基于可变形模型的轮廓提取与表面重建
|
摘要
轮廓提取与表面重建是计算机视觉中的重要研究课题，其在虚拟现实、自控车辆、机器人环境分析、监控系统中的物体跟踪与识别、生物医学图像处理、工业在线自动检测、形状反求等方面有着广泛的应用前景。基于断层数据的三维重建是通过提取被测物体的截面轮廓曲线以实现被测对象的三维反求和重构的一种测量方法，是目前国内外研究的热点。根据断层图像提取的实体轮廓可以通过表面重建得到物体的CAD模型，或直接应用于快速成形系统。在此过程中，从断层测量图像中提取实体轮廓是关键的一步。
近年来，将物理原理引入物体的形状恢复吸引了学者们的研究兴趣，基于可变形模型的轮廓提取与表面重建就是其中的一类。可变形模型可视为在内力和外力作用下的能量极小化样条模型，内力来自几何模型，约束它的形状；外力来自图像特征，引导它的行为，将其吸引至图像特征处。可变形模型的提出给传统的计算机视觉理论及应用研究带来了新的观点和思维方式，已经被越来越多的研究者成功地应用于图像分割、运动跟踪、3D重建、立体匹配等许多领域，并已发展成为计算机视觉与模式识别中最为活跃和成功的研究领域之一。
因此，基于可变形模型的断层图像轮廓提取与表面重建研究，在科学研究及工程应用中有着重要的意义。本论文主要针对医学断层图像的轮廓提取与表面重建，重点对B样条可变形模型进行了研究。本文在系统地分析了国内外关于可变形模型理论与应用研究的基础上，提出了一种基于有限元法的B样条主动轮廓模型；并将其应用于由断层图像导出的截面轮廓数字曲线的拟合，利用自适应有限元技术提高拟合精度；最后将其推广到三维，提出了基于有限元法的B样条主动曲面模型。本文主要工作和结论如下：
（1）对可变形模型的基础理论进行了论述，并从弹性理论的观点对可变形模型的物理本质进行讨论分析；然后从几何表达、能量方程及优化方法三个方面对可变形模型的理论研究与发展进行了综述，指出了可变形模型理论上存在的问题及进一步研究的方向。
（2）提出了一种基于有限元法的B样条主动轮廓模型。B样条方法是当前自由曲线曲面造型最为流行的方法，而有限元法是一种有效地求解泛函极值问题的数值方法。该模型以三次B样条曲线段作为有限单元，运用有限元法对B样条主动轮廓的能量泛函极值问题进行求解，从而实现对图像的轮廓提取。该模型结合了B样条方法与有限元法的优点，可快速稳定地收敛到目标轮廓，得到的B样条轮廓有利于进一步的表面重建处理。
（3）提出了基于自适应有限元的B样条主动轮廓模型，并将其应用于基于断层图像的表面重建。首先对断层图像的边缘提取方法进行了论述，针对得到的截面轮廓数字曲线，提出了一种基于自适应B样条主动轮廓拟合方法。该方法避免了传统插值、近似方法中的采样点数量、分布及参数化过程所带来的问题，通过在拟合过程中插入新的控制点，提高拟合精度，实现对数字轮廓的整体逼近。最后，由各断层截面的B样条轮廓重构出物体的三维模型。
（4）将二维的B样条主动轮廓模型扩展到三维，提出了基于有限元法的B样条主动曲面模型。通过在数据点与模型之间连接虚拟弹簧，建立光顺模型，并以双三次B样条曲面片作为单元，运用有限元法对模型在虚拟弹簧作用下的变形问题进行求解，得到光顺后的B样条曲面；并在此基础上，提出了一种新的基于断层数据的表面重建方法，可很好地解决不均匀截面族的曲面生成问题。
Contour extraction and surface reconstruction is an important problem in computer vision, and can be used extensively in many fields such as virtual reality, autonomous guided vehicles, robot environment analysis, object tracking and recognition in monitor system, biology medical image processing, industry online automatic checking and reverse engineering etc. 3D reconstruction based on layer data has been a hot point of research and new direction of reverse engineering. The contours extracted from cross-sections can be used to surface reconstruction to obtain the object’s CAD model, or can be used directly to prototyping system. During this process, contour extraction form cross-section is the key step.
In the last few years, introducing physics principle into shape recovery has attracted researchers’ attention, and deformable model based contour extraction and shape reconstruction is one kind. A deformable model is an energy-minimizing spline model controlled by internal forces and external forces. Internal forces come from geometry model which constrain its shape, and external forces come from image data which guide its movement and pull it toward image feature. Deformable models which bring a new viewpoint to traditional computer vision have been successfully applied to many fields in computer vision such as image segmentation, motion tracking, 3D reconstruction and stereo matching etc., and have become one of the most active and successful fields in computer vision and pattern recognition.
Therefore, cross-sectional contour extraction and surface reconstruction based on deformable models is important signification of theory and application. Aim at medical cross-sections, the research work of this dissertation put emphasis on B-spline deformable model. In this dissertation, based on the systematically analysis of theory and application research on deformable model, a new B-spline active contour based on the finite element method (FEM) is proposed; then B-spline active contour is applied to fitting the digital curve cross-sectional contour, and adaptive FEM is adopted to improve the precision; lastly the model is generalized to 3D, and a B-spline active surface model based on FEM is proposed. The main work and conclusions in this dissertation are as follows:
(1) The basic theory of deformable model is expatiated and the physical essence of deformable model is discussed from the viewpoint of elasticity theory. The research,
development and application of deformable model is reviewed from geometry representation, energy function and optimization method, and the existing problems and possible future research orientations are presented.
(2) A B-spline active contour model based on FEM is presented. B-spline is the most popular method to represent free-form curve and surface, and FEM provides an efficient way to solve the functional minima problem. In this model, a cubic B-spline curve segment is used as one element, and the FEM is adopted to find the B-spline active contour which minimize its energy. Experiment results showes that this medel could effectively combine the merits of B-spline and EFM for active contour model, yielding stable, accurate and faster convergence, and its result is favorable for surface reconstruction.
(3) A B-spline active contour model based on adaptive FEM is presented and applied to surface reconstruction from layer data. Firstly edge detection method for cross-sections is expatiated, and a B-spline active contour based on adaptive FEM is proposed to fit the digital contour curve. This method can avoid the sampling and parametrization problems result from traditional interpolation and approximation methods and improve fitting accuracy is gradually improved during fitting process by insert new control points to realize the satisfying result. Lastly 3D model is reconstructed from a set of planar B-spline contour.
(4) To generalize the B-spline active contour to 3D, a B-spline active surface model based on FEM is proposed. By anchoring an imaginary spring between each data p
近年来，将物理原理引入物体的形状恢复吸引了学者们的研究兴趣，基于可变形模型的轮廓提取与表面重建就是其中的一类。可变形模型可视为在内力和外力作用下的能量极小化样条模型，内力来自几何模型，约束它的形状；外力来自图像特征，引导它的行为，将其吸引至图像特征处。可变形模型的提出给传统的计算机视觉理论及应用研究带来了新的观点和思维方式，已经被越来越多的研究者成功地应用于图像分割、运动跟踪、3D重建、立体匹配等许多领域，并已发展成为计算机视觉与模式识别中最为活跃和成功的研究领域之一。
因此，基于可变形模型的断层图像轮廓提取与表面重建研究，在科学研究及工程应用中有着重要的意义。本论文主要针对医学断层图像的轮廓提取与表面重建，重点对B样条可变形模型进行了研究。本文在系统地分析了国内外关于可变形模型理论与应用研究的基础上，提出了一种基于有限元法的B样条主动轮廓模型；并将其应用于由断层图像导出的截面轮廓数字曲线的拟合，利用自适应有限元技术提高拟合精度；最后将其推广到三维，提出了基于有限元法的B样条主动曲面模型。本文主要工作和结论如下：
（1）对可变形模型的基础理论进行了论述，并从弹性理论的观点对可变形模型的物理本质进行讨论分析；然后从几何表达、能量方程及优化方法三个方面对可变形模型的理论研究与发展进行了综述，指出了可变形模型理论上存在的问题及进一步研究的方向。
（2）提出了一种基于有限元法的B样条主动轮廓模型。B样条方法是当前自由曲线曲面造型最为流行的方法，而有限元法是一种有效地求解泛函极值问题的数值方法。该模型以三次B样条曲线段作为有限单元，运用有限元法对B样条主动轮廓的能量泛函极值问题进行求解，从而实现对图像的轮廓提取。该模型结合了B样条方法与有限元法的优点，可快速稳定地收敛到目标轮廓，得到的B样条轮廓有利于进一步的表面重建处理。
（3）提出了基于自适应有限元的B样条主动轮廓模型，并将其应用于基于断层图像的表面重建。首先对断层图像的边缘提取方法进行了论述，针对得到的截面轮廓数字曲线，提出了一种基于自适应B样条主动轮廓拟合方法。该方法避免了传统插值、近似方法中的采样点数量、分布及参数化过程所带来的问题，通过在拟合过程中插入新的控制点，提高拟合精度，实现对数字轮廓的整体逼近。最后，由各断层截面的B样条轮廓重构出物体的三维模型。
（4）将二维的B样条主动轮廓模型扩展到三维，提出了基于有限元法的B样条主动曲面模型。通过在数据点与模型之间连接虚拟弹簧，建立光顺模型，并以双三次B样条曲面片作为单元，运用有限元法对模型在虚拟弹簧作用下的变形问题进行求解，得到光顺后的B样条曲面；并在此基础上，提出了一种新的基于断层数据的表面重建方法，可很好地解决不均匀截面族的曲面生成问题。
Contour extraction and surface reconstruction is an important problem in computer vision, and can be used extensively in many fields such as virtual reality, autonomous guided vehicles, robot environment analysis, object tracking and recognition in monitor system, biology medical image processing, industry online automatic checking and reverse engineering etc. 3D reconstruction based on layer data has been a hot point of research and new direction of reverse engineering. The contours extracted from cross-sections can be used to surface reconstruction to obtain the object’s CAD model, or can be used directly to prototyping system. During this process, contour extraction form cross-section is the key step.
In the last few years, introducing physics principle into shape recovery has attracted researchers’ attention, and deformable model based contour extraction and shape reconstruction is one kind. A deformable model is an energy-minimizing spline model controlled by internal forces and external forces. Internal forces come from geometry model which constrain its shape, and external forces come from image data which guide its movement and pull it toward image feature. Deformable models which bring a new viewpoint to traditional computer vision have been successfully applied to many fields in computer vision such as image segmentation, motion tracking, 3D reconstruction and stereo matching etc., and have become one of the most active and successful fields in computer vision and pattern recognition.
Therefore, cross-sectional contour extraction and surface reconstruction based on deformable models is important signification of theory and application. Aim at medical cross-sections, the research work of this dissertation put emphasis on B-spline deformable model. In this dissertation, based on the systematically analysis of theory and application research on deformable model, a new B-spline active contour based on the finite element method (FEM) is proposed; then B-spline active contour is applied to fitting the digital curve cross-sectional contour, and adaptive FEM is adopted to improve the precision; lastly the model is generalized to 3D, and a B-spline active surface model based on FEM is proposed. The main work and conclusions in this dissertation are as follows:
(1) The basic theory of deformable model is expatiated and the physical essence of deformable model is discussed from the viewpoint of elasticity theory. The research,
development and application of deformable model is reviewed from geometry representation, energy function and optimization method, and the existing problems and possible future research orientations are presented.
(2) A B-spline active contour model based on FEM is presented. B-spline is the most popular method to represent free-form curve and surface, and FEM provides an efficient way to solve the functional minima problem. In this model, a cubic B-spline curve segment is used as one element, and the FEM is adopted to find the B-spline active contour which minimize its energy. Experiment results showes that this medel could effectively combine the merits of B-spline and EFM for active contour model, yielding stable, accurate and faster convergence, and its result is favorable for surface reconstruction.
(3) A B-spline active contour model based on adaptive FEM is presented and applied to surface reconstruction from layer data. Firstly edge detection method for cross-sections is expatiated, and a B-spline active contour based on adaptive FEM is proposed to fit the digital contour curve. This method can avoid the sampling and parametrization problems result from traditional interpolation and approximation methods and improve fitting accuracy is gradually improved during fitting process by insert new control points to realize the satisfying result. Lastly 3D model is reconstructed from a set of planar B-spline contour.
(4) To generalize the B-spline active contour to 3D, a B-spline active surface model based on FEM is proposed. By anchoring an imaginary spring between each data p
引文
林瑶，田捷.医学图像分割综述.模式识别与人工智能，2002，15（2）：192-204
王爱民，沈兰荪.图像分割研究综述.测控技术，2000，19（5）：1-6
李培华，张田文.主动轮廓线模型（蛇模型）综述.软件学报，2000，11（6）：751-757
Leymarie F, Levine MD. Tracking deformable objects in the plane using an active contour model.IEEE Transactions on Pattern Analysis and Machine Intelligence,1993,15(6): 617-634
Terzopoulos D, Fleischer K. Deformable models. The Visual Computer, 1988, 4(6): 306-331
Gibson S, Mirtich B. A survey of deformable modeling in computer graphics. Technical Report, No.TR97-19, Department of Computer Science, University of Toronto, Canada, 1997
Yuille A L,Hallinan PW, Cohen DS. Feature extraction from faces using deformable templates. International Journal of Computer Vision,1992,8(2):99-111
史文中，朱长青，王昱.从遥感影像中提取道路特征的方法综述与展望.测绘学报，2001，30（3）：257-262
Dhond UR, Aggarwal JK.Structure from stereo—a review. IEEE Transactions on Systems, Man,and Cybernetics, 1989, 19(6): 1489-1510
Cham TJ,Cipolla R. Stereo coupled active contours.In:Grewe L ed.Proceedings of the International Conference on CVPR, San Juan, DuertoRico, IEEE Computer Society Press, 1997: 1094-1099
马颂德，张正友.计算机视觉—计算理论与算法基础.北京：科学出版社，1999：1-10
王润生.图像理解.湖南：国防科技大学出版社，1995：1-6
Kass M, Witkin A, Terzopoulos D.Snakes: active contour models. International Journal of computer vision, 1988, 1(4):321-331
McInerney T, Terzopoulos D. Deformable models in medical image analysis:a survey.Medical Image Analysis, 1996, 1(2): 91-108
Cohen LD. On active contour models and balloons. CVGIP:Image Understanding, 1991, 53(2): 211-218
Gavrila DM. Hermite deformable contours. in Proc. International Conference on Pattern Recognition, Vienna, Austria, 1996: 130-135
Xu Chengyang. Deformable models with application to human cerebral cortex reconstruction from magnetic resonance images. PhD thesis, Baltimore, Maryland, the Johns Hopkins University, 2000
Cohen I, Cohen LD, Ayache N. Using deformable surfaces to segment 3-D images and infer
differential structures. CVGIP:Image Understanding, 1992, 56(2):242-263
Várady T, Martin RR, Cox J. Reverse engineering of geometric models—an introduction. Computer-Aided Design, 1997, 29(4): 255-268
刘振凯，陈剑虹，乔志林等.一种基于断层测量的反求工程.中国机械工程，2000，11（4）：393-397
王宗彦，梁远蕾，李奇敏等.断层数据三维重构技术的研究进展.工程图学学报，2002，23(1):125-130
Montanari U. On the optimal detection of curves in noisy pictures. ACM Comminications, 1971, 14(5): 335-345
Ivins JP. Statistical snakes:active region models. PhD thesis, Western Bank, Sheffield, University of Sheffield, 1996
Cohen LD, Cohen I. Finite element methods for active contour models and balloons for 2-D and 3-d images. IEEE Transaction on Pattern Analysis and Machine Intelligence, 1993, 15(11): 1131-1147
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. Computerized Medical Imaging and Graphics, 1995, 19(1): 69-83
Singh A, Kurowski L, Chiu MY. Cardiac MR image segmentation using deformable models. In Biomedical Image Processing and Biomedical Visualization, 1993, volume 1905 of SPIE proceeding, 8-28
MacDonald D, Kabani N,?Avis D. Automated 3-D extraction of inner and outer surfaces of cerebral cortex from MRI. Neuroimage, 2000, 12(3): 340-356
Davatzikos C, Bryan RN. Using a deformable surface model to obtain a shape representation of the cortex. IEEE Transactions on Medical Imaging, 1996, 15(6): 785-795
Xu C, Prince JL. Snakes,shape,and gradient vector flow. IEEE Transaction on Image Processing, 1998, 7(3): 359-369
贺士娟，李颖，杨雅妹等.基于可变形模型算法提取MRI图像脑区域的方法.河北工业大学学报，2002，31（2），1-5
Staib LH, Duncan JS. Boundary finding with parametrically deformable models. IEEE Transaction on Pattern Analysis and Machine Intelligence, 1992, 14(11): 1061-1075
McInerkey T, Terzopoulos D. Topologically adaptable snakes. Proceedings of the Fifth International Conference on Computer Vision, Cambridge, USA, 1995: 840-845
Ivins J, Porrill J. Active region models for segmenting medical images. Image and Vision Computing, 1995, 13(5): 431-438
Jones TD, Plassmann P. An active contour model for measuring the area of leg ulcers. IEEE Transactions on Medical Imaging, 2000, 19(12): 1202-1210
Lu Aidong, Tang Long, Xu Yuhua etc. Segmentation of ultrasound images with interactive B-spline snakes and its application. Journal of Software, 2001, 12(12): 1760-1768
Hjelmas E, Low B K. Face detection:A survey. Computer Vision and Image Understanding, 2001, 83(3): 236-274
Huang CL, Chen CW. Human facial feature extraction for face interpretation and recognition. Pattern Recognition, 1992, 25(12): 1435-1444
Gunn SR, Nixon M S. A dual active contour for head and boundary extraction. in IEE Colloquium on Image Processing for Biometric Measurement, London,Apr. 1994,6-11
Yuille AL, Hallinan PW, Cohen DS. Feature extraction from faces using deformable templates. International Journal of Computer Vision, 1992, 8(2): 99-111
Coots TF, Taylor CJ, Lanitis A. Active shape models—‘smart snakes’. in Proceedings of British Machine Vision Conterence, Berlin, Germany, 1992: 266-275
史文中，朱长青，王昱.从遥感影像中提取道路特征的方法综述与展望.测绘学报，2001，30（3）：257-262
Gruen A, Li H.Semiautomatic linear feature extraction with dynamic programming and LSB-Snakes. Photogrammetric Engineering and Romote Sensing, 1997, 63 (8): 985-995
刘少创，林宗坚.航空影象分割的snake方法. 武汉测绘科技大学学报，1995，20（1）：7-11
刘少创，林宗坚.基于动态规划的航空影象中的目标提取.中国图象图形学报，1996，1（1）：30-35
孔祥维，石浩.形状约束的Snake算法在探地雷达图像目标自动提取中的应用.物探化探计算技术，2001，23（4）：333-337
Wang Y, Teoh EK. Lane detection using B-snake. IEEE International Conference on Information, Intelligence and Systems, Washington DC, 1999: 438-443
Wang Y, Shen Dinggang, Teoh E K. Lane detection using spline model. Pattern Recognition letters, 2000, 21(8): 677-689
Terzopoulos D, Witkin A, Kass M. Constraints on deformable models:recovering 3D shape and nonrigid motion. Artificial Intelligence, 1988, 36(1): 91-123
Sclaroff S, Pentland A. Closed-form solutions for physically-based shape modeling and recognition. IEEE Transaction on Pattern Analysis and Machine Intelligence, 1991, 13 (7): 715-729
Delingette H, Hebert M, Ikeuchi K. Shape representation and image segmentation using
deformable surface. Image and Vision Computing, 1992, 10(3): 132-144
McInerney T, Terzopoulos D. Finite element techniques for fitting a deformable model to 3D data. Proc. Vision Interface '93, Toronto, Canada, May, 1993, 70-76
Shen Xinquan, Hogg D. 3D shape recovery using a deformable model. Image and Vision Computing, 1995, 13(5): 377-383
夏利民，谷士文，曾三友.基于形变模型由立体序列图象恢复物体的形状.计算机工程，2001，27（10）：22-23
夏利民，谷士文，沈新权.基于形变模型的3D表面自适应重建.中国图象图形学报，2000，5（5）：396-400
Unser M. Spline: A prefect fit for Signal and Image Processing, IEEE Signal Processing Magazine. 1999, 16(6): 22-38
施法中.计算机辅助几何设计与非均匀有理B样条.北京：北京航空航天大学出版社，1994
陆明万，罗学富.弹性理论基础.北京：清华大学出版社，1990
丁学成.弹性力学中的变分方法.北京：高等教育出版社，1986
Chenyang Xu, Pham DL, Prince JL .Image Segmentation Using Deformable Models. Handbook of Medical Imaging -- Volume 2: Medical Image Processing and Analysis, edited by J.M. Fitzpatrick and M. Sonka, SPIE Press, May 2000: 129-174
柯朗，希尔伯特著，熊振翔，杨应辰译.数学物理方法.北京：科学出版社，1977
Rückert D. Segmentation and tracking in cardiovascular MR images using geometrically deformable models and templates. PhD thesis, London, University of London, 1997
Menet S, Saint-Marc P, Medioni G. B-snakes: implementation and application to stereo. In: Proceedings of image understanding workshop. Pittsburgh: IEEE Computer Society Press, 1990, 720-726
Amini A, Weymouth T, Jain R. Using dynamic programming for solving variational problems in vision. IEEE Transaction on Pattern Analysis and Machine Intelligence, 1990, 12(9): 855-867
Williams DJ, Shah M. A fast algorithm for active contours and curvature estimation. CVGIP: Image Understanding, 1992, 55(1): 14-26
Gunn SR, Nixon MS. A robust snake implementation: a dual active contour. IEEE Transaction on Pattern Analysis and Machine Intelligence, 1997, 19(1): 63-68
Velasco FA, Marroquín JL. Robust parametric active contours:the sandwich snakes. Machine Vision and Application, 2001, 12(5): 238-242
Poon CS, Braun M, Fahrig R, etc. Segmentation of medical images using an active contour model incorporating region-based images features. In Visualization in Biomedical Computing
1994, volume 2359,90-97
Szeliski R. Bayesian modeling of uncertainty in low-level vision. International Journal of computer vision, 1990, 5(3): 271-301
Storvik G. A Bayesian approach to dynamic contours through stochastic sampling and simulates annealing. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1994, 16(10): 970-986
Brigger P, Hoeg J, Unser M. B-spline snakes:a flexible tool for parametric contour detection. IEEE Transactions on Image Processing, 2000, 9(9): 1484-1496
Wang M, Evans J, Hassebrook L, et al. A multistage optimal active contour model. IEEE Transaction on Image Process, 1996, 5 (11): 1586-1591
Wang Y, Teoh EK, Shen DG. Structure-adaptive B-snake for segmenting complex objects. IEEE International Conference on Image Processing , Thessaloniki,Greece, 2001,IEEE press
朱心雄等.自由曲线曲面造型技术.北京：科学出版社，2000
Liang JM, McInerney T, Terzopoulos D. United snakes. In: Proceedings of the IEEE international conference on computer vision. Corfu, Greece, 1999. 923-940
王勖成，邵敏.有限单元法基本原理和数值方法.北京：清华大学出版社，1997
刘正兴，孙雁，王国庆.计算固体力学.上海：上海交通大学出版社，2000
Terzopoulos D, Qin H. Dynamic NURBS with geometric constraints for interactive sculpting. ACM Transaction on Graphics, 1994, 13(2): 103-135
经玲，席平，唐荣锡.有限元方法在变形曲线曲面造型中的应用.计算机学报，1998，21（3）：245-251
Flickner M, Sawhney H, Pryor D etc. Intelligent interactive image outlining using spline snakes. in 28th Asilomar Conference on Signals, Systems and Computers, vol 1, 1994: 731-735
Xu C, Prince JL. Gradient Vector Flow: A New External Force for Snakes. Proc. IEEE Conf. on Comp. Vis. Patt. Recog. (CVPR), Los Alamitos: Comp. Soc. Press, June 1997: 66-71
Gupta SN, Prince JL. Stochastic models for DIV-CURL optical flow methods. IEEE signal processing letters, 1996, 3(2): 32-35
周彦博，张志广.可变形物体的轮廓的提取.电子学报，1998，26（7）：133-137
Castleman KR著，朱志刚，林学訚，石定机等译.数学图像处理.北京：电子工业出版社，2002
Lobregt S, Viergever MA. A discrete dynamic contour model. IEEE Transactions on Medical Imaging, 1995, 14(1): 12-24
张爱东，张田文.一种B样条主动轮廓线模型.哈尔滨师范大学自然科学学报，1999，15
（4）：62-66
Brigger P, Engel R, Unser M. B-spline Snakes and a JAVA interface: An Intuitive Tool for General Contour Outlining. Proceedings of the 1998 IEEE International Conference on Image Processing (ICIP'98), Chicago IL, USA, October 4-7, 1998, vol. II, pp. 277-281
罗希平，田捷，诸葛婴等.图像分割方法综述.模式识别与人工智能，1999，12（3）：300-312
Kim DK. B-spline representation of active contours. Fifth international Symposium on Signal Processing and its Applications, Brisbane, Australia, 22-25 August, 1998, 813-816
梁远蕾，王宗彦.基于断层数据的三维重构技术.华北工学院学报，2002，23（1）：38-41
郭开波，周钢，王从军等.一种基于断层测量图片的实体轮廓提取方法.计算机辅助工程，2001,10（4）：50-54
谢红，张树生，张定华.产品工业计算机断层扫描成象切片的图象处理.机械科学与技术，1997，16（6）：1115-1118
林丽华，卢清萍，颜永年等.面向RP的CT图象反求技术.计算机辅助设计与制造，1998，（9）：48-50
刘振凯，陈剑虹，乔志林等.层去图像法反求工程中的数据处理.中国机械工程，2000，11（8）：929-932
张英杰，栗全庆，白作霖等.面向逆工程的实体再现方法的研究—图象滤波和边界提取.机床与液压，1999（2）：55-56
谷口庆治著，朱虹译.数字图像处理：基础篇.北京：科学出版社，2002
Canny J. A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1986, 8(6): 679-698
徐建华.图像处理与分析.北京：科学出版社，1992
蒋剑峰，何永辉，赵万生.利用CCD进行光切法测量的研究.计量技术，1999，（8）：7-10
贾云得.机器视觉.北京：科学出版社，2000
Wu JS, Leou PE. New polygonal approximation schemes for object shape representation. Pattern Recognition, 1993, 26(4): 471-484
Pikaz A, Dinstein I. Optimal polygonal approximation of digital curves. Pattern Recognition, 1995, 28(3): 373-379
肖轶军，丁明跃，彭嘉雄.基于迭代最近点的B样条曲线拟合方法研究.中国图象图形学报，2000，5A（7）：585-588
曲军，孙烨，曲刚.B样条曲线参数重载的非线性整体逼近拟合法.机械设计与制造，2001（5）：36-37
Lu Feng,Milios EE. Optimal spline fitting to planar shape. Signal Processing, 1994, vol. 37: 129-140
余学进，Redekop D.自适应有限元的应用和发展.南昌航空工业学院学报，1998,12（4）：78-83
郭书祥.自适应有限元方法用其工程应用.力学进展，1997，27(4)：479-488
周辉，李涛，邢启江等.数字曲线的线性逼近和分段识别.大连理工大学学报，1997，37（5）：576-580
文贡坚，王润生.数字曲线上特征点的检测.计算机学报，1998，21（6）：520-526
徐飞，施晓红.MATLAB应用图像处理.西安：西安电子科技大学出版社，2002
Boehm W. Inserting new knots into B-spline curve. Computer Aided Design, 13(6): 199-201
Dierckx P. Curve and surface fitting with splines. Clarendon press, 1993
Cham TJ, Cipolla R. Automated B-spline curve representation incorporating MDL and error- minimizing control point insertion strategies. IEEE transactions on pattern analysis and machine intelligence, 1999, 21(1): 49-53
张习文，李佐，蔡士杰等.基于遗传算法的以线段和圆弧为基元的曲线拟合.计算机辅助设计与图形学学报，2002，14（2）：1-4
余学军，彭立中.二值图象曲线轮廓提取的新算法.中国图象图形学报，2002，7A（3）：272-275
李斌，庄天戈.图像分割中采用自适应三次B样条修饰轮廓线.光学技术，2001，27（5）：477-480
Xu Meihe, Tang Zesheng. General surface reconstruction from a set of planar contours. Journal of Tsinghua University, 1997, 37(4): 59-64
Meyers D, Skinner S. Surface from contours. ACM Transactions on Graphics, 1992, 11 (3): 228-258
秦绪佳，纪凤欣，吴良武等.由基于轮廓重建的表面模型构建实体几何模型.机械设计与研究，2001，17（2）：15-17
普雷帕拉塔，沙莫斯著，庄心谷译.计算几何导论.北京：科学出版社，1990
张英杰，许长军.一种适用于逆真设计的特征提取方法的研究.机床与液压，2001（1）：36-37
吕铁英，彭嘉雄.图象轮廓特征提取新方法研究.中国图象图形学报，1999，4A（8）：655-658
Dumitras A, Venetsanopoulkos AN. A comparative study of snake models with application to object shape description in bi-level and gray-level images. Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing,Baltimore, USA, 3-6 June, 2001,1-5
经玲，席平，唐荣锡.应用可变形模型进行曲线曲面光顺.软件学报，1998，9（6）：464-468
Guan Zhidong, Jin Ling, Ning Tao etc. Study and application of physics-based deformable curves and surfaces. Computer & Graphics, 1997, 21(3): 305-313
张庆丰，乐清洪，马泽恩.织物变形的物理仿真技术.计算机辅助设计与图形学学报，2001，13（5）：1-6
冯康，石钟慈.弹性结构的数学理论.北京：科学出版社，第二版，1987
杨桂通.弹塑性力学.北京：人民教育出版社，1980
吴家龙.弹性力学.上海：同济大学出版社，1987
Celniker G, Gossard D. Deformable curve and surface finite-elements for free- form shape design. Computer Graphics, 1991, 25(4): 257-266
李永林，王启付，钟毅芳等.可变形曲面的有限元模型.计算机辅助设计与制造，1998（3）：46-49
Fang L, Dossard DC. Fitting 3D curves to unorganized data points using deformable curves. Proceedings of the International Conference’92 on Computer Graphics, North- Holland, 1992: 535-543
Fang L,Dossard DC.Multidimensional curve fitting to unorganized data points by nonlinear minimization.Computer-Aided Design,1995,27(1):48-58
Qin Hong, Terzopoulos D. Dynamic NURBS swung surfaces for physics-based shape design. Computer-Aided Design, 1995, 27(2): 111-127
纪玉春，李鹏.基于物理的可变形模型.系统仿真学报，2001，13（增刊）：19-22
朱东波，张舜德，李涤尘等.密集散乱测量数据点的B样条曲面拟合研究.计算机辅助设计与图形学学报，2001，13（12）：1123-1128
Liao Chia-Wei, Medioni G. Representation of range data with B-spline surface patches. 11th International Conference on Pattern Recognition (ICPR'92), The Hague , Netherlands, August 1992, 745-748
程久平，史翔.用B样条曲线构造几何物体模型的方法.合肥工业大学学报，1998，21（5）：88-91
侯宇，李刚，崔晨阳等.自由型曲面的测量与重建.计量学报，1999，20（4）：252-255
寇淑清，杨慎华，金文明.B样条曲面在模具型腔描述及3 维网格生成中的应用.中国机械工程，2000，11（11）：1268-1271
张丽艳，周来水，周儒荣.逆向工程中曲面重构算法研究与实现.航空学报，1999，20（3）：242-244
Lee Seungyong, Wolberg G, Shin SY. Scattered data interpolation with multilevel B-splines. IEEE Transactions on Visualization and Computer Graphics, 3(3): 1-17
蔡宣三.最优化与最优控制.北京：清华大学出版社，1982
彭芳瑜，周济，周艳红等.基于最小二乘法的曲面生成算法研究.工程图学学报，1999,20(3)：41-46
吕科，耿国华，周明全.医学图象三维表面的快速重构方法.计算机工程，2002，28（2）：68-78
Hanselman D,Littlefield B著，张航，黄攀译.精通MATLAB 6.北京：清华大学出版社，2002
Hanselman D,Littlefield B著,李人厚，张平安等译.精通MATLAB—综合辅导与指南.西安：西安交通大学出版社，1998
Magrab EB等著，高会生，李新叶，胡智奇等译.MATLAB原理与工程应用.北京：电子工业出版社，2002
Pfeifle R, Seidel HP. Fitting Triangular B-Splines to Functional Scattered Data. Computer Graphics Forum, 1996, 15(1): 15-23
孙延奎，朱心雄.B样条曲线的小波光顺法.工程图学学报，1998，19（4）：51-58
孙延奎，朱心雄.任意B样条曲面的多分辨率表示及光顺.工程图学学报，1998，19（3）：49-54
王爱民，沈兰荪.图像分割研究综述.测控技术，2000，19（5）：1-6
李培华，张田文.主动轮廓线模型（蛇模型）综述.软件学报，2000，11（6）：751-757
Leymarie F, Levine MD. Tracking deformable objects in the plane using an active contour model.IEEE Transactions on Pattern Analysis and Machine Intelligence,1993,15(6): 617-634
Terzopoulos D, Fleischer K. Deformable models. The Visual Computer, 1988, 4(6): 306-331
Gibson S, Mirtich B. A survey of deformable modeling in computer graphics. Technical Report, No.TR97-19, Department of Computer Science, University of Toronto, Canada, 1997
Yuille A L,Hallinan PW, Cohen DS. Feature extraction from faces using deformable templates. International Journal of Computer Vision,1992,8(2):99-111
史文中，朱长青，王昱.从遥感影像中提取道路特征的方法综述与展望.测绘学报，2001，30（3）：257-262
Dhond UR, Aggarwal JK.Structure from stereo—a review. IEEE Transactions on Systems, Man,and Cybernetics, 1989, 19(6): 1489-1510
Cham TJ,Cipolla R. Stereo coupled active contours.In:Grewe L ed.Proceedings of the International Conference on CVPR, San Juan, DuertoRico, IEEE Computer Society Press, 1997: 1094-1099
马颂德，张正友.计算机视觉—计算理论与算法基础.北京：科学出版社，1999：1-10
王润生.图像理解.湖南：国防科技大学出版社，1995：1-6
Kass M, Witkin A, Terzopoulos D.Snakes: active contour models. International Journal of computer vision, 1988, 1(4):321-331
McInerney T, Terzopoulos D. Deformable models in medical image analysis:a survey.Medical Image Analysis, 1996, 1(2): 91-108
Cohen LD. On active contour models and balloons. CVGIP:Image Understanding, 1991, 53(2): 211-218
Gavrila DM. Hermite deformable contours. in Proc. International Conference on Pattern Recognition, Vienna, Austria, 1996: 130-135
Xu Chengyang. Deformable models with application to human cerebral cortex reconstruction from magnetic resonance images. PhD thesis, Baltimore, Maryland, the Johns Hopkins University, 2000
Cohen I, Cohen LD, Ayache N. Using deformable surfaces to segment 3-D images and infer
differential structures. CVGIP:Image Understanding, 1992, 56(2):242-263
Várady T, Martin RR, Cox J. Reverse engineering of geometric models—an introduction. Computer-Aided Design, 1997, 29(4): 255-268
刘振凯，陈剑虹，乔志林等.一种基于断层测量的反求工程.中国机械工程，2000，11（4）：393-397
王宗彦，梁远蕾，李奇敏等.断层数据三维重构技术的研究进展.工程图学学报，2002，23(1):125-130
Montanari U. On the optimal detection of curves in noisy pictures. ACM Comminications, 1971, 14(5): 335-345
Ivins JP. Statistical snakes:active region models. PhD thesis, Western Bank, Sheffield, University of Sheffield, 1996
Cohen LD, Cohen I. Finite element methods for active contour models and balloons for 2-D and 3-d images. IEEE Transaction on Pattern Analysis and Machine Intelligence, 1993, 15(11): 1131-1147
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. Computerized Medical Imaging and Graphics, 1995, 19(1): 69-83
Singh A, Kurowski L, Chiu MY. Cardiac MR image segmentation using deformable models. In Biomedical Image Processing and Biomedical Visualization, 1993, volume 1905 of SPIE proceeding, 8-28
MacDonald D, Kabani N,?Avis D. Automated 3-D extraction of inner and outer surfaces of cerebral cortex from MRI. Neuroimage, 2000, 12(3): 340-356
Davatzikos C, Bryan RN. Using a deformable surface model to obtain a shape representation of the cortex. IEEE Transactions on Medical Imaging, 1996, 15(6): 785-795
Xu C, Prince JL. Snakes,shape,and gradient vector flow. IEEE Transaction on Image Processing, 1998, 7(3): 359-369
贺士娟，李颖，杨雅妹等.基于可变形模型算法提取MRI图像脑区域的方法.河北工业大学学报，2002，31（2），1-5
Staib LH, Duncan JS. Boundary finding with parametrically deformable models. IEEE Transaction on Pattern Analysis and Machine Intelligence, 1992, 14(11): 1061-1075
McInerkey T, Terzopoulos D. Topologically adaptable snakes. Proceedings of the Fifth International Conference on Computer Vision, Cambridge, USA, 1995: 840-845
Ivins J, Porrill J. Active region models for segmenting medical images. Image and Vision Computing, 1995, 13(5): 431-438
Jones TD, Plassmann P. An active contour model for measuring the area of leg ulcers. IEEE Transactions on Medical Imaging, 2000, 19(12): 1202-1210
Lu Aidong, Tang Long, Xu Yuhua etc. Segmentation of ultrasound images with interactive B-spline snakes and its application. Journal of Software, 2001, 12(12): 1760-1768
Hjelmas E, Low B K. Face detection:A survey. Computer Vision and Image Understanding, 2001, 83(3): 236-274
Huang CL, Chen CW. Human facial feature extraction for face interpretation and recognition. Pattern Recognition, 1992, 25(12): 1435-1444
Gunn SR, Nixon M S. A dual active contour for head and boundary extraction. in IEE Colloquium on Image Processing for Biometric Measurement, London,Apr. 1994,6-11
Yuille AL, Hallinan PW, Cohen DS. Feature extraction from faces using deformable templates. International Journal of Computer Vision, 1992, 8(2): 99-111
Coots TF, Taylor CJ, Lanitis A. Active shape models—‘smart snakes’. in Proceedings of British Machine Vision Conterence, Berlin, Germany, 1992: 266-275
史文中，朱长青，王昱.从遥感影像中提取道路特征的方法综述与展望.测绘学报，2001，30（3）：257-262
Gruen A, Li H.Semiautomatic linear feature extraction with dynamic programming and LSB-Snakes. Photogrammetric Engineering and Romote Sensing, 1997, 63 (8): 985-995
刘少创，林宗坚.航空影象分割的snake方法. 武汉测绘科技大学学报，1995，20（1）：7-11
刘少创，林宗坚.基于动态规划的航空影象中的目标提取.中国图象图形学报，1996，1（1）：30-35
孔祥维，石浩.形状约束的Snake算法在探地雷达图像目标自动提取中的应用.物探化探计算技术，2001，23（4）：333-337
Wang Y, Teoh EK. Lane detection using B-snake. IEEE International Conference on Information, Intelligence and Systems, Washington DC, 1999: 438-443
Wang Y, Shen Dinggang, Teoh E K. Lane detection using spline model. Pattern Recognition letters, 2000, 21(8): 677-689
Terzopoulos D, Witkin A, Kass M. Constraints on deformable models:recovering 3D shape and nonrigid motion. Artificial Intelligence, 1988, 36(1): 91-123
Sclaroff S, Pentland A. Closed-form solutions for physically-based shape modeling and recognition. IEEE Transaction on Pattern Analysis and Machine Intelligence, 1991, 13 (7): 715-729
Delingette H, Hebert M, Ikeuchi K. Shape representation and image segmentation using
deformable surface. Image and Vision Computing, 1992, 10(3): 132-144
McInerney T, Terzopoulos D. Finite element techniques for fitting a deformable model to 3D data. Proc. Vision Interface '93, Toronto, Canada, May, 1993, 70-76
Shen Xinquan, Hogg D. 3D shape recovery using a deformable model. Image and Vision Computing, 1995, 13(5): 377-383
夏利民，谷士文，曾三友.基于形变模型由立体序列图象恢复物体的形状.计算机工程，2001，27（10）：22-23
夏利民，谷士文，沈新权.基于形变模型的3D表面自适应重建.中国图象图形学报，2000，5（5）：396-400
Unser M. Spline: A prefect fit for Signal and Image Processing, IEEE Signal Processing Magazine. 1999, 16(6): 22-38
施法中.计算机辅助几何设计与非均匀有理B样条.北京：北京航空航天大学出版社，1994
陆明万，罗学富.弹性理论基础.北京：清华大学出版社，1990
丁学成.弹性力学中的变分方法.北京：高等教育出版社，1986
Chenyang Xu, Pham DL, Prince JL .Image Segmentation Using Deformable Models. Handbook of Medical Imaging -- Volume 2: Medical Image Processing and Analysis, edited by J.M. Fitzpatrick and M. Sonka, SPIE Press, May 2000: 129-174
柯朗，希尔伯特著，熊振翔，杨应辰译.数学物理方法.北京：科学出版社，1977
Rückert D. Segmentation and tracking in cardiovascular MR images using geometrically deformable models and templates. PhD thesis, London, University of London, 1997
Menet S, Saint-Marc P, Medioni G. B-snakes: implementation and application to stereo. In: Proceedings of image understanding workshop. Pittsburgh: IEEE Computer Society Press, 1990, 720-726
Amini A, Weymouth T, Jain R. Using dynamic programming for solving variational problems in vision. IEEE Transaction on Pattern Analysis and Machine Intelligence, 1990, 12(9): 855-867
Williams DJ, Shah M. A fast algorithm for active contours and curvature estimation. CVGIP: Image Understanding, 1992, 55(1): 14-26
Gunn SR, Nixon MS. A robust snake implementation: a dual active contour. IEEE Transaction on Pattern Analysis and Machine Intelligence, 1997, 19(1): 63-68
Velasco FA, Marroquín JL. Robust parametric active contours:the sandwich snakes. Machine Vision and Application, 2001, 12(5): 238-242
Poon CS, Braun M, Fahrig R, etc. Segmentation of medical images using an active contour model incorporating region-based images features. In Visualization in Biomedical Computing
1994, volume 2359,90-97
Szeliski R. Bayesian modeling of uncertainty in low-level vision. International Journal of computer vision, 1990, 5(3): 271-301
Storvik G. A Bayesian approach to dynamic contours through stochastic sampling and simulates annealing. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1994, 16(10): 970-986
Brigger P, Hoeg J, Unser M. B-spline snakes:a flexible tool for parametric contour detection. IEEE Transactions on Image Processing, 2000, 9(9): 1484-1496
Wang M, Evans J, Hassebrook L, et al. A multistage optimal active contour model. IEEE Transaction on Image Process, 1996, 5 (11): 1586-1591
Wang Y, Teoh EK, Shen DG. Structure-adaptive B-snake for segmenting complex objects. IEEE International Conference on Image Processing , Thessaloniki,Greece, 2001,IEEE press
朱心雄等.自由曲线曲面造型技术.北京：科学出版社，2000
Liang JM, McInerney T, Terzopoulos D. United snakes. In: Proceedings of the IEEE international conference on computer vision. Corfu, Greece, 1999. 923-940
王勖成，邵敏.有限单元法基本原理和数值方法.北京：清华大学出版社，1997
刘正兴，孙雁，王国庆.计算固体力学.上海：上海交通大学出版社，2000
Terzopoulos D, Qin H. Dynamic NURBS with geometric constraints for interactive sculpting. ACM Transaction on Graphics, 1994, 13(2): 103-135
经玲，席平，唐荣锡.有限元方法在变形曲线曲面造型中的应用.计算机学报，1998，21（3）：245-251
Flickner M, Sawhney H, Pryor D etc. Intelligent interactive image outlining using spline snakes. in 28th Asilomar Conference on Signals, Systems and Computers, vol 1, 1994: 731-735
Xu C, Prince JL. Gradient Vector Flow: A New External Force for Snakes. Proc. IEEE Conf. on Comp. Vis. Patt. Recog. (CVPR), Los Alamitos: Comp. Soc. Press, June 1997: 66-71
Gupta SN, Prince JL. Stochastic models for DIV-CURL optical flow methods. IEEE signal processing letters, 1996, 3(2): 32-35
周彦博，张志广.可变形物体的轮廓的提取.电子学报，1998，26（7）：133-137
Castleman KR著，朱志刚，林学訚，石定机等译.数学图像处理.北京：电子工业出版社，2002
Lobregt S, Viergever MA. A discrete dynamic contour model. IEEE Transactions on Medical Imaging, 1995, 14(1): 12-24
张爱东，张田文.一种B样条主动轮廓线模型.哈尔滨师范大学自然科学学报，1999，15
（4）：62-66
Brigger P, Engel R, Unser M. B-spline Snakes and a JAVA interface: An Intuitive Tool for General Contour Outlining. Proceedings of the 1998 IEEE International Conference on Image Processing (ICIP'98), Chicago IL, USA, October 4-7, 1998, vol. II, pp. 277-281
罗希平，田捷，诸葛婴等.图像分割方法综述.模式识别与人工智能，1999，12（3）：300-312
Kim DK. B-spline representation of active contours. Fifth international Symposium on Signal Processing and its Applications, Brisbane, Australia, 22-25 August, 1998, 813-816
梁远蕾，王宗彦.基于断层数据的三维重构技术.华北工学院学报，2002，23（1）：38-41
郭开波，周钢，王从军等.一种基于断层测量图片的实体轮廓提取方法.计算机辅助工程，2001,10（4）：50-54
谢红，张树生，张定华.产品工业计算机断层扫描成象切片的图象处理.机械科学与技术，1997，16（6）：1115-1118
林丽华，卢清萍，颜永年等.面向RP的CT图象反求技术.计算机辅助设计与制造，1998，（9）：48-50
刘振凯，陈剑虹，乔志林等.层去图像法反求工程中的数据处理.中国机械工程，2000，11（8）：929-932
张英杰，栗全庆，白作霖等.面向逆工程的实体再现方法的研究—图象滤波和边界提取.机床与液压，1999（2）：55-56
谷口庆治著，朱虹译.数字图像处理：基础篇.北京：科学出版社，2002
Canny J. A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1986, 8(6): 679-698
徐建华.图像处理与分析.北京：科学出版社，1992
蒋剑峰，何永辉，赵万生.利用CCD进行光切法测量的研究.计量技术，1999，（8）：7-10
贾云得.机器视觉.北京：科学出版社，2000
Wu JS, Leou PE. New polygonal approximation schemes for object shape representation. Pattern Recognition, 1993, 26(4): 471-484
Pikaz A, Dinstein I. Optimal polygonal approximation of digital curves. Pattern Recognition, 1995, 28(3): 373-379
肖轶军，丁明跃，彭嘉雄.基于迭代最近点的B样条曲线拟合方法研究.中国图象图形学报，2000，5A（7）：585-588
曲军，孙烨，曲刚.B样条曲线参数重载的非线性整体逼近拟合法.机械设计与制造，2001（5）：36-37
Lu Feng,Milios EE. Optimal spline fitting to planar shape. Signal Processing, 1994, vol. 37: 129-140
余学进，Redekop D.自适应有限元的应用和发展.南昌航空工业学院学报，1998,12（4）：78-83
郭书祥.自适应有限元方法用其工程应用.力学进展，1997，27(4)：479-488
周辉，李涛，邢启江等.数字曲线的线性逼近和分段识别.大连理工大学学报，1997，37（5）：576-580
文贡坚，王润生.数字曲线上特征点的检测.计算机学报，1998，21（6）：520-526
徐飞，施晓红.MATLAB应用图像处理.西安：西安电子科技大学出版社，2002
Boehm W. Inserting new knots into B-spline curve. Computer Aided Design, 13(6): 199-201
Dierckx P. Curve and surface fitting with splines. Clarendon press, 1993
Cham TJ, Cipolla R. Automated B-spline curve representation incorporating MDL and error- minimizing control point insertion strategies. IEEE transactions on pattern analysis and machine intelligence, 1999, 21(1): 49-53
张习文，李佐，蔡士杰等.基于遗传算法的以线段和圆弧为基元的曲线拟合.计算机辅助设计与图形学学报，2002，14（2）：1-4
余学军，彭立中.二值图象曲线轮廓提取的新算法.中国图象图形学报，2002，7A（3）：272-275
李斌，庄天戈.图像分割中采用自适应三次B样条修饰轮廓线.光学技术，2001，27（5）：477-480
Xu Meihe, Tang Zesheng. General surface reconstruction from a set of planar contours. Journal of Tsinghua University, 1997, 37(4): 59-64
Meyers D, Skinner S. Surface from contours. ACM Transactions on Graphics, 1992, 11 (3): 228-258
秦绪佳，纪凤欣，吴良武等.由基于轮廓重建的表面模型构建实体几何模型.机械设计与研究，2001，17（2）：15-17
普雷帕拉塔，沙莫斯著，庄心谷译.计算几何导论.北京：科学出版社，1990
张英杰，许长军.一种适用于逆真设计的特征提取方法的研究.机床与液压，2001（1）：36-37
吕铁英，彭嘉雄.图象轮廓特征提取新方法研究.中国图象图形学报，1999，4A（8）：655-658
Dumitras A, Venetsanopoulkos AN. A comparative study of snake models with application to object shape description in bi-level and gray-level images. Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing,Baltimore, USA, 3-6 June, 2001,1-5
经玲，席平，唐荣锡.应用可变形模型进行曲线曲面光顺.软件学报，1998，9（6）：464-468
Guan Zhidong, Jin Ling, Ning Tao etc. Study and application of physics-based deformable curves and surfaces. Computer & Graphics, 1997, 21(3): 305-313
张庆丰，乐清洪，马泽恩.织物变形的物理仿真技术.计算机辅助设计与图形学学报，2001，13（5）：1-6
冯康，石钟慈.弹性结构的数学理论.北京：科学出版社，第二版，1987
杨桂通.弹塑性力学.北京：人民教育出版社，1980
吴家龙.弹性力学.上海：同济大学出版社，1987
Celniker G, Gossard D. Deformable curve and surface finite-elements for free- form shape design. Computer Graphics, 1991, 25(4): 257-266
李永林，王启付，钟毅芳等.可变形曲面的有限元模型.计算机辅助设计与制造，1998（3）：46-49
Fang L, Dossard DC. Fitting 3D curves to unorganized data points using deformable curves. Proceedings of the International Conference’92 on Computer Graphics, North- Holland, 1992: 535-543
Fang L,Dossard DC.Multidimensional curve fitting to unorganized data points by nonlinear minimization.Computer-Aided Design,1995,27(1):48-58
Qin Hong, Terzopoulos D. Dynamic NURBS swung surfaces for physics-based shape design. Computer-Aided Design, 1995, 27(2): 111-127
纪玉春，李鹏.基于物理的可变形模型.系统仿真学报，2001，13（增刊）：19-22
朱东波，张舜德，李涤尘等.密集散乱测量数据点的B样条曲面拟合研究.计算机辅助设计与图形学学报，2001，13（12）：1123-1128
Liao Chia-Wei, Medioni G. Representation of range data with B-spline surface patches. 11th International Conference on Pattern Recognition (ICPR'92), The Hague , Netherlands, August 1992, 745-748
程久平，史翔.用B样条曲线构造几何物体模型的方法.合肥工业大学学报，1998，21（5）：88-91
侯宇，李刚，崔晨阳等.自由型曲面的测量与重建.计量学报，1999，20（4）：252-255
寇淑清，杨慎华，金文明.B样条曲面在模具型腔描述及3 维网格生成中的应用.中国机械工程，2000，11（11）：1268-1271
张丽艳，周来水，周儒荣.逆向工程中曲面重构算法研究与实现.航空学报，1999，20（3）：242-244
Lee Seungyong, Wolberg G, Shin SY. Scattered data interpolation with multilevel B-splines. IEEE Transactions on Visualization and Computer Graphics, 3(3): 1-17
蔡宣三.最优化与最优控制.北京：清华大学出版社，1982
彭芳瑜，周济，周艳红等.基于最小二乘法的曲面生成算法研究.工程图学学报，1999,20(3)：41-46
吕科，耿国华，周明全.医学图象三维表面的快速重构方法.计算机工程，2002，28（2）：68-78
Hanselman D,Littlefield B著，张航，黄攀译.精通MATLAB 6.北京：清华大学出版社，2002
Hanselman D,Littlefield B著,李人厚，张平安等译.精通MATLAB—综合辅导与指南.西安：西安交通大学出版社，1998
Magrab EB等著，高会生，李新叶，胡智奇等译.MATLAB原理与工程应用.北京：电子工业出版社，2002
Pfeifle R, Seidel HP. Fitting Triangular B-Splines to Functional Scattered Data. Computer Graphics Forum, 1996, 15(1): 15-23
孙延奎，朱心雄.B样条曲线的小波光顺法.工程图学学报，1998，19（4）：51-58
孙延奎，朱心雄.任意B样条曲面的多分辨率表示及光顺.工程图学学报，1998，19（3）：49-54