自由曲面逆向工程技术的研究
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摘要
逆向工程是近年来发展非常迅速的一个研究领域，它结合了测量技术、计算机图形学、图像处理等快速发展的技术，是各学科交叉的新兴学科。与传统的将设计概念或CAD模型转化为实际产品的设计方法不同，逆向工程是将实际的零件转变为产品的概念或CAD模型。它不仅提供了产品设计的新方法，而且可以显著缩短设计时间，加快产品开发周期。逆向工程已开始在汽车、家电等领域得到广泛的应用，成为企业增强产品竞争力的关键技术之一。逆向工程处理的零件，很多都具有复杂的自由曲面，自由曲面的反求是逆向工程解决的主要问题。本文研究了逆向工程中B样条曲面重构的关键技术，包括B样条曲线的拟合、散乱数据的B样条曲面拟合、逐层扫描数据的B样条曲面拟合以及B样条曲线的光顺。
作者完成的主要工作如下：
1)研究了B样条曲线拟合的问题，讨论分析了数种常用的曲线拟合方法，并提出了一种给定允差下B样条曲线拟合的方法。
2)研究了对散乱数据进行B样条曲面拟合的问题，讨论了数种常用的曲面拟合方法，分析了散乱数据的参数化、节点矢量的选取、精度控制、数值稳定性等问题。提出了一种给定允差下对散乱数据进行拟合的方法。特别地，为保证曲面间的光滑连接，提出了一种插值于给定边界曲线且逼近内部散乱数据点的曲面拟合算法：提出了一种插值于给定边界曲线及其跨界导矢且逼近内部散乱数据点的曲面拟合算法。
3)研究了逐层扫描数据的曲面拟合问题，给出了一种是利用统一的节点矢量进行拟合的算法，用该法得到的B样条曲面不仅可以满足精度要求，而且具有较少的控制顶点和较好的光顺性，计算速度也有较大的提高。
4)研究了B样条曲线光顺的问题，在传统能量法的基础上提出了B样条曲线的一种光顺方法。该方法通过对局部控制顶点的修改来进行曲线光顺。与能量法相比，不但可提高计算速度而且可使曲线的形状变化较小，光顺效果较好。
上述所提出的方法都用实例进行了验证。
Reverse engineering is a rapidly developing discipline, which integrates the developing technologies such as measurement technology, computer geometry a nd graphics processing. While conventional engineering transforms design cone epts and CAD models into real parts, in reverse engineering real parts are tra nsformed into design concepts and CAD models. Reverse engineering not only provides a new methods for product design, but also greatly shortens the desi gn cycle. In industrial areas such as automobiles, domestic electronics and so on, reverse engineering begins to play an important role. The parts reverseeng ineering deals with mostly have freeform surfaces.The regeneration of the free form surfaces is the main task of reverse engineering. In this paper some key techniques for NURBS surface reconstruction in reverse engineering such as B -spline curve fitting , B-spline surface fitting to scattered data ,B-spline surfa ce fitting to contour data and B-spline curve fairing are studied.
The dissertation is summarized as follows:
1) B-spline curve fitting is studied. Several curve fitting methods are disc ussed, and a method for B-spline curve fitting in the specified tolerance is pr esented.
2) B-spline surface fitting is studied. Several surface fitting methods are d iscussed. the parametrization of scattered data ,the selection of knot vector, th e control of precision and the stability of the methods are analyzed, and the algorithm for B-spline surface fitting to scattered data in specified tolerance i s given. Especially, in order to assure the smooth connection between the adj acent surfaces, the surface fitting algorithm of interpolating four boundary cur ves and approximating inner scattered data is presented, the surface fitting al gorithm of interpolating four boundary curves and cross boundary derivative v ectors and approximating inner scattered data is also presented.
3) B-spline surface fitting to contour data is studied. A fitting method of using uniform knot vector is given. Using this method, the resulted B-spline s urface not only satisfy the requirements of precision, but also possess fewer c ontrol vertices and good fairness, the computation speed is greatly improved t oo.
4) B-spline curve fairing is studied. A fairing method based on energy me thod for cubic B-spline curves is presented.Since only local control vertices ar e modified, this method, compared with energy method, not only makes the c omputation faster, but also makes the changes in shape of curve less . Using this method ,we can obtain good B-spline curves.
All the algorithms are verified.
作者完成的主要工作如下：
1)研究了B样条曲线拟合的问题，讨论分析了数种常用的曲线拟合方法，并提出了一种给定允差下B样条曲线拟合的方法。
2)研究了对散乱数据进行B样条曲面拟合的问题，讨论了数种常用的曲面拟合方法，分析了散乱数据的参数化、节点矢量的选取、精度控制、数值稳定性等问题。提出了一种给定允差下对散乱数据进行拟合的方法。特别地，为保证曲面间的光滑连接，提出了一种插值于给定边界曲线且逼近内部散乱数据点的曲面拟合算法：提出了一种插值于给定边界曲线及其跨界导矢且逼近内部散乱数据点的曲面拟合算法。
3)研究了逐层扫描数据的曲面拟合问题，给出了一种是利用统一的节点矢量进行拟合的算法，用该法得到的B样条曲面不仅可以满足精度要求，而且具有较少的控制顶点和较好的光顺性，计算速度也有较大的提高。
4)研究了B样条曲线光顺的问题，在传统能量法的基础上提出了B样条曲线的一种光顺方法。该方法通过对局部控制顶点的修改来进行曲线光顺。与能量法相比，不但可提高计算速度而且可使曲线的形状变化较小，光顺效果较好。
上述所提出的方法都用实例进行了验证。
Reverse engineering is a rapidly developing discipline, which integrates the developing technologies such as measurement technology, computer geometry a nd graphics processing. While conventional engineering transforms design cone epts and CAD models into real parts, in reverse engineering real parts are tra nsformed into design concepts and CAD models. Reverse engineering not only provides a new methods for product design, but also greatly shortens the desi gn cycle. In industrial areas such as automobiles, domestic electronics and so on, reverse engineering begins to play an important role. The parts reverseeng ineering deals with mostly have freeform surfaces.The regeneration of the free form surfaces is the main task of reverse engineering. In this paper some key techniques for NURBS surface reconstruction in reverse engineering such as B -spline curve fitting , B-spline surface fitting to scattered data ,B-spline surfa ce fitting to contour data and B-spline curve fairing are studied.
The dissertation is summarized as follows:
1) B-spline curve fitting is studied. Several curve fitting methods are disc ussed, and a method for B-spline curve fitting in the specified tolerance is pr esented.
2) B-spline surface fitting is studied. Several surface fitting methods are d iscussed. the parametrization of scattered data ,the selection of knot vector, th e control of precision and the stability of the methods are analyzed, and the algorithm for B-spline surface fitting to scattered data in specified tolerance i s given. Especially, in order to assure the smooth connection between the adj acent surfaces, the surface fitting algorithm of interpolating four boundary cur ves and approximating inner scattered data is presented, the surface fitting al gorithm of interpolating four boundary curves and cross boundary derivative v ectors and approximating inner scattered data is also presented.
3) B-spline surface fitting to contour data is studied. A fitting method of using uniform knot vector is given. Using this method, the resulted B-spline s urface not only satisfy the requirements of precision, but also possess fewer c ontrol vertices and good fairness, the computation speed is greatly improved t oo.
4) B-spline curve fairing is studied. A fairing method based on energy me thod for cubic B-spline curves is presented.Since only local control vertices ar e modified, this method, compared with energy method, not only makes the c omputation faster, but also makes the changes in shape of curve less . Using this method ,we can obtain good B-spline curves.
All the algorithms are verified.
引文
1.Computer Aided Design, 1997,29(4).
2.Varady T, Martin R R, Cox J.Reverse engineering of geometric models-an introduction.CAD, 1997,29(4):255-268.
3.Sarkar B,Menq C H.Parameter optimization in approximating curves and surfaces to measurement data. CAGD, 1991,8:267-290.
4.Motavalli S,Bidanda B.A part image reconstruction system for reverse engineering of design modifications.Journal of Manufacturing Systems, 1991,10(5):383-395.
5．许智钦，孙长库．3D逆向工程技术．中国计量出版社，2002．
6.Farin G. Smooth Interpolation to Scattered Data. In Barnhill R E,Boehm w, surface in C A GD,North-Holland,Amsterdam, 1983:43-63.
7.Green P J and Sibson R.Computing Dirichlet Tesssellations in the plane.Cpomputer Journal, 1978,21(2): 168-173.
8.Bowyer A.Computing Dirichlet Tesssellations. Cpomputer Journal, 1981,24(2): 162-166.
9.Lawson C L.Software for C~1 surface Interpolation,in Methematical Sofeware Ⅲ.Academic Press, 1977.
10.Lee D T and Schachter B J.Two Algorithms for Constructing a Delaunay Triangulation. Int'l J of Comp and Info Sci, 1980,9(3):219-242.
11.Cline A K and Renka R L.Storage Efficient Method for Construction of a Thiessen Triangulation. Rockey Mountain Journal of Methematics, 1984,14(1): 119-139.
12.Choi B K,Shin H Y, Yoon Y I and Lee J W. Triangulation of Scattered Data in 3D Space. CAD, 1988,20(5):239-248.
13．朱本富．CAD／CAM中基于三角域的散乱数据几何造型研究．北京航空航天大学博士学位论文．1997．
14．周晓云．散乱数据几何造型技术研究．北京航空航天大学硕士论文．1993．
15．张鲜．CAD／CAM中参数Bernstein-Bezier三角曲面造型方法的研究．北京航空航天大学科学研究报告，1992．
16.Piper B R.Visually Smooth Tnterpolation with Triangular B-B Patches,in Geometric Modeling:Algorithms and Trends,SIAM, 1987.
17．姜寿山．关于CAD／CAM中三角曲面造型方法的研究．西北工业大学博士学位论文，1987．
18．柯映林．离散数据的几何造型技术及其应用研究．南京航空航天大学博士学位论文，1992．
19．张和明，复杂曲面设计和数控加工的研究，浙江大学博士论文，1995．
20．李江雄．反求工程中曲面建模技术及相关软件(模块)分析．计算机辅助设计与制造．1999，10：14-16．
21．管伟光．体视化技术及应用．电子工业出版社，1998．
22.Sarkar B,Menq C H.Smooth-surface approximation and reverse engineering. CAD, 1991, 23(9):623-628.
23.Yang M,Lee E.Segmentation of measured point data using a parametric quadric surface approximation. CAD, 1999,31:449-457.
24.Pietilainen M,Rosenfeld A,Walter I.Split-and-link algorithms for image segmentation.Pattern Recognition, 1982,15:287-298.
25.Leonardis A,Gupta A,Bajcsy R.Segmentation of range image as the search for geometric parametric models.International Jounal of Computer Vision, 199514:253-277.
26.Fitzgbbon A W, Eggert D,Fisher R B.High-level CAD model acquisition from range images. CA D, 1997,29(4):321-330.
27.Ghosal S,Mehrotra R.Range surface characterization and segmentation using neural networks.Pattern Recognition, 1995,28(5):711-727.
28.Chen Y H,Liu C Y. Quadric surface extraction using genetic algorithms.CAD, 1999,31:101-110.
29．宋德军．基于能量优化法的曲线曲面造型理论及应用研究．北京航空航天大学博士学位论
文，1998．
30.Floater MS.How to approximate scattered data. SINTEF Report No. STF42 A96019,Oslo,1998.
31.Gu P, Yan X.Neural network approach to the reconstruction of freeform surface for reverse engineering. CAD, 1995,27(1):59-64.
32.Lee S,Wolberg G,Shin S Y. Scattered Data Interpolation with Multilevel B-splines. IEEE Transaction on visualization and Computer Graphics,1997,3(3):228-244.
33.Piegl L,Tiller W. Algorithm for approximate NURBS skinning. CAD, 1996,28(9):699-706.
34.Park H,Kim K.Smooth surface approximation to serial cross-sections. CAD, 1996,28(12):995-1005.
35．王拉柱，朱心雄，唐泽圣．C~n连续Coons类混合B样条曲面．计算机学报，1996，19：201-209．
36.Wang Lazhu,Zhu Xinxiong and Tang Zesheng. Coons Type Blended B-spline Surface and Its Conversion to NURBS Surface. Computer & Graphics, 1997,21(3):297-303.
37．王拉柱等．曲面转换中的一个重要定理及其证明．清华大学自然科学学报，1998．
38．宋德军，朱心雄．用能量法构造N边域曲面．工程图学学报，1997，4．
39.Kjellander, J A P. Smoothing of Cubic Parametric Spline. CAD,1983,15(5):288-293.
40.Farin G,Rein G,Sapids N and Worsey A J.Fairing Cubic B-Spline curves. CAGD,1987,4:91-103.
41.Lott N J and Pullin D I.Method for Fairing B-Spline Surfaces. CAD, 1988,20(10),597.604.
42．宁涛．NURBS曲面造型应用技术研究．北京航空航天大学博士学位论文，1996．
43．经玲．用有限元法整体构造B样条曲面的理论与应用研究．北京航空航天大学博士学位论文，1997．
44．孙延奎，朱心雄．B样条曲线的小波光顾法．工程图学学报，1998，4：51-58．
45．朱心雄．自由曲线曲面造型技术．科学出版社，2000．
46.Terzopoulos D,Platt J and Barr A.Elastically Deformable models.ACM Computer Graphics, 1987,24(4):205-214.
47.Celniker G and Grossard D.Deformable Curve and Surface Finite-Elements for Free-From Shape Design. ACM Computer Graphics, 1991,25(4):257-266.
48.Welch W and Witckin A.Variational Surface Modelling. ACM Computer Graphics, 1992,26(2):157-166.
49.Moreton H P and Sequin C H.Functional Optimization for Fair Surface Design. ACM Computer Graphics, 1992,26:167-176.
50.Qin H and Terzopoulos D.Dynamic NURBS Swung Surfaces for Physics-Based Shape Design. CAD, 1995,27(2): 111-127.
51.Kallay M.Constrained optimization in Surface Design in modeling in Computer Graphics.B Falcidieno and T L Lunii,Eds. Springer-Verlag, 1993:85-93.
52.Ma W, Kruth J P. Parametrization of randomly measured points for least squares fitting of B-spline curve and surfaces.CAD, 1995,27(9):663-675.
53．苏步青，刘鼎元．计算几何．上海科学技术出版社，1981．
54．施法中．计算机辅助几何设计与非均匀有理B样条．北京航空航天大学出版社，1994．
55．马利庄，石教英．曲线曲面的几何光顺算法．计算机学报，1996，19：210-216．
56．曹智军，侯伯杰，郑鹏，蒋文兵．参数三次B样条曲线的一种光顺方法．郑州轻工业学院学报，2003，18(1)：7-9．
2.Varady T, Martin R R, Cox J.Reverse engineering of geometric models-an introduction.CAD, 1997,29(4):255-268.
3.Sarkar B,Menq C H.Parameter optimization in approximating curves and surfaces to measurement data. CAGD, 1991,8:267-290.
4.Motavalli S,Bidanda B.A part image reconstruction system for reverse engineering of design modifications.Journal of Manufacturing Systems, 1991,10(5):383-395.
5．许智钦，孙长库．3D逆向工程技术．中国计量出版社，2002．
6.Farin G. Smooth Interpolation to Scattered Data. In Barnhill R E,Boehm w, surface in C A GD,North-Holland,Amsterdam, 1983:43-63.
7.Green P J and Sibson R.Computing Dirichlet Tesssellations in the plane.Cpomputer Journal, 1978,21(2): 168-173.
8.Bowyer A.Computing Dirichlet Tesssellations. Cpomputer Journal, 1981,24(2): 162-166.
9.Lawson C L.Software for C~1 surface Interpolation,in Methematical Sofeware Ⅲ.Academic Press, 1977.
10.Lee D T and Schachter B J.Two Algorithms for Constructing a Delaunay Triangulation. Int'l J of Comp and Info Sci, 1980,9(3):219-242.
11.Cline A K and Renka R L.Storage Efficient Method for Construction of a Thiessen Triangulation. Rockey Mountain Journal of Methematics, 1984,14(1): 119-139.
12.Choi B K,Shin H Y, Yoon Y I and Lee J W. Triangulation of Scattered Data in 3D Space. CAD, 1988,20(5):239-248.
13．朱本富．CAD／CAM中基于三角域的散乱数据几何造型研究．北京航空航天大学博士学位论文．1997．
14．周晓云．散乱数据几何造型技术研究．北京航空航天大学硕士论文．1993．
15．张鲜．CAD／CAM中参数Bernstein-Bezier三角曲面造型方法的研究．北京航空航天大学科学研究报告，1992．
16.Piper B R.Visually Smooth Tnterpolation with Triangular B-B Patches,in Geometric Modeling:Algorithms and Trends,SIAM, 1987.
17．姜寿山．关于CAD／CAM中三角曲面造型方法的研究．西北工业大学博士学位论文，1987．
18．柯映林．离散数据的几何造型技术及其应用研究．南京航空航天大学博士学位论文，1992．
19．张和明，复杂曲面设计和数控加工的研究，浙江大学博士论文，1995．
20．李江雄．反求工程中曲面建模技术及相关软件(模块)分析．计算机辅助设计与制造．1999，10：14-16．
21．管伟光．体视化技术及应用．电子工业出版社，1998．
22.Sarkar B,Menq C H.Smooth-surface approximation and reverse engineering. CAD, 1991, 23(9):623-628.
23.Yang M,Lee E.Segmentation of measured point data using a parametric quadric surface approximation. CAD, 1999,31:449-457.
24.Pietilainen M,Rosenfeld A,Walter I.Split-and-link algorithms for image segmentation.Pattern Recognition, 1982,15:287-298.
25.Leonardis A,Gupta A,Bajcsy R.Segmentation of range image as the search for geometric parametric models.International Jounal of Computer Vision, 199514:253-277.
26.Fitzgbbon A W, Eggert D,Fisher R B.High-level CAD model acquisition from range images. CA D, 1997,29(4):321-330.
27.Ghosal S,Mehrotra R.Range surface characterization and segmentation using neural networks.Pattern Recognition, 1995,28(5):711-727.
28.Chen Y H,Liu C Y. Quadric surface extraction using genetic algorithms.CAD, 1999,31:101-110.
29．宋德军．基于能量优化法的曲线曲面造型理论及应用研究．北京航空航天大学博士学位论
文，1998．
30.Floater MS.How to approximate scattered data. SINTEF Report No. STF42 A96019,Oslo,1998.
31.Gu P, Yan X.Neural network approach to the reconstruction of freeform surface for reverse engineering. CAD, 1995,27(1):59-64.
32.Lee S,Wolberg G,Shin S Y. Scattered Data Interpolation with Multilevel B-splines. IEEE Transaction on visualization and Computer Graphics,1997,3(3):228-244.
33.Piegl L,Tiller W. Algorithm for approximate NURBS skinning. CAD, 1996,28(9):699-706.
34.Park H,Kim K.Smooth surface approximation to serial cross-sections. CAD, 1996,28(12):995-1005.
35．王拉柱，朱心雄，唐泽圣．C~n连续Coons类混合B样条曲面．计算机学报，1996，19：201-209．
36.Wang Lazhu,Zhu Xinxiong and Tang Zesheng. Coons Type Blended B-spline Surface and Its Conversion to NURBS Surface. Computer & Graphics, 1997,21(3):297-303.
37．王拉柱等．曲面转换中的一个重要定理及其证明．清华大学自然科学学报，1998．
38．宋德军，朱心雄．用能量法构造N边域曲面．工程图学学报，1997，4．
39.Kjellander, J A P. Smoothing of Cubic Parametric Spline. CAD,1983,15(5):288-293.
40.Farin G,Rein G,Sapids N and Worsey A J.Fairing Cubic B-Spline curves. CAGD,1987,4:91-103.
41.Lott N J and Pullin D I.Method for Fairing B-Spline Surfaces. CAD, 1988,20(10),597.604.
42．宁涛．NURBS曲面造型应用技术研究．北京航空航天大学博士学位论文，1996．
43．经玲．用有限元法整体构造B样条曲面的理论与应用研究．北京航空航天大学博士学位论文，1997．
44．孙延奎，朱心雄．B样条曲线的小波光顾法．工程图学学报，1998，4：51-58．
45．朱心雄．自由曲线曲面造型技术．科学出版社，2000．
46.Terzopoulos D,Platt J and Barr A.Elastically Deformable models.ACM Computer Graphics, 1987,24(4):205-214.
47.Celniker G and Grossard D.Deformable Curve and Surface Finite-Elements for Free-From Shape Design. ACM Computer Graphics, 1991,25(4):257-266.
48.Welch W and Witckin A.Variational Surface Modelling. ACM Computer Graphics, 1992,26(2):157-166.
49.Moreton H P and Sequin C H.Functional Optimization for Fair Surface Design. ACM Computer Graphics, 1992,26:167-176.
50.Qin H and Terzopoulos D.Dynamic NURBS Swung Surfaces for Physics-Based Shape Design. CAD, 1995,27(2): 111-127.
51.Kallay M.Constrained optimization in Surface Design in modeling in Computer Graphics.B Falcidieno and T L Lunii,Eds. Springer-Verlag, 1993:85-93.
52.Ma W, Kruth J P. Parametrization of randomly measured points for least squares fitting of B-spline curve and surfaces.CAD, 1995,27(9):663-675.
53．苏步青，刘鼎元．计算几何．上海科学技术出版社，1981．
54．施法中．计算机辅助几何设计与非均匀有理B样条．北京航空航天大学出版社，1994．
55．马利庄，石教英．曲线曲面的几何光顺算法．计算机学报，1996，19：210-216．
56．曹智军，侯伯杰，郑鹏，蒋文兵．参数三次B样条曲线的一种光顺方法．郑州轻工业学院学报，2003，18(1)：7-9．