扭曲自由曲面的测量数据处理与误差评定方法研究
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摘要
包括航空发动机的叶盘和叶片等零件在内的复杂自由曲面，其扭曲角度较大，制造难度远大于一般的自由曲面，而这些零件往往决定着装备的总体性能和水平，甚至影响着一个国家的科技水平和国防实力。现有的自由曲面测量、数据处理、曲面拟合及误差评定等方法主要针对一般的自由曲面，虽然在点云数据建模及误差评定方面已有一些商业软件和文献报道，但对于扭曲的自由曲面来说，不少商业软件和文献方法的实际使用效果不佳，且极少针对具有较大扭曲转角和极薄边缘的自由曲面进行点云建模与评定。研究适用于扭曲自由曲面的测量数据处理与误差评定方法对于提高复杂零件的加工精度，提高我国科技水平都具有重要的意义。
本论文针对扭曲自由曲面，对测量数据的预处理、自由曲面的连续性拼接、自由曲面的拟合技术和基于坐标系统一的误差评定方法进行了深入的研究。论文的主要工作包括：
简单介绍了天津大学精密测试技术及仪器国家重点实验室自主开发、研制的在线原位高精度测量机的结构，及测量机基本参数与误差的标定和补偿原理。该测量机能够测量包括直径在1000mm以内、曲面片之间空隙可窄至10mm、扭转角可高达65°的扭曲自由曲面，且测量不确定度不超过0.01mm。
对在线原位测量机所测得的扭曲自由曲面型值点数据进行了预处理。首先，对多次测量得到的数据进行坐标变换和坐标归一化的处理，使其成为统一的点云数据，并运用双正交小波实现点云数据的去噪，并提出了基于第二代小波变换上升格式的数据删减算法，此方法计算量小，计算速度快，适用于自适应、非线性的变换，而且可以确保变换的可逆性，文中用实验结果比较了此方法和其他数据删减方法的效果，表明其能更好的适应和实现扭曲自由曲面点云数据的删减。
通过深入分析分片Bezier曲面的拼接理论，考虑到Bezier曲线是NURBS（非均匀有理B样条）曲线的特殊格式，提出了借助分片Bezier曲面的G1连续性条件进行NURBS曲面拼接的思想，推导并实现了NURBS曲面间的G1光滑拼接。通过求解多片NURBS曲面G1光滑连续的约束方程，获得了多片NURBS曲面G1光滑连续时角点处的控制顶点和两张相邻NURBS曲面片G1光滑连续约束方程的初始条件，为预处理后的数据点进行参数化和曲面拟合提供了基础。
在对已有的数据参数化方法进行深入研究的基础上创立了一种数据参数化的算法。经过分析参数化的两种方案，选择直接在由折线围成的空间四边区域内进行参数化。通过坐标变换、点面投影把空间四边区域的数据参数化问题转化为了平面四边区域网格划分问题，通过提出平面四边区域两向弹性网格生成算法快速实现了由折线围成的空间区域的参数化；提出了基于B样条基函数快速计算和控制顶点数目优化的张量积双向NURBS拟合方法，通过NURBS曲面分片拟合可得到整体G1光滑连续的曲面，并通过比较本文的实验结果和文献中的结果，说明了本文方法针对扭曲自由曲面拟合的有效性和优势。
对拟合所得到的G1光滑连续的扭曲自由曲面进行了测头半径补偿和误差评定。对测量数据直接进行测头半径补偿的办法会给测量带来负担，且误差较大，本文在得到曲面拟合模型之后再进行测头半径补偿，研究了求取已知模型上任意点法矢量的算法，在法矢量方向进行测头半径补偿。为了评定产品是否符合原始设计要求，需要与设计给定的数学模型进行比较，即进行误差的分析；经过测头半径补偿后的拟合曲面与理论数据间存在着位置误差，即二者的坐标系往往是不统一的，需要先进行坐标系的配准才能进行误差评定，本文认为每个理论数据到被测曲面的距离平方和为最小时，理论坐标系与测量坐标系就匹配到了一起，在此基础上根据点到曲面的法向最短距离可得到扭曲自由曲面的形状误差。
The complex free-form surfaces, blisk and blade of aero engine included, arebig-twisted, they have more manufacture difficulties than the common free-formsurface, but the complex part decides the overall performance of the equipment, eveneffectes the technological level and national defense capabilities. Existing methods offree-form surface measurement、surface fitting and error evaluation mainly targetedthe common free-form surface, some methods in business software and literature wereineffective for twisted free-form surface, and barely studied for the surface withbig-twist and thin edge. So the study on data processing and evaluation for twistedfree-form surface is very important for improving the accuracy of complex part andtechnological level of our country.
For the twisted free-form surface, this paper studied the data preprocessing, thecontinued montage of free-form surface, fitting and error evaluation under theuniform coordinate system, based on the point cloud measured by the on-line in-situhigh-precision machine. The main work is as follows:
The structural design and error calibration of on-line in-situ measuring machinedeveloped by State Key Laboratory of precision measurement technology andinstruments in Tianjin University was described, and the error compensation ofthermal deformation and optical distortion was proposed the measuring machinecould measure all kinds of complex twisted free-form surfaces, including the surfacewas smaller than1000mm in diameter, with the gap between patches only10mm, thetorsion angle can be65°,and the measuring error was no more than0.01mm.
The deleting method of abnormal noises is researched. Ineorporating the3δ lawof mathematical statistics, by means of calculating mean and standard error todeterminate threshold, the adaptive counting threshold method is putforward, whichhas well adaptivity. Denoising cloudy data based wavelet technology is researched.Using wavelet technology to denoise is the process of decomposing signal withwavelet, acquiring approximate coeffieients and detail coeffieients, acting detailedcoefficients with soft threshold to suppress noises, then reconstructing signal to getthe goal of denoising. The key of simplifying data technique is retaining originalfeatures of cloudy data utmost. On the basis of reseaching current simplification technique, new method using second generation wavelet based on lifting tosimplification data is proposed.
TheG1smooth merging theory of the piecewise Bezier surface is analyzed, theBezier curve is a special NURBS curve is considered, the idea of resort to BeziersurfaceG1continuous condition for solving theG1continuous condition ofNURBS surfaces is presented.G1continuous condition of the NURBS is deducedand realized. Linear system ofG1continuous condition is solved, which providedthe basis for the following surface fitting.
The existing grid generation theory is studied, incapability of grid generation thatthese theory to the four sided region constructed by four discontinuous broken line isfound. A rapid parametrization algorithm which is used to spatial four-sided region ontriangular mesh is presented in the paper. By comparing two kinds of parametrizationprojects of the spatial four-sided region, directly grid generation algorithm is chose. Aplanar two-directional flexing grid generation algorithm is presented which canquickly parametrizing the planar four-sided region, in fact, a spatial four-sided regionis parametrizated. Comparing two NURBS surface fitting methods, in order toenhance the fitting precision and obtain more detail information of the NURBSsurface fitting, two-directional fitting method is adopted to fitting the NURBS surface.For the points on the common boundaries and corners is restricted base on theG1smooth connection condition. So by fitting every patches and a wholeG1smoothsurface is obtained.
Based on non-uniform B-splines, a new algorithm of calculating the normalvector at random position was deduced, and a formula with probe radius correctionwas proposed. An optimization algorithm named simplex method was applied in orderto eliminating the orientation error in the stage of workpiece measurement. It cansettle surfaces matching well by shifting and rotating the measuring coordinatesystem.
本论文针对扭曲自由曲面，对测量数据的预处理、自由曲面的连续性拼接、自由曲面的拟合技术和基于坐标系统一的误差评定方法进行了深入的研究。论文的主要工作包括：
简单介绍了天津大学精密测试技术及仪器国家重点实验室自主开发、研制的在线原位高精度测量机的结构，及测量机基本参数与误差的标定和补偿原理。该测量机能够测量包括直径在1000mm以内、曲面片之间空隙可窄至10mm、扭转角可高达65°的扭曲自由曲面，且测量不确定度不超过0.01mm。
对在线原位测量机所测得的扭曲自由曲面型值点数据进行了预处理。首先，对多次测量得到的数据进行坐标变换和坐标归一化的处理，使其成为统一的点云数据，并运用双正交小波实现点云数据的去噪，并提出了基于第二代小波变换上升格式的数据删减算法，此方法计算量小，计算速度快，适用于自适应、非线性的变换，而且可以确保变换的可逆性，文中用实验结果比较了此方法和其他数据删减方法的效果，表明其能更好的适应和实现扭曲自由曲面点云数据的删减。
通过深入分析分片Bezier曲面的拼接理论，考虑到Bezier曲线是NURBS（非均匀有理B样条）曲线的特殊格式，提出了借助分片Bezier曲面的G1连续性条件进行NURBS曲面拼接的思想，推导并实现了NURBS曲面间的G1光滑拼接。通过求解多片NURBS曲面G1光滑连续的约束方程，获得了多片NURBS曲面G1光滑连续时角点处的控制顶点和两张相邻NURBS曲面片G1光滑连续约束方程的初始条件，为预处理后的数据点进行参数化和曲面拟合提供了基础。
在对已有的数据参数化方法进行深入研究的基础上创立了一种数据参数化的算法。经过分析参数化的两种方案，选择直接在由折线围成的空间四边区域内进行参数化。通过坐标变换、点面投影把空间四边区域的数据参数化问题转化为了平面四边区域网格划分问题，通过提出平面四边区域两向弹性网格生成算法快速实现了由折线围成的空间区域的参数化；提出了基于B样条基函数快速计算和控制顶点数目优化的张量积双向NURBS拟合方法，通过NURBS曲面分片拟合可得到整体G1光滑连续的曲面，并通过比较本文的实验结果和文献中的结果，说明了本文方法针对扭曲自由曲面拟合的有效性和优势。
对拟合所得到的G1光滑连续的扭曲自由曲面进行了测头半径补偿和误差评定。对测量数据直接进行测头半径补偿的办法会给测量带来负担，且误差较大，本文在得到曲面拟合模型之后再进行测头半径补偿，研究了求取已知模型上任意点法矢量的算法，在法矢量方向进行测头半径补偿。为了评定产品是否符合原始设计要求，需要与设计给定的数学模型进行比较，即进行误差的分析；经过测头半径补偿后的拟合曲面与理论数据间存在着位置误差，即二者的坐标系往往是不统一的，需要先进行坐标系的配准才能进行误差评定，本文认为每个理论数据到被测曲面的距离平方和为最小时，理论坐标系与测量坐标系就匹配到了一起，在此基础上根据点到曲面的法向最短距离可得到扭曲自由曲面的形状误差。
The complex free-form surfaces, blisk and blade of aero engine included, arebig-twisted, they have more manufacture difficulties than the common free-formsurface, but the complex part decides the overall performance of the equipment, eveneffectes the technological level and national defense capabilities. Existing methods offree-form surface measurement、surface fitting and error evaluation mainly targetedthe common free-form surface, some methods in business software and literature wereineffective for twisted free-form surface, and barely studied for the surface withbig-twist and thin edge. So the study on data processing and evaluation for twistedfree-form surface is very important for improving the accuracy of complex part andtechnological level of our country.
For the twisted free-form surface, this paper studied the data preprocessing, thecontinued montage of free-form surface, fitting and error evaluation under theuniform coordinate system, based on the point cloud measured by the on-line in-situhigh-precision machine. The main work is as follows:
The structural design and error calibration of on-line in-situ measuring machinedeveloped by State Key Laboratory of precision measurement technology andinstruments in Tianjin University was described, and the error compensation ofthermal deformation and optical distortion was proposed the measuring machinecould measure all kinds of complex twisted free-form surfaces, including the surfacewas smaller than1000mm in diameter, with the gap between patches only10mm, thetorsion angle can be65°,and the measuring error was no more than0.01mm.
The deleting method of abnormal noises is researched. Ineorporating the3δ lawof mathematical statistics, by means of calculating mean and standard error todeterminate threshold, the adaptive counting threshold method is putforward, whichhas well adaptivity. Denoising cloudy data based wavelet technology is researched.Using wavelet technology to denoise is the process of decomposing signal withwavelet, acquiring approximate coeffieients and detail coeffieients, acting detailedcoefficients with soft threshold to suppress noises, then reconstructing signal to getthe goal of denoising. The key of simplifying data technique is retaining originalfeatures of cloudy data utmost. On the basis of reseaching current simplification technique, new method using second generation wavelet based on lifting tosimplification data is proposed.
TheG1smooth merging theory of the piecewise Bezier surface is analyzed, theBezier curve is a special NURBS curve is considered, the idea of resort to BeziersurfaceG1continuous condition for solving theG1continuous condition ofNURBS surfaces is presented.G1continuous condition of the NURBS is deducedand realized. Linear system ofG1continuous condition is solved, which providedthe basis for the following surface fitting.
The existing grid generation theory is studied, incapability of grid generation thatthese theory to the four sided region constructed by four discontinuous broken line isfound. A rapid parametrization algorithm which is used to spatial four-sided region ontriangular mesh is presented in the paper. By comparing two kinds of parametrizationprojects of the spatial four-sided region, directly grid generation algorithm is chose. Aplanar two-directional flexing grid generation algorithm is presented which canquickly parametrizing the planar four-sided region, in fact, a spatial four-sided regionis parametrizated. Comparing two NURBS surface fitting methods, in order toenhance the fitting precision and obtain more detail information of the NURBSsurface fitting, two-directional fitting method is adopted to fitting the NURBS surface.For the points on the common boundaries and corners is restricted base on theG1smooth connection condition. So by fitting every patches and a wholeG1smoothsurface is obtained.
Based on non-uniform B-splines, a new algorithm of calculating the normalvector at random position was deduced, and a formula with probe radius correctionwas proposed. An optimization algorithm named simplex method was applied in orderto eliminating the orientation error in the stage of workpiece measurement. It cansettle surfaces matching well by shifting and rotating the measuring coordinatesystem.
引文
[1] Lai J Y, Ueng W D, Yao C Y. Registration and Data Merging for MultipleSets of Scan Data[J]. The International Journal of Advanced ManufacturingTechnology,1999,15:54~63
[1] Lai J Y, Ueng W D, Yao C Y. Registration and Data Merging for MultipleSets of Scan Data[J]. The International Journal of Advanced ManufacturingTechnology,1999,15:54~63
[2]单岩，魏志刚，梁建国.反向工程中三坐标测量重定位整合[J].模具工业，2001,246(8):8~11
[3]王磊，邢渊.反向工程中数据点云的拼合[J].模具技术，2004,(1):47~49,56
[4]Ristic M, Brujic D. Efficient registration of NURBS geometry[J].InternationalJournal of Image and Vision Computing,1997,15:925~935
[5]Brujic D, Ristic M. Analysis of free form surface registration[J].ProceedingsInstitution of Mechanical Engineers, Part B,1997,211:605~617
[6]Williams J, Bennamoun M. Simultaneous Registration of MultipleCorresponding Point Sets[J]. Computer Vision and Image Understanding,2001,81:117~142
[7]Tian X D, Wang H, Zhou X H, Ruan X Y. Object Modelling of MultipleViews Using Dual Quaternion in Reverse Engineering[J].The International Journal ofAdvanced Manufacturing Technology,2002,20:495~502
[8]Fitzgibbon A W. Robust registration of2D and3D point sets[J].Image andVision Computing,2003,21:1145~1153
[9]冯雷，陈康宁.多视角下数据的配准[J].西安交通大学学报，2002,36(3):270~273
[10]Chuang C M, Chen C Y, Yau H T. A Reverse Engineering Approach toGenerating Interference-Free Tool Paths in Three-Axis Machining from Scanned Dataof Physical Models[J].The International Journal of Advanced ManufacturingTechnology,2002,19:23~31
[11]罗先波，钟约先，李仁举.三维扫描系统中的数据配准技术[J].清华大学学报（自然科学版）,2004,44(8):1104~1106
[12]金涛，童水光.逆向工程技术[M].北京：机械工业出版社，2003.8
[13]Lee K H, Woo H, Suk T. Data reduction Methods for Reverse Engineering[J].International Journal of Advanced Manufacturing Technology,2001,17(11):735~743
[14]刘军强，高建民，李言等.基于逆向工程的点云数据预处理技术研究[J].现代制造过程.2005,(7):73~75
[15]B. Hamman. A Data Reduction Scheme for Triangulation Surfaces.Computer Aided Geometric Design,1994,11:197~214
[16] A. C. LIN，C. F. CHEN. Point-data processing and error analysis in reverseengineering. International Journal of Advanced Manufacturing Technology,1998,14(11):824~834
[17] Finkelstein. A. and Salesin D. H., Multiresolution Curves, SIGGRAPH’94,261~268
[18] Lounsbery，M.，DeRose，T. and warren J.，Mulitiresolution surfaces ofArbitrary Topological Type, ACM Transactions on Graphics.1997,16(l):34~73
[19] Eek, M., DeRose, T. and Warren J., Multiresolution Analysis of ArbitraryMeshes, SIGGRAPH’95，173~182
[20]Certain. A. et al.， Interactive Multiresolution Surface Viewing，SIGGRAPH’96,91~98
[21]Gortler, S. J. and Cohen, M.F.，Hierachical and Variational GeometricModeling with Wavelets, In Proc. Symp. On Interactive3D Graphics,May1995
[22]孙延奎，朱心雄，唐泽圣.准均匀B样条曲面小波分解的快速算法[J].清华大学学报（自然科学版）,2001,41(4/5):209~213
[23]孙延奎，朱心雄.曲线分层表示的小波方法[J].工程图学学报,1999，1：40~44
[24]孙延奎，朱心雄.任意B样条曲面的多分辨率表示及光顺[J].工程图学学报,1998,3：49~54
[25]赵罡，朱心雄等.均匀B样条曲线曲面的小波表示[J].工程图学学报，2001，1：80~88
[26]吴维勇，王小椿.基于小波分析的散乱测量数据的压缩算法[J].小型微型计算机系统，2002，23(11):1384~1386
[27] C. de. Boor,“On calculation with B-Spline”, J. Approx, Theor,Vol.6(1992),P50~62
[28]陆润民，《计算机图形学教程》（清华大学计算机基础教育课程系列教材），北京：清华大学出版社，2003年，P142~154
[29] Les Piegl and Wayne Tiller，“The Nurbs Book”,1997,P117~138
[30]L. A. Piegl, W. Tiller, Fitting NURBS spherical patches to measured data,Engineering with Computers (2008)24:97~106
[31]L. A. Piegl, W. Tiller, Parametrization for surface fitting in reverseengineering, Computer-Aided Design33(2001)593~603
[32]L. A. Piegl, W. Tiller, Surface approximation to scanned data, The VisualComputer (2000)16:386–395
[33]L. A. Piegl, W. Tiller, Algorithm for approximate NURBS skinning,Computer-Aided Design, Vol.26, No.9, pp.699-706,1996
[34]L. A. Piegl, WeiyinMa, W. Tiller, An alternative method of curveinterpolation, The Visual Computer (2005)21:104–117
[35]L. A. Piegl, W. Tiller, Computing offsets of NURBS curves and surfaces,Computer-Aided Design31(1999)147–156
[36]L. A. Piegl, W. Tiller, Computing the derivative of NURBS with respect to aknot, Computer Aided Geometric Design15(1998)925~934
[37]L. A. Piegl, W. Tiller, Surface skinning revisited, The Visual Computer(2002)18:273–283
[38]L. A. Piegl, W. Tiller, Symbolic operators for NURBS, Computer-AidedDesign. Vol.29. No.5, pp.361~368,1997
[39]Weber T, Motavalli S, et al. A Unified Approach to Form ErrorEvaluation[J].Precision Engineering: Journal of the International Societies forPrecision Engineering and Nanotechnology,2002,26:269~278
[40]Chen L C, Lin G C. Reverse Engineering in the Design of Turbine Blades-ACase Study in Applying the MAMDP, Robotics and Computer IntegratedManufacturing[J],2000,16:161~167
[41]Wolfa K, Rollerb D, Schafer D. An Approach to Computer-Aided QualityControl Based on3D Coordinate Metrology[J].Journal of Materials ProcessingTechnology,2000(107):96~110
[42]Yau H T. Evaluation and Uncertainty Analysis of Vectorial Tolerances[J],Precision Engineering,1997,20:123~137
[43] Yau H T, Meng C H. A Unified Least Squares Approach to the Evaluation ofGeometric Errors Using Discrete Measurement[J].International Journal of MechanicalTools Manufacture,1996,36(11):1269~1290
[44]Corbo P, Germani M, Mandorli F. Aesthetic and Functional Analysis forProduct Model Validation in Reverse Engineering Applications[J].Computer-AidedDesign,2004,36:65~74
[45]Choi W, Kurfess T R, Cagant J. Sampling Uncertainty in CoordinateMeasurement Data Analysis[J].Precision Engineering,1998,22:153~163
[46]Choi W, Kurfess T R, Uncertainty of Extreme Fit Evaluation forThree-Dimensional Measurement Data Analysis[J]. Computer-Aided Design,1998,30(7):549~557
[47]Narayanan Namboothiri V N, Shunmugam M S. Function-oriented FormEvaluation of Engineering Surfaces[J].Precision Engineering,1998,22:98~109
[48]孙秀慧，周来水，刘胜兰.基于交互的B样条曲面重建与误差计算[J].机械制造与自动化，2005，（2）：51~54,68
[49]吴华杰，杜玉明，国永红.实物反求设计中尺寸精度的控制[J].机械设计，2001（2）：15~17
[50]刘之生，黄纯颖.反求工程技术[M].北京：机械工业出版社，1991,4
[51]乐英，基于NURBS曲面的汽轮机叶片重构及插补算法的应用研究[博士论文]，华北电力大学，2011
[52]H. Kunzmann, T. Pfeifer, R. Schmitt, H. Schwenke, A. Weckenmann.Productive Metrology: Adding Value to Manufacture [J]. Annals of the CIRP,54(4):691-704,2005
[53]M.Galetto， E.Vezzetti. Reverse engineering of free-form surfaces; Amethodology for threshold definition in selective sampling[J].International Journal ofMaehine Tools and Manufacture，2006，46(10):1079~1086
[54] V Carbone，M Carocci, E Savio et al. Combination of a vision system and acoordinate measuring maehine for the reverse engineering of freeformsurfaees.International Journal of Advanced Manufacturing Technology,2000,17(4):263~271
[55]Anon. High-speed optical measuring boosts reverse engineering.Professionalengineering，1997，10(9):32~32
[56]C.K.SONG，S.W.Kim. Reverse engineering. Autonomous digitization offree-formed surfaces on a CNC coordinate measuring maehine. International Journalof Machine Tools&Manufaeture，1997，37(7):1041~1051
[57]H.Aoyama.Y. Suzuki. Autonomous measurement of Physical model shapefor reverse engineering. Journal of Manufacturing System，2001，19(6):375~382
[58]Vincent H. Chan，Colin Bradley，Geoffrey.W Vickers. Automating laserscanning of3D surfaces for reverse engineering. SPIE，1999，32(4):156~164
[59]V.H. CHAN， C.Bradley， G.W.Vickers. A multi-sensor approach toautomating coordinate measuring maehine-based reverse engineering. Computer inIndustry，2001，44(2):105~115
[60] T Wohler.3D digitizing for engineering.Computer Graphics World，1995，18(2):1~3
[61]高晓辉，蔡鹤皋.三维数字化测量系统[J].中国机械工程,2000,1l(10):1161~1164
[62]E.X. Shi, J.J. Guo, H.J. Zhou, W. Shao. Study on On-line MeasurementTechnology for Large-scale Sheet Parts with Free-form Surface [J]. Chinese Journalof Scientific Instrument,30(9):1795-1800,2009
[63]ASME B89.4.22-2004. Methods for Performance Evaluation of ArticulatedArm Coordinate Measuring Machines [S]. New York, America Society of MechanicalEngineers,2004
[64]L. Werner. Scan Max-a Novel3D Coordinate Measuring Machine for theShop-floor Environment [J]. Measurement,18(1):17-25,1997
[65]http:www.renishaw.com, REVO five-axis measurement system
[66]R.J. Hocken, P.H. Pereira. Coordinate Measuring Machines and Systems,second edition [M]. CRC Press,2011
[67]G. Zhang, R. Veale, T. Charlton, B. Borchardt, R. Hocken. ErrorCompensation of Coordinate Measuring Machines [J]. Annals of the CIRP,34(1):445-448,1985
[68]S. Sartori, G. Zhang. Geometric Error Measurements and Compensation ofMachines [J]. Annals of the CIRP,44(2):599-609,1995
[69]G. Zhang, S.G. Liu, Z.R. Qiu, F.S. Yu, Y.L. Na, C.L. Leng. Non-contactMeasurement of Sculptured Surface of Rotation [J], Chinese Journal of MechanicalEngineering (English Edition),17(4):571-574,2004
[70]H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, F.Delbressine. Geometric Error Measurement and Compensation of Machines–anUpdate [J]. Annals of the CIRP,57(2):660-675,2008
[71]ASME B89.3.4-2010. Axes of Rotation, Methods for Specifying and Testing[S]. New York, America Society of Mechanical Engineers,2010
[72]汪平平，费业泰，林慎旺.柔性三坐标测量臂精度的优化设计[J].应用科学学报,4(23),2006
[73]王学影，刘书桂，张国雄，王斌.多关节柔性三坐标测量系统标定技术研究[J].哈尔滨工业大学学报,2008.09
[74]C. Evans, R. Hocken, W. Estler. Self-Calibration: Reversal, Redundancy,Error Separation, and ‘Absolute Testing’[J]. Annals of the CIRP,45(2):617-634,1996
[75]林述温,吴昭同,李刚.三坐标测量机非刚性效应测量误差分布特征[J].仪器仪表学报,2001.02
[76]邹璇,李德华,多关节机械臂的坐标模型和参数标定[J].光学精密工程,2001
[77]叶东,黄庆成,车仁生.多关节坐标测量机结构参数的校准[J].宇航计测技术,1999,19(6):12-16
[78]T. Ohnishi, S. Takase, K. Takamasu. Study on Improvement Methods ofCMM in Workshop Environment-Evaluation of Temperature Correction UsingLow-expansion Gauge Block [J]. Seimitsu Kogaku Kaishi/Journal of the JapanSociety for Precision Engineering,76(5):541-545,2010
[79]张国雄，三坐标测量机，天津：天津大学出版社，1999：1~17
[80]熊有伦，机器人技术基础，武汉：华中科技大学出版社，1996:121~259
[81]阎华，刘桂雄，郑时雄，机器人位姿误差建模方法综述，机床与液压，2000,(1):3~5
[82]R.R. Donaldson. A Simple Method for Separating Spindle Error from TestBall Roundness Error [J]. Annals of the CIRP,21(1):125-128,1972
[83]张国雄.三坐标测量机的发展趋势[J].中国机械工程,11(2):222-226,2000
[84]F. Franceschini, M. Galetto, L.Settineri. On-line Diagnostic Tools for CMMPerformance [J]. International Journal of Advanced Manufacturing Technology,19(2):125-130,2002
[85]H. Qiu, H. Nisitani, A. Kubo, Y. Yong. Autonomous Form Measurement onMachining Centers for Free-form Surfaces [J]. International Journal of Machine Toolsand Manufacture,44(9):961-968,2004
[86]王宵，刘会霞．逆向工程技术及其应用[M]．化学工业出版社，2004：10～55
[87]周长发译计算机图形学几何工具算法详解Geometric tools forcomputer graphics（美）Philip J. Schneider, David H. Eberly著
[88]董锦菊.逆向工程中数据测量和点云处理研究，西安理工大学硕士论文，西安，2007.3
[89]张瑞乾.逆向工程中对测量数据进行重定位的研究[J].烟台大学学报(自然科学与工程版).2004，17(l):55~58，64
[90] K S Arum，T S Huang，S D Blistein. Least square fitting of two3D pointsets[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence.1987,9(5):698~700
[91]D Faugeras, M Hebert. The representation, recognition, and Locating3Dobjects. International Journal of Robotic Research.1986,5(3):27~52
[92]韦志辉，小波分析讲义，南京理工大学，2004.
[93]Cristobal, G.， Chagoyen， M. Wavelet-based denoising methods. Acooparative study with applications in microscopy. Proc. SPIE’s1996InternationalSymposium on Optical Seience， Engineering and Instrumentation， WaveletApplications in Signal and Image Processing IV，2825.
[94]D.Donoho，De-noising by soft-thresholding. IEEE Trans. Inf. Theory，1995，41(3):613~627.
[95]Albert Boggess，Rrancis J. Narcowich.芮国胜，康健等译.小波与傅立叶分析基础〔M].北京:电子工业出版社，2004.
[96]单岩，谢斌飞. Imageware逆向造型技术基础[M].清华大学出版社，北京，2006
[97]单岩，谢龙汉. CATIA V5自由曲面造型[M].清华大学出版社，北京，2004
[98]R R Martin, I A Stroud and A D Marshall. Data reduction for reverseengineering. RECCAD, Deliverable Document ICOPERUNICUS projeet, No.1068，Computer and Automation Institute of Hungarian Academy of Science，January，1996
[99]姜寿山，杨海成，候增选.用空间形状优化标准完成散乱数据的三角剖分[J].计算机辅助设计与图形学学报，1995，7(5):241~249
[100]Chen Y H, NegC T，wang Y Z. Data reduction in integrated reverseengineering and rapid prototyping[J].International Journal of Computer IntegratedManufacturing，1999，12(2):97~103.
[101]W Sweldens. The lifting scheme: A construction of second generationwavelets[J] SIAM J.Math.，1997，29(2):511~546.
[102]W Sweldens. The lifting scheme: A philosophy of in biorthogonal waveletconstruction[A]. Pro. SPIE[C].1995，25~69.
[103]于丕强. NURBS曲面重构中的几何连续性的问题.大连理工大学博士学位论文.2002
[104]施法中.计算机辅助几何设计与非均匀有理B样条（CAGD&NURBS）.高等教育出版社.2001
[105]李建坤.多边参数域曲面设计方法的实现及应用研究.吉林大学硕士论文.2003
[106]潘日晶,潘日红,姚志强. B样条曲线同时插入多个节点的快速算法.2003,24(12):2295~2298
[107]Xiquan Shi, Tianjun Wang, Peiru Wu, Fengshan Liu. Reconstruction ofConvergentG1Smooth B-Spline Surfaces. CAGD.2004,(21):893~913
[108]彭芳瑜,严思杰,周云飞.基于能量法的大型叶片毛坯曲面重构[J].机械设计与制造工程,2002
[109]Floater M. S. Parameterization And Smooth Approximation of SurfaceTriangulations. Computer Aided Geometric Design,1997,l(14):231~250
[110]Floater M. S. Parametric Tilings and Scattered Data Approximation.International Journal of Shape Modeling,1998,l(4):165~182
[111]Floater M. S. and Hormann K. Parameterization of Triangulations andUnorganized Points. Berlin Heidelberg: Springer-Verlag Press.2002:287~316
[112]Floater M. S. and Hormann K. Surface Parameterization: a Tutorial andSurvey. Advances in Multiresolution for Geonetric Modeling. Heidelberg.2005:157~186
[113]Hormann K. and Greiner G. MIPS: An Efficient Global ParameterizationMethod. Curves and Surface Design Saint-Malo. Vancouver.2000:153~162
[114]李基拓,陆国栋.基于边折叠和质点-弹簧模型的网格简化优化算法.计算机辅助设计与图形学学报.2006,18(3):426~432
[115]甘家付,杨勋年,赵艳.一种网格参数化的优化算法.浙江大学学报（理学版）.2004,31(5):538-543
[116]Robert T.Bailey, C.K. Hsieh and H. Li: Grid Generation in Two DimensionsUsing the Complex Variable Boundary Element Method. Applied MathematicalModelling.1995,19(6):322~332
[117]Ma W., Kruth J. P. Parameterization of Randomly Measured Points forLeast Squares Fitting of B-Spline Curves and Surfaces. Computer-Aided Design.1995,(27):663~675
[118]Piegl L. A., Tiller W. Parameterization for Surface Fitting in ReverseEngineering. Computer-Aided Design.2001,(33):593~603
[119]LIN Hongwei, WANG Guojin, Liu Ligang, BAO Hujun. Parameterizationfor Fitting Triangular Mesh. Progress in Natural Science.2006,16(11):1214~1221
[120]L. Chacon, G. Lapenta. A Fully Implicit, Nonlinear Adaptive Grid Strategy.Journal of Computational Physics.2006,(212):703~717
[121]B.A. Souza, E.M. Matos, L.T. Furlan, J.R. Nunhez. A SimpleTwo-Dimensional Method for Orthogonal and Nonorthogonal Grid Generation.Computers and Chemical Engineering.2007,(31):800~807
[122]Hoschek J. Intrinsic parametrization for approximation[J]. Computer AidedGeometric Design,1988,5(1):27～31
[123]Golub G, Pereyra V. The differentiation of p seudo2inverse and nonlinearleast2squares problems whose variables separate [J]. Society for Industrial andApplied Mathematics, Journal of Numerical Analysis,1973,10(2):413～432
[124]Golub G, Van L C. Matrix computations[M]. Baltimore, Maryland, USA:The Johns Hopkins University Press,1996:38～56
[125]Rao C, Mitra S. Generalized inverse of matrices and its applications[EB/OL]. http://shum. huji. ac. il/～ritov/Lab/lib. htm,1971/2002
[126]Nielson G. Coordinate free scattered data interpolation [A]. In: SchumakerL, Chui C, Utreras F (Eds): Topics in Multivariate Approximation[C], New York:Academic Press,1987:175～184
[127]Carlos F B, Tim P. Total least squares fitting of Bézier and B2sp line curvesto ordered data [J]. Computer Aided Geometric Design,2002,19(4):275～289.
[128]Piegl L A, TillerW. Parametrization for surface fitting in reverseengineering[J]. Computer-Aided Design,2001,33(8):593～603
[129]WANG Guo-jin, WANG Guo-zhao, ZHENG Jian-min. Computer aidedgeometric design[M]. Beijing Higher Education Press,2001:124～126.[王国瑾,汪国昭,郑建民.计算机辅助几何设计[M].北京:高等教育-施普林格出版社,2001:124～126
[130]刘胜兰.逆向工程中自由曲面与规则曲面重建关键技术研究.南京航空航天大学博士学位论文.2004
[131]deBoor, C.,On Calculating with B-SPlines, J.Approx.Theory,1972,6,50~60
[132]DeBoor, C. A practical Guide of Splines. Applied Mathematical SciencesSeries.1978(7):20~40
[133]Cox, M. G., The Numerical Evaluation of B-Splines, ReportNo.NPL-DNACS-4, National Physical Laboratory,1971
[134]国家自然科学基金委员会，先进制造技术基础(优先领域战略研究告).1997.7
[135]白明光.我国机械制造业发展前景及策励.制造技术与床,1999,3:5-6,34
[136]李卫国.逆向工程中的曲面重构技术研究及基于Web的应用系统开发，南京航空航天大学博士论文，南京，2001.4
[137]周长发译计算机图形学几何工具算法详解=Geometric tools forcomputer graphics (美) Philip J. Schneider, David H. Eberly著
[138]卢红，张仲甫.测头半径补偿的方法。组合机床与自动化加工技术，2001，（010）：39~41
[139]王增强，蔺小军，任军学. CMM测量曲面测头半径补偿与路径规划研究,《机床与液压》2006.No.3
[140]严庆光，李明哲，李东成.多点成形件检测中三维数据配准方法的研究[J].中国机械工程,2003
[141]郭旗.三坐标测量机在汽轮机叶片测量中的应用[J].数字技术与应用,2010(12)
[1] Lai J Y, Ueng W D, Yao C Y. Registration and Data Merging for MultipleSets of Scan Data[J]. The International Journal of Advanced ManufacturingTechnology,1999,15:54~63
[2]单岩，魏志刚，梁建国.反向工程中三坐标测量重定位整合[J].模具工业，2001,246(8):8~11
[3]王磊，邢渊.反向工程中数据点云的拼合[J].模具技术，2004,(1):47~49,56
[4]Ristic M, Brujic D. Efficient registration of NURBS geometry[J].InternationalJournal of Image and Vision Computing,1997,15:925~935
[5]Brujic D, Ristic M. Analysis of free form surface registration[J].ProceedingsInstitution of Mechanical Engineers, Part B,1997,211:605~617
[6]Williams J, Bennamoun M. Simultaneous Registration of MultipleCorresponding Point Sets[J]. Computer Vision and Image Understanding,2001,81:117~142
[7]Tian X D, Wang H, Zhou X H, Ruan X Y. Object Modelling of MultipleViews Using Dual Quaternion in Reverse Engineering[J].The International Journal ofAdvanced Manufacturing Technology,2002,20:495~502
[8]Fitzgibbon A W. Robust registration of2D and3D point sets[J].Image andVision Computing,2003,21:1145~1153
[9]冯雷，陈康宁.多视角下数据的配准[J].西安交通大学学报，2002,36(3):270~273
[10]Chuang C M, Chen C Y, Yau H T. A Reverse Engineering Approach toGenerating Interference-Free Tool Paths in Three-Axis Machining from Scanned Dataof Physical Models[J].The International Journal of Advanced ManufacturingTechnology,2002,19:23~31
[11]罗先波，钟约先，李仁举.三维扫描系统中的数据配准技术[J].清华大学学报（自然科学版）,2004,44(8):1104~1106
[12]金涛，童水光.逆向工程技术[M].北京：机械工业出版社，2003.8
[13]Lee K H, Woo H, Suk T. Data reduction Methods for Reverse Engineering[J].International Journal of Advanced Manufacturing Technology,2001,17(11):735~743
[14]刘军强，高建民，李言等.基于逆向工程的点云数据预处理技术研究[J].现代制造过程.2005,(7):73~75
[15]B. Hamman. A Data Reduction Scheme for Triangulation Surfaces.Computer Aided Geometric Design,1994,11:197~214
[16] A. C. LIN，C. F. CHEN. Point-data processing and error analysis in reverseengineering. International Journal of Advanced Manufacturing Technology,1998,14(11):824~834
[17] Finkelstein. A. and Salesin D. H., Multiresolution Curves, SIGGRAPH’94,261~268
[18] Lounsbery，M.，DeRose，T. and warren J.，Mulitiresolution surfaces ofArbitrary Topological Type, ACM Transactions on Graphics.1997,16(l):34~73
[19] Eek, M., DeRose, T. and Warren J., Multiresolution Analysis of ArbitraryMeshes, SIGGRAPH’95，173~182
[20]Certain. A. et al.， Interactive Multiresolution Surface Viewing，SIGGRAPH’96,91~98
[21]Gortler, S. J. and Cohen, M.F.，Hierachical and Variational GeometricModeling with Wavelets, In Proc. Symp. On Interactive3D Graphics,May1995
[22]孙延奎，朱心雄，唐泽圣.准均匀B样条曲面小波分解的快速算法[J].清华大学学报（自然科学版）,2001,41(4/5):209~213
[23]孙延奎，朱心雄.曲线分层表示的小波方法[J].工程图学学报,1999，1：40~44
[24]孙延奎，朱心雄.任意B样条曲面的多分辨率表示及光顺[J].工程图学学报,1998,3：49~54
[25]赵罡，朱心雄等.均匀B样条曲线曲面的小波表示[J].工程图学学报，2001，1：80~88
[26]吴维勇，王小椿.基于小波分析的散乱测量数据的压缩算法[J].小型微型计算机系统，2002，23(11):1384~1386
[27] C. de. Boor,“On calculation with B-Spline”, J. Approx, Theor,Vol.6(1992),P50~62
[28]陆润民，《计算机图形学教程》（清华大学计算机基础教育课程系列教材），北京：清华大学出版社，2003年，P142~154
[29] Les Piegl and Wayne Tiller，“The Nurbs Book”,1997,P117~138
[30]L. A. Piegl, W. Tiller, Fitting NURBS spherical patches to measured data,Engineering with Computers (2008)24:97~106
[31]L. A. Piegl, W. Tiller, Parametrization for surface fitting in reverseengineering, Computer-Aided Design33(2001)593~603
[32]L. A. Piegl, W. Tiller, Surface approximation to scanned data, The VisualComputer (2000)16:386–395
[33]L. A. Piegl, W. Tiller, Algorithm for approximate NURBS skinning,Computer-Aided Design, Vol.26, No.9, pp.699-706,1996
[34]L. A. Piegl, WeiyinMa, W. Tiller, An alternative method of curveinterpolation, The Visual Computer (2005)21:104–117
[35]L. A. Piegl, W. Tiller, Computing offsets of NURBS curves and surfaces,Computer-Aided Design31(1999)147–156
[36]L. A. Piegl, W. Tiller, Computing the derivative of NURBS with respect to aknot, Computer Aided Geometric Design15(1998)925~934
[37]L. A. Piegl, W. Tiller, Surface skinning revisited, The Visual Computer(2002)18:273–283
[38]L. A. Piegl, W. Tiller, Symbolic operators for NURBS, Computer-AidedDesign. Vol.29. No.5, pp.361~368,1997
[39]Weber T, Motavalli S, et al. A Unified Approach to Form ErrorEvaluation[J].Precision Engineering: Journal of the International Societies forPrecision Engineering and Nanotechnology,2002,26:269~278
[40]Chen L C, Lin G C. Reverse Engineering in the Design of Turbine Blades-ACase Study in Applying the MAMDP, Robotics and Computer IntegratedManufacturing[J],2000,16:161~167
[41]Wolfa K, Rollerb D, Schafer D. An Approach to Computer-Aided QualityControl Based on3D Coordinate Metrology[J].Journal of Materials ProcessingTechnology,2000(107):96~110
[42]Yau H T. Evaluation and Uncertainty Analysis of Vectorial Tolerances[J],Precision Engineering,1997,20:123~137
[43] Yau H T, Meng C H. A Unified Least Squares Approach to the Evaluation ofGeometric Errors Using Discrete Measurement[J].International Journal of MechanicalTools Manufacture,1996,36(11):1269~1290
[44]Corbo P, Germani M, Mandorli F. Aesthetic and Functional Analysis forProduct Model Validation in Reverse Engineering Applications[J].Computer-AidedDesign,2004,36:65~74
[45]Choi W, Kurfess T R, Cagant J. Sampling Uncertainty in CoordinateMeasurement Data Analysis[J].Precision Engineering,1998,22:153~163
[46]Choi W, Kurfess T R, Uncertainty of Extreme Fit Evaluation forThree-Dimensional Measurement Data Analysis[J]. Computer-Aided Design,1998,30(7):549~557
[47]Narayanan Namboothiri V N, Shunmugam M S. Function-oriented FormEvaluation of Engineering Surfaces[J].Precision Engineering,1998,22:98~109
[48]孙秀慧，周来水，刘胜兰.基于交互的B样条曲面重建与误差计算[J].机械制造与自动化，2005，（2）：51~54,68
[49]吴华杰，杜玉明，国永红.实物反求设计中尺寸精度的控制[J].机械设计，2001（2）：15~17
[50]刘之生，黄纯颖.反求工程技术[M].北京：机械工业出版社，1991,4
[51]乐英，基于NURBS曲面的汽轮机叶片重构及插补算法的应用研究[博士论文]，华北电力大学，2011
[52]H. Kunzmann, T. Pfeifer, R. Schmitt, H. Schwenke, A. Weckenmann.Productive Metrology: Adding Value to Manufacture [J]. Annals of the CIRP,54(4):691-704,2005
[53]M.Galetto， E.Vezzetti. Reverse engineering of free-form surfaces; Amethodology for threshold definition in selective sampling[J].International Journal ofMaehine Tools and Manufacture，2006，46(10):1079~1086
[54] V Carbone，M Carocci, E Savio et al. Combination of a vision system and acoordinate measuring maehine for the reverse engineering of freeformsurfaees.International Journal of Advanced Manufacturing Technology,2000,17(4):263~271
[55]Anon. High-speed optical measuring boosts reverse engineering.Professionalengineering，1997，10(9):32~32
[56]C.K.SONG，S.W.Kim. Reverse engineering. Autonomous digitization offree-formed surfaces on a CNC coordinate measuring maehine. International Journalof Machine Tools&Manufaeture，1997，37(7):1041~1051
[57]H.Aoyama.Y. Suzuki. Autonomous measurement of Physical model shapefor reverse engineering. Journal of Manufacturing System，2001，19(6):375~382
[58]Vincent H. Chan，Colin Bradley，Geoffrey.W Vickers. Automating laserscanning of3D surfaces for reverse engineering. SPIE，1999，32(4):156~164
[59]V.H. CHAN， C.Bradley， G.W.Vickers. A multi-sensor approach toautomating coordinate measuring maehine-based reverse engineering. Computer inIndustry，2001，44(2):105~115
[60] T Wohler.3D digitizing for engineering.Computer Graphics World，1995，18(2):1~3
[61]高晓辉，蔡鹤皋.三维数字化测量系统[J].中国机械工程,2000,1l(10):1161~1164
[62]E.X. Shi, J.J. Guo, H.J. Zhou, W. Shao. Study on On-line MeasurementTechnology for Large-scale Sheet Parts with Free-form Surface [J]. Chinese Journalof Scientific Instrument,30(9):1795-1800,2009
[63]ASME B89.4.22-2004. Methods for Performance Evaluation of ArticulatedArm Coordinate Measuring Machines [S]. New York, America Society of MechanicalEngineers,2004
[64]L. Werner. Scan Max-a Novel3D Coordinate Measuring Machine for theShop-floor Environment [J]. Measurement,18(1):17-25,1997
[65]http:www.renishaw.com, REVO five-axis measurement system
[66]R.J. Hocken, P.H. Pereira. Coordinate Measuring Machines and Systems,second edition [M]. CRC Press,2011
[67]G. Zhang, R. Veale, T. Charlton, B. Borchardt, R. Hocken. ErrorCompensation of Coordinate Measuring Machines [J]. Annals of the CIRP,34(1):445-448,1985
[68]S. Sartori, G. Zhang. Geometric Error Measurements and Compensation ofMachines [J]. Annals of the CIRP,44(2):599-609,1995
[69]G. Zhang, S.G. Liu, Z.R. Qiu, F.S. Yu, Y.L. Na, C.L. Leng. Non-contactMeasurement of Sculptured Surface of Rotation [J], Chinese Journal of MechanicalEngineering (English Edition),17(4):571-574,2004
[70]H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, F.Delbressine. Geometric Error Measurement and Compensation of Machines–anUpdate [J]. Annals of the CIRP,57(2):660-675,2008
[71]ASME B89.3.4-2010. Axes of Rotation, Methods for Specifying and Testing[S]. New York, America Society of Mechanical Engineers,2010
[72]汪平平，费业泰，林慎旺.柔性三坐标测量臂精度的优化设计[J].应用科学学报,4(23),2006
[73]王学影，刘书桂，张国雄，王斌.多关节柔性三坐标测量系统标定技术研究[J].哈尔滨工业大学学报,2008.09
[74]C. Evans, R. Hocken, W. Estler. Self-Calibration: Reversal, Redundancy,Error Separation, and ‘Absolute Testing’[J]. Annals of the CIRP,45(2):617-634,1996
[75]林述温,吴昭同,李刚.三坐标测量机非刚性效应测量误差分布特征[J].仪器仪表学报,2001.02
[76]邹璇,李德华,多关节机械臂的坐标模型和参数标定[J].光学精密工程,2001
[77]叶东,黄庆成,车仁生.多关节坐标测量机结构参数的校准[J].宇航计测技术,1999,19(6):12-16
[78]T. Ohnishi, S. Takase, K. Takamasu. Study on Improvement Methods ofCMM in Workshop Environment-Evaluation of Temperature Correction UsingLow-expansion Gauge Block [J]. Seimitsu Kogaku Kaishi/Journal of the JapanSociety for Precision Engineering,76(5):541-545,2010
[79]张国雄，三坐标测量机，天津：天津大学出版社，1999：1~17
[80]熊有伦，机器人技术基础，武汉：华中科技大学出版社，1996:121~259
[81]阎华，刘桂雄，郑时雄，机器人位姿误差建模方法综述，机床与液压，2000,(1):3~5
[82]R.R. Donaldson. A Simple Method for Separating Spindle Error from TestBall Roundness Error [J]. Annals of the CIRP,21(1):125-128,1972
[83]张国雄.三坐标测量机的发展趋势[J].中国机械工程,11(2):222-226,2000
[84]F. Franceschini, M. Galetto, L.Settineri. On-line Diagnostic Tools for CMMPerformance [J]. International Journal of Advanced Manufacturing Technology,19(2):125-130,2002
[85]H. Qiu, H. Nisitani, A. Kubo, Y. Yong. Autonomous Form Measurement onMachining Centers for Free-form Surfaces [J]. International Journal of Machine Toolsand Manufacture,44(9):961-968,2004
[86]王宵，刘会霞．逆向工程技术及其应用[M]．化学工业出版社，2004：10～55
[87]周长发译计算机图形学几何工具算法详解Geometric tools forcomputer graphics（美）Philip J. Schneider, David H. Eberly著
[88]董锦菊.逆向工程中数据测量和点云处理研究，西安理工大学硕士论文，西安，2007.3
[89]张瑞乾.逆向工程中对测量数据进行重定位的研究[J].烟台大学学报(自然科学与工程版).2004，17(l):55~58，64
[90] K S Arum，T S Huang，S D Blistein. Least square fitting of two3D pointsets[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence.1987,9(5):698~700
[91]D Faugeras, M Hebert. The representation, recognition, and Locating3Dobjects. International Journal of Robotic Research.1986,5(3):27~52
[92]韦志辉，小波分析讲义，南京理工大学，2004.
[93]Cristobal, G.， Chagoyen， M. Wavelet-based denoising methods. Acooparative study with applications in microscopy. Proc. SPIE’s1996InternationalSymposium on Optical Seience， Engineering and Instrumentation， WaveletApplications in Signal and Image Processing IV，2825.
[94]D.Donoho，De-noising by soft-thresholding. IEEE Trans. Inf. Theory，1995，41(3):613~627.
[95]Albert Boggess，Rrancis J. Narcowich.芮国胜，康健等译.小波与傅立叶分析基础〔M].北京:电子工业出版社，2004.
[96]单岩，谢斌飞. Imageware逆向造型技术基础[M].清华大学出版社，北京，2006
[97]单岩，谢龙汉. CATIA V5自由曲面造型[M].清华大学出版社，北京，2004
[98]R R Martin, I A Stroud and A D Marshall. Data reduction for reverseengineering. RECCAD, Deliverable Document ICOPERUNICUS projeet, No.1068，Computer and Automation Institute of Hungarian Academy of Science，January，1996
[99]姜寿山，杨海成，候增选.用空间形状优化标准完成散乱数据的三角剖分[J].计算机辅助设计与图形学学报，1995，7(5):241~249
[100]Chen Y H, NegC T，wang Y Z. Data reduction in integrated reverseengineering and rapid prototyping[J].International Journal of Computer IntegratedManufacturing，1999，12(2):97~103.
[101]W Sweldens. The lifting scheme: A construction of second generationwavelets[J] SIAM J.Math.，1997，29(2):511~546.
[102]W Sweldens. The lifting scheme: A philosophy of in biorthogonal waveletconstruction[A]. Pro. SPIE[C].1995，25~69.
[103]于丕强. NURBS曲面重构中的几何连续性的问题.大连理工大学博士学位论文.2002
[104]施法中.计算机辅助几何设计与非均匀有理B样条（CAGD&NURBS）.高等教育出版社.2001
[105]李建坤.多边参数域曲面设计方法的实现及应用研究.吉林大学硕士论文.2003
[106]潘日晶,潘日红,姚志强. B样条曲线同时插入多个节点的快速算法.2003,24(12):2295~2298
[107]Xiquan Shi, Tianjun Wang, Peiru Wu, Fengshan Liu. Reconstruction ofConvergentG1Smooth B-Spline Surfaces. CAGD.2004,(21):893~913
[108]彭芳瑜,严思杰,周云飞.基于能量法的大型叶片毛坯曲面重构[J].机械设计与制造工程,2002
[109]Floater M. S. Parameterization And Smooth Approximation of SurfaceTriangulations. Computer Aided Geometric Design,1997,l(14):231~250
[110]Floater M. S. Parametric Tilings and Scattered Data Approximation.International Journal of Shape Modeling,1998,l(4):165~182
[111]Floater M. S. and Hormann K. Parameterization of Triangulations andUnorganized Points. Berlin Heidelberg: Springer-Verlag Press.2002:287~316
[112]Floater M. S. and Hormann K. Surface Parameterization: a Tutorial andSurvey. Advances in Multiresolution for Geonetric Modeling. Heidelberg.2005:157~186
[113]Hormann K. and Greiner G. MIPS: An Efficient Global ParameterizationMethod. Curves and Surface Design Saint-Malo. Vancouver.2000:153~162
[114]李基拓,陆国栋.基于边折叠和质点-弹簧模型的网格简化优化算法.计算机辅助设计与图形学学报.2006,18(3):426~432
[115]甘家付,杨勋年,赵艳.一种网格参数化的优化算法.浙江大学学报（理学版）.2004,31(5):538-543
[116]Robert T.Bailey, C.K. Hsieh and H. Li: Grid Generation in Two DimensionsUsing the Complex Variable Boundary Element Method. Applied MathematicalModelling.1995,19(6):322~332
[117]Ma W., Kruth J. P. Parameterization of Randomly Measured Points forLeast Squares Fitting of B-Spline Curves and Surfaces. Computer-Aided Design.1995,(27):663~675
[118]Piegl L. A., Tiller W. Parameterization for Surface Fitting in ReverseEngineering. Computer-Aided Design.2001,(33):593~603
[119]LIN Hongwei, WANG Guojin, Liu Ligang, BAO Hujun. Parameterizationfor Fitting Triangular Mesh. Progress in Natural Science.2006,16(11):1214~1221
[120]L. Chacon, G. Lapenta. A Fully Implicit, Nonlinear Adaptive Grid Strategy.Journal of Computational Physics.2006,(212):703~717
[121]B.A. Souza, E.M. Matos, L.T. Furlan, J.R. Nunhez. A SimpleTwo-Dimensional Method for Orthogonal and Nonorthogonal Grid Generation.Computers and Chemical Engineering.2007,(31):800~807
[122]Hoschek J. Intrinsic parametrization for approximation[J]. Computer AidedGeometric Design,1988,5(1):27～31
[123]Golub G, Pereyra V. The differentiation of p seudo2inverse and nonlinearleast2squares problems whose variables separate [J]. Society for Industrial andApplied Mathematics, Journal of Numerical Analysis,1973,10(2):413～432
[124]Golub G, Van L C. Matrix computations[M]. Baltimore, Maryland, USA:The Johns Hopkins University Press,1996:38～56
[125]Rao C, Mitra S. Generalized inverse of matrices and its applications[EB/OL]. http://shum. huji. ac. il/～ritov/Lab/lib. htm,1971/2002
[126]Nielson G. Coordinate free scattered data interpolation [A]. In: SchumakerL, Chui C, Utreras F (Eds): Topics in Multivariate Approximation[C], New York:Academic Press,1987:175～184
[127]Carlos F B, Tim P. Total least squares fitting of Bézier and B2sp line curvesto ordered data [J]. Computer Aided Geometric Design,2002,19(4):275～289.
[128]Piegl L A, TillerW. Parametrization for surface fitting in reverseengineering[J]. Computer-Aided Design,2001,33(8):593～603
[129]WANG Guo-jin, WANG Guo-zhao, ZHENG Jian-min. Computer aidedgeometric design[M]. Beijing Higher Education Press,2001:124～126.[王国瑾,汪国昭,郑建民.计算机辅助几何设计[M].北京:高等教育-施普林格出版社,2001:124～126
[130]刘胜兰.逆向工程中自由曲面与规则曲面重建关键技术研究.南京航空航天大学博士学位论文.2004
[131]deBoor, C.,On Calculating with B-SPlines, J.Approx.Theory,1972,6,50~60
[132]DeBoor, C. A practical Guide of Splines. Applied Mathematical SciencesSeries.1978(7):20~40
[133]Cox, M. G., The Numerical Evaluation of B-Splines, ReportNo.NPL-DNACS-4, National Physical Laboratory,1971
[134]国家自然科学基金委员会，先进制造技术基础(优先领域战略研究告).1997.7
[135]白明光.我国机械制造业发展前景及策励.制造技术与床,1999,3:5-6,34
[136]李卫国.逆向工程中的曲面重构技术研究及基于Web的应用系统开发，南京航空航天大学博士论文，南京，2001.4
[137]周长发译计算机图形学几何工具算法详解=Geometric tools forcomputer graphics (美) Philip J. Schneider, David H. Eberly著
[138]卢红，张仲甫.测头半径补偿的方法。组合机床与自动化加工技术，2001，（010）：39~41
[139]王增强，蔺小军，任军学. CMM测量曲面测头半径补偿与路径规划研究,《机床与液压》2006.No.3
[140]严庆光，李明哲，李东成.多点成形件检测中三维数据配准方法的研究[J].中国机械工程,2003
[141]郭旗.三坐标测量机在汽轮机叶片测量中的应用[J].数字技术与应用,2010(12)