational Gaussian Curves and surfaces. International Journal of Computer Vision.199310(3):233-256P
[69] Zhu S C,MuMford D.Grade:Gibbs Reaction and Diffusion Equations.In:Proc of the 6th International Conference on Computer Vision.Bombay. India. 1998:847-854P
[70] Xu G,Segawa E,Tsuji S. Robust. Active contours with insensitive parameters. Pattern Recognition. 1994,27 (7) :879-884P
[71] 张建伟，罗剑，夏德深．一种基于遗传算法的双T-Snake模型图像分割方法．中国图象图形学报．2005，10(1)：38-42页
[72] 王福生，齐国清．二值图像中目标物体轮廓的边界跟踪算法．大连海事大学学报．2006，32(1)：62-63，67页
[73] 毋立芳 沈兰荪等．基于曲线拟合的人脸分割算法．计算机学报．2003，26(7)：893-896，1003页
[74] 许燕，胡广书，耿进朝，商丽华．基于形态学和动态规划的冠状动脉树分割方法．清华大学学报，2006，46(12)：2061-2064，2076页
[75] F.Hoppner,F. Klawonn. A Contribution to Convergence Theory of Fuzzy c-Means and Derivatives. IEEE Trans,on Fuzzy Systems.2003,11(5):682-694P
[76] 余锦华，汪源源，施心陵．基于空间邻域信息的二维模糊聚类图像分割．光电工程．2007 34(4)：114-119页
[77] 左洪福，陈果．图像的自适应模糊阈值分割法．自动化学报．2003，29(5)：791-796页
[78] 宗晓萍，徐艳，董江涛．多信息融合的模糊边缘检测技术．物理学报．2006，55(7)：3223-3228页
[79] Ma Yide,Dai Rolan,Li Lian. Image segmentation of embryonic plant cell using pulse-coupled neural networks. Chinese Science Bulletin. 2002,47(2):167-172P
[80] 王润民，钱盛友，宋平，许慧燕．基于神经网络和颜色特征的车牌字符分割方法．计算机工程与应用．2006，42(6)：57-59，118页
[81] B D Lucas,T Kanade. An iterative Image Registration Technique with an Application to Stereo Vision. Processing of the International Joint Conference on Artificial Intelligence,1981:121-130P
[82] H-H Nagel.Displacement Vectors Derived from Second-Order Intensity Variations in image Sequence. Computer Graphics and image Processing. 1983 (21) :299-324P
[83] E P Ong. Robust Optical Flow Computation Based on Least Median of Squares Regression. Computer Vision. 1999,31(1):51-82P
[84] A B Hadiashar D Suter. Robust Optical Flow Computation. Computer Vision.1998,29(1):59-77P
[85] Roland Mech,Michael Wollbom.A Noise Robust Method for 2D Shape Estimation of Moving Objects in Video Sequences Considering a Moving Camera. Signal Processing. 1998,66(2):203-217P
[86] Frederic Dufaux,Fabrice Mothem,Andrew Lippman. Spatio-Temporal Segmentation Based on Motion and Static Segmentation.Proc.Internal Conference on Image Processing voll.Washington 1995: 306-309P
[87] S N Jayaramamurthy,Ramesh Jain. An Approach to the Segmentation of Textured Dynamic Scenes. Computer Graphics and Image Processing. 1983 (21) : 239-61P
[88] Robert Thoma,Matthias Bierling. Motion Compensating Interpolation Considering Covered and Uncovered Background.Signal Processing: Image Communication.1989(1):191?12P
[89] Til Aach, Andre Kan, et al. Statistical Model-based Change Detection in Moving Video. Signal Processing. 1993,31(2):165-180P
[90] A Div G. Determining Three-Dimensional Motion and Structure form Optical How Generated by Several Moving Objects. IEEE Trans. on Pattern Analysis Machine Intelligence. 1985, 7(4): 384-401P
[91] Wang J, Adelson E. Representing Moving Images with Layers. IEEE Trans. on Image Processing, 1994, 3(5):625-638P
[92] Hotter M, Thoma R. Image Segmentation Based on Object Oriented Figureping Parameter Estimation. Signal Processing. 1988, 15(3):315-334P
[93] Musmann H G, Hotter M, Ostermann J. Object-Oriented Analysis Synthesis Coding of Moving Image. Signal Processing: Image Communication. 1989, 1(2): 117-138P
[94] Diehl N. Object-Oriented Motion Estimation and Segmentation in Image Sequences. Signal Processing: Image Communication. 1991,3(1):23-56P
[95] Murray D W, Buxton B F. Scene Segmentation form Visual Motion Using Global Optimization. IEEE Trans. on Pattern Analysis and Machine Intelligence.1987, 9(2): 220-228P
[96] Michael M Chang, A Murat Tekalp, M Ibrahim Sezan. An Algorithm for Simultaneous Motion Estimation and Segmentation. IEEE Internal Conference on Acoustics. Speech and Signal Processing.1994:221-224P
[97] A Neri, S Colonnese, et al. Automatic Moving Object and Background Separation. Signal Processing. 1998 (66) :219-232P
[98] M Pard6s. Relative Depth Estimation and Segmentation in Monocular Schemes. Picture Coding Symposium (PCS 97), Berlin, Germany.1997: 367-372P
[99] Y Deng, B Manjunath. Unsupervised Segmentation of Color-Texture Regions in Images and Video. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2001, 23(8):800-810P
[100] Del Bimbo, P Pala, L Tanganeli. Video Retrieval Based on Dynamics of Color Flows. International Conference on Pattern Recognition.2000: 851-854P
[101]H Greonspan, d al. A Probabilistic Framework for Spatio-Temporal Video Representation and Indexing . European Conference on Computer Vision,LNCS1614,Springer-Verlag, Berlin, Germany. 2002:461-475P
[102]D DeMenthon.Spatio-Temporal Segmentation of Video by Hierarchical Mean Shift Analysis. Statistical Methods in Video Processing. 2002: 800-810P
[103]Marques F,Molina C.Object Tracking for Content based Functionalities. SPIE Visual Communication Image Processing.1997: 190-198P
[104]Ed Chalom,et al.Segmentation of an Image sequence Using Multi-Dimensional Image Attributes. Proc.Internal Conference Image Processing(ICIP). 1996: 525-528P
[105]S Kruse,et al.A New Tracking Scheme for Semi-Automatic Video Object Segmentation. Proc. Workshop Image Analysis for Multi-media Application Services(WIAMIS'99).1999:93-99P
[106] 张昊，黄战华，郁道银等．基于差分图像的运动目标跟踪与分割方法的应用研究．光学技术．2005，20(4)：470-475页
[107] 朱明旱，罗大庸．基于帧间差分背景模型的运动物体检测与跟踪．计算机测量与控制．2006，14(8)：1004-1006，1009页
[108] 陈凤东，洪炳镕．基于动态阈值背景差分算法的目标检测方法．哈尔滨工业大学学报．2005，37(7)：883-884，955页
[109] 李波，郑锦，李景颉，周兵．基于累积光流的运动对象检测．高技术通讯．2006,16(11)：1101-1106页
[110] 邵文坤，黄爱民，韦庆．动态场景下的运动目标跟踪方法研究．计算机仿真．2006，23(5)：181-184页
[111] 聂(?)，赵荣椿．一种视频序列图像中区域匹配方法．西北工业大学学报．2005，23(3)：352-355页
[112] 顾海军，赵晓晖，王洪革．基于小波模板匹配的目标跟踪．吉林大学学报：信息科学版．2003，21(4)：347-351页
[113] 王东升，李在铭．空域视频场景监视中运动对象的实时检测与跟踪技术．信号处理．2005，21(2)：195-198，135页
[114] 张伟伟，王森，王阳生．基于中心点的指纹细节结构匹配算法．自动化学报．2003，29(6)：842-850页
[115] 郭美娜，李波．基于树搜索的一种动态空间调度方法．计算机工程与应用．2007，43(14)：180-183，198页
[116] 陈建祥，桑农．已知最大定位误差的点特征松弛匹配．宇航学报．2006，27(4)：794-797页
[117] Natan Peterfreund. Robust Tracking of Position and Velocity with Kalman Snakes. IEEE transactions on pattern analysis and machine intelligence. 1999,21 (6): 165-169P
[118]Xu C,Prince P, L. Snakes, Shapes and Gradient Vector Flow. IEEE,Trans on Image Processing, 1998(7):359-369P
[119]Kennedy J,Eberhart R C.Particle Swarm Optimization. Proceedings of IEEE International Conference on Neural Networks.1995:1942-1948P
[120]Wirth M. A., Stapinski A. Using active contours to extract the breast region in mammograms. Technical Report. 2003
[121]V.Caselles. and B. Coll. Snake in Movement. Numercial Analysis.1996 (33): 2445-2456P
[122]张桂林，徐捷．频域相关技术在图像匹配中的应用．模式识别与人工智能．1997，10(1)：87-92页
[123]D. Laneuville, FDufour,Pbertrand. Image based maneuvering target tracking. Proc.of TheAmerican Control Conference. Philadelphia,Pennsylvania.1998:2444-2449P
[124]E Mazor, Y Bar-Shalom. Interacting multiple Model Methods in Target Tracking: A Survey. IEEE Transactions on AES. 1998, 34(1):103-121P
[125]X R Li,Y Bar-Shalom. Model-Set Adaptation in Multiple-Model Estimation for Hybrid Systems. 1992 American Control Conf.1992:1794-1799P
[126]李辉，张安，何胜强，沈莹．基于神经网络的多机动目标跟踪算法．西北工业大学学报．2006，24(5)：552-557页
[127]毕靖 王青．神经网络数据融合机动目标跟踪算法．北京航空航天大学学报．2002，28(6)：692-694页
[128]Comaniciu D,Meer P.Mean shift: A robust approach toward feature space analysis. IEEE Trans,on Pattern Analysis and Machine Intelligence. 2002, 24(5):603-619P
[129]M.Rosenblatt. Remarks on some nonparametric estimates of a density functionl Annals of Mathematical Statistics. 1956,27(6):832-837P
[130]徐成贤，陈志平等，近代优化方法．北京：科学出版社．2002：11页
[131]文志强，蔡自兴．Mean Shift算法的收敛性分析．软件学报．2007，18(2)：205-212页
[132]Comaniciu D, Ramesh V,Meer P. Real-Time tracking of non-rigid objects using mean shift. In: Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). 2000: 142-149P
[133]同济大学数学教研室．高等数学．高等教育出版社．1996：235页
[134]周则明，王平安，夏德深．基于Snake模型的低对比度噪声心脏MRI图像 分割．系统仿真学报．2004，16(11)：2493-2496,2500页
[135]WILLIAMS D J, SHAH M. A fast algorithm for active contours and curvature estimation. CVGIP: Image Understanding.1992,55(1):14-26P
[136]Kim Changick, Hwang Jenq Neng. Fast and automatic video object segmentation and tracking for content-based applications. IEEE Transactions on Circuits and Systems for video Technology. 2002,12(2):122-129P
[137]梁在中，沈永欢等．实用数学手册．北京：科学出版社.1999:653页