基于工业CT的管道圆柱度测量与内表面显示算法研究
|
摘要
工业CT,即工业计算机层析成像(Industrial Computerized Tomography),是指在工业中用射线扫描待测物体,获得物体断层的投影数据,经重建后得到一系列能够反映待测物体内部结构的切片图像。基于工业CT的检测技术有着无破坏性的显著优点,能够解决许多传统方法无法解决的检测难题。工件中常包含各种各样的管道,对管道的圆柱度误差测量和内表面显示对评定工件的整体性能和使用寿命有着重要意义。本文以工业CT为基础,研究了工件内部管道的圆柱度测量与内表面显示算法。
圆柱度误差是指实际被测圆柱面对其理想圆柱的变动量。由于测量工具的限制,传统方法对无法充分接触的内部管道的圆柱度难以测量。本文研究了一种借助工业CT进行圆柱度测量的方法,首先对待检测工件进行CT扫描,得到工件的CT图像,然后提取待测目标的边缘点,采用最小二乘圆柱法建立数学模型,通过优化求解得到圆柱度误差。实验结果表明,该方法可以在无接触无破坏的前提下得到较为准确的测量结果。
对工业CT图像进行分析和处理的过程中,经常需要对工件管道的内表面进行显示,为后续的分析与检测做准备。研究了内表面显示的两种方法:剖开显示与虚拟内窥显示。针对剖开显示,首先采用区域生长法分割目标,经轮廓提取和轮廓跟踪定位目标边缘点,然后计算各边缘点到形心的距离。将此距离经过线性变换转换为灰度信息,写入图像。实验结果表明,剖开显示简单有效,可以观测到内表面的缺陷。
虚拟内窥显示能够模拟传统光学内窥镜更加立体地显示目标内表面。首先对目标对象进行分割,同时定位目标边缘点坐标。然后提取目标的中心轴径,为了使虚拟镜头能够沿着中心轴平稳前进,用B样条插值对中心轴径作平滑处理。在已知目标边缘点坐标的基础上,采用最短对角线法完成轮廓拼接并最终实现绘制。针对管道的柱状结构,研究了一种特定的轮廓线拼接方法。虚拟内窥显示可动态地观测到管道内表面,更为形象直接。
Industrial computerized tomography (Industrial CT) acquires projection data, which the x-ray transmits the cross-section of the tested object without damage to the object. The cross-section images can be obtained with special image reconstruction algorithm. Industrial CT, which has the obvious advantage of non-destructivity, can solve many problems that the traditional methods can not solve. There are many kinds of pipelines in real workpiece. The cylindricity error evaluation and inner surface display of pipe are significant for evaluating performance and service life. Based on industrial CT, this dissertation studied the cylindricity error evaluation and inner surface display of pipeline.
Cylindricity error is defined as the changes between measured column and ideal column. Because of the limit of measuring tools, the traditional methods can not evaluate the cylindricity error of internal pipe. This dissertation presents cylindricity error measure according to industrial CT. The inspected area is scanned by industrial CT, which obtains the CT images of the inspected area. The edge is detected with Facet algorithm. Then the mathematical model is established by the least-square method. The cylindricity error can be got by optimization. The results show that this method can get cylindricity error without damage and touch.
In industrial CT image analysis and process, it is necessary to display the inner surface of the pipeline for subsequent analysis and defect detection. There are usually section display and virtual endoscope for inner surface display. At first, we get the target with region growing for section display, then use contour extracting and contour tracking to get the target’s contour points. Finally, we calculate the distances between the centroid and the contour points and transform the distances to gray for writing to an image. The experimental results show that section display is simple and effective. It can observe the defects on the inner surface.
Virtual endoscope display is a kind technology of simulating traditional optical endoscope. It can display inner face of target with three-dimensional quality. At first we use image segmentation to get the target and locate the edge points, then extract the central path of the target. In order to move the virtual camera lens slowly we smooth the path with B-spline. After that we use the shortest diagonal method to complete the contour tiling and render the surface at last. Also we study a special method for the contour tiling of the pipeline. Virtual endoscope display can observe the inner surface of the pipe dynamically and directly.
圆柱度误差是指实际被测圆柱面对其理想圆柱的变动量。由于测量工具的限制,传统方法对无法充分接触的内部管道的圆柱度难以测量。本文研究了一种借助工业CT进行圆柱度测量的方法,首先对待检测工件进行CT扫描,得到工件的CT图像,然后提取待测目标的边缘点,采用最小二乘圆柱法建立数学模型,通过优化求解得到圆柱度误差。实验结果表明,该方法可以在无接触无破坏的前提下得到较为准确的测量结果。
对工业CT图像进行分析和处理的过程中,经常需要对工件管道的内表面进行显示,为后续的分析与检测做准备。研究了内表面显示的两种方法:剖开显示与虚拟内窥显示。针对剖开显示,首先采用区域生长法分割目标,经轮廓提取和轮廓跟踪定位目标边缘点,然后计算各边缘点到形心的距离。将此距离经过线性变换转换为灰度信息,写入图像。实验结果表明,剖开显示简单有效,可以观测到内表面的缺陷。
虚拟内窥显示能够模拟传统光学内窥镜更加立体地显示目标内表面。首先对目标对象进行分割,同时定位目标边缘点坐标。然后提取目标的中心轴径,为了使虚拟镜头能够沿着中心轴平稳前进,用B样条插值对中心轴径作平滑处理。在已知目标边缘点坐标的基础上,采用最短对角线法完成轮廓拼接并最终实现绘制。针对管道的柱状结构,研究了一种特定的轮廓线拼接方法。虚拟内窥显示可动态地观测到管道内表面,更为形象直接。
Industrial computerized tomography (Industrial CT) acquires projection data, which the x-ray transmits the cross-section of the tested object without damage to the object. The cross-section images can be obtained with special image reconstruction algorithm. Industrial CT, which has the obvious advantage of non-destructivity, can solve many problems that the traditional methods can not solve. There are many kinds of pipelines in real workpiece. The cylindricity error evaluation and inner surface display of pipe are significant for evaluating performance and service life. Based on industrial CT, this dissertation studied the cylindricity error evaluation and inner surface display of pipeline.
Cylindricity error is defined as the changes between measured column and ideal column. Because of the limit of measuring tools, the traditional methods can not evaluate the cylindricity error of internal pipe. This dissertation presents cylindricity error measure according to industrial CT. The inspected area is scanned by industrial CT, which obtains the CT images of the inspected area. The edge is detected with Facet algorithm. Then the mathematical model is established by the least-square method. The cylindricity error can be got by optimization. The results show that this method can get cylindricity error without damage and touch.
In industrial CT image analysis and process, it is necessary to display the inner surface of the pipeline for subsequent analysis and defect detection. There are usually section display and virtual endoscope for inner surface display. At first, we get the target with region growing for section display, then use contour extracting and contour tracking to get the target’s contour points. Finally, we calculate the distances between the centroid and the contour points and transform the distances to gray for writing to an image. The experimental results show that section display is simple and effective. It can observe the defects on the inner surface.
Virtual endoscope display is a kind technology of simulating traditional optical endoscope. It can display inner face of target with three-dimensional quality. At first we use image segmentation to get the target and locate the edge points, then extract the central path of the target. In order to move the virtual camera lens slowly we smooth the path with B-spline. After that we use the shortest diagonal method to complete the contour tiling and render the surface at last. Also we study a special method for the contour tiling of the pipeline. Virtual endoscope display can observe the inner surface of the pipe dynamically and directly.
引文
[1] Jiang Hsieh. Computed tomography:principle, design, artifacts and recent advances [M].北京:科学出版社, 2006.
[2]叶云长.计算机层析成像检测[M].北京:机械工业出版社, 2006.
[3] Chye Hwang Yan, T. Robert Whalen, S. Gary Beaupré, Y. Shin Yen, and Sandy Napel. Reconstruction Algorithm for Polychromatic CT Imaging: Application to Beam Hardening Correction [J]. IEEE Transactions on Medical Imaging, 2000, 19(1): 1-11.
[4]庄天戈. CT原理与算法[M].上海:上海交通大学出版社, 1992.
[5]曹玉玲,陈存柱.铸件的工业CT三维检测技术[J]. CT理论与应用研究. 2003, 8(12):36-39.
[6]李俊杰,韩焱,王黎明.基于ICT的石墨密度均匀性检测方法研究[J].无损检测. 2008, 30(3):163-165.
[7]郭履灿,曲韵笙.论CT科技在某些工程项目中的应用[J]. CT理论与应用研究. 1999, 2(8):44-46.
[8]刘丰林,程森林,王珏.工业CT技术[J].现代制造工程. 2003, 3(5): 38-40.
[9]叶海霞.工业CT窄角扇束卷积反投影并行图像重建研究[D].重庆:重庆大学, 2003.
[10]李学军,常智勇等.基于遗传算法的圆柱几何特征信息的测量[J].计算机工程与应用,2006, 22.
[11]邹斌.基于工业CT的几何特征反求与测量算法研究[D].重庆:重庆大学, 2009.
[12]王凯.工业CT切片图像高精度轮廓提取技术研究[D].西安:西北工业大学. 2005.
[13]马睿.工业CT高精度图像测量算法研究[D].重庆:重庆大学, 2008.
[14]王凯,张定华.基于3-D Facet模型的亚体素边缘检测算法研究[J].机械科学与技术, 2005, 24(7).
[15]何斌. Visual C++数字图像处理[M].北京:人民邮电出版社, 2001
[16]张娇娜,郭伟伟,曹衍龙等.圆柱度误差评价方法研究[J].机床与液压, 2008, 36(2).
[17]刘国光.基于Matlab评定圆柱度误差[J].工程设计学报, 2005, 12(4).
[18]倪骁骅.形状误差评定和测量不确定度估计[M].北京:化学工业出版社: 2008.
[19] GB/T 11336-2004,国家标准[S].
[20]陈永鹏.基于MATLAB优化工具箱的机械产品形状误差评定系统研究[D].四川:四川大学, 2003.
[21]张晓光,高顶.射线检测焊接缺陷的提取和自动识别[M].北京:国防工业出版社, 2004.
[22]李弼程,彭天强,彭波.智能图像处理技术[M].北京:电子工业出版社, 2004.
[23]李俊山,李旭辉.数字图像处理[M].北京:清华大学出版社, 2007.
[24]求是科技.数字图像处理典型算法及实现[M].北京:人民邮电出版社, 2006.
[25]朱文忠.一种基于VC实现数字图像的轮廓提取法[J].微计算机信息, 2007, 23(1-3).
[26]杨大地,谈骏渝.实用数值分析[M].重庆:重庆大学出版社, 2000.
[27] A. Marsh, F. Simistria, R. Robb: VR in medicine: Virtual endoscopy, Future Generation Computer Systems.1998, 14(5):253-264.
[28] S. Kourelea, T. Vontetsianos, V. Maniatis, etc: The Application of Virtual Bronchoscopy in the Evaluation of Hemoptysis: Comparative Evaluation with Real Fiberoptic Bronchoscopy. Information Technology Applications in Biomedicine, 2000:246-249.
[29] A. Radetzky, A. Nu. Visualization and simulation techniques for surgical simulators using actualpatient’s data, Artificial Intelligence in Medicine 26(4), 2002:255–279.
[30] Virtual Endoscopy Medical Application. http://www.crd.ge.com/esl/cgsp/fact sheet/virtendo/index.html
[31] S. F. Sheikh, D. S. Paik, C. F. Beaulieu, G. D. Rubin, R. B. Jeffrey, S. Napel. "Wide-Angle Virtual Endoscopyusing Multiple-View Rendering: The Virtual Cockpit." RSNA EJ 1998; 2.
[32] The Visualization Lab of the Computer Science Department at Stony Brook University. http://www.cs.sunvsb.edu/wislab/nroiects/colonosconv/.
[33] T. Nakasatoa, M. Sasakia, S. Ehara, etal. Virtual CT endoscopy of ossicles in the middle ear, Journal of Clinical Imaging, 2001, 25(3):171-177.
[34] Freeffight software. http://www.vec.wfubmc.edu/software/.
[35] The Multidimensional Image Processing Lab at Penn State University. http://cobb.ee.nsu.edu/nroiects/auisk}e/auirksee.htm.
[36] G. C. Kagadis, V. Patrinou, C. P. Kaloger, etal. Virtual endoscopy in the diagnosis of an adult doubletracheal bronchi case, European Journal of Radiology, 2001, 40 (9): 50-53.
[37]刘保权,周明全,赵宏安.一种基于体绘制的交互式实时虚拟内窥镜算法[J].西北大学学报, 2003, 33(2): 155-159.
[38]袁非牛,周荷琴,冯焕清等.虚拟内窥镜系统中的自动导航[J].航天医学与医学工程, 2003, 16(3): 201-205.
[39]彭延军,王元红,石教英等.虚拟内窥镜关键技术研究[J].计算机应用研究, 2005, 5: 37-39.
[40]卢艳平,喻洪麟,王珏.工业虚拟内窥关键技术研究[J].计算机应用研究, 2008, 44(33).
[41] H. BLUM. Biological shapes and visual science[J]. Jour2 nal of TheoreticalBiology, 1973, 38: 205-287.
[42]王怡,周明全,耿国华.虚拟内窥镜中心路径抽取技术[J].西北大学学报(自然科学版) ,2005, 35(6).
[43] M. Wan, F. Dachille, A. Kaufman. Distance-Field Based Skeletons for Virtual Navigation, Visualization 2001, San Diego, 2001: 239 -245.
[44] Son Tran, Liwen Shih. Efficient 3D Binary Image Skeletonization. Proceedings of the 2005 IEEE Computational Systems Bioinformatics Conference Workshops, 2005: 364-372.
[45]袁非牛,周荷琴,冯焕清等.虚拟内窥镜系统中的自动导航[J].航天医学与医学工程,2003, 16(3).
[46] M. Unser, Ten Good Reasons for Using Spline Wavelets[J], Wavelet Applications in Signal and Image Processing , 1997: 422-431.
[47] M. Unser, A. Aldroubi, M. Eden. B-Spline Signal Processing:Part II—Efficient Design and Applications, IEEE Trans. Signal Processing, 1993, 41(2): 834-847.
[48] M. Unser, A. Aldroubi, and M. Eden, The Polynomial Spline Pyramid, IEEE Trans. Pattern Analysis and Machine Intelligence,vol. 15, no. 4: 364-378, Apr. 1993.
[49] M. Unser,“Ten Good Reasons for Using Spline Wavelets,”Proc.SPIE, vol. 3,169, Wavelet Applications in Signal and Image Processing,vol. 5:422-431, 1997.
[50] M. Unser, A. Aldroubi, and M. Eden,“A Family of Polynomial Spline Wavelet Transforms,”Signal Processing, vol. 30, no. 2:141-162, 1993.
[51] Yu-Ping Wang, S. L. Lee. Scale-Space Derived From B-Splines. IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 10: 1040-1055, 1998.
[52] Herman G. T, Liu H. K. Three-dimensional display of human organs from computed tomograms. Computer Graphics Image Processing, 1979,9.
[53] W. E. Lorensen, Cline construction algorithm. H. E. Marching cubes: a high resolution 3D surface Computer Graphics, 1987,V21(4):163-169.
[54]王敏科.基于面绘制的虚拟内窥镜系统[D].浙江:浙江大学, 2006.
[55] G. Nielson, B. Hamann. The asymptotic decider: resolving the ambiguitu in marching cube. Visualization'91, 1991:83-91.
[56] J. Wilhelms, Von fielder A. Topologicalon Volume considerations in isosurface generation. San Diego Workshop Visualization, 1990.
[57]纪凤欣.基于断层图像的几何重建理论与技术研究[D].大连:大连理工大学, 2001.
[58]王世文.基于CT的虚拟内窥系统[D].山东:山东大学, 2005.
[59] H. Fuchs, Z. M. Kedem, S.P. Uselton. 1977. Optimal Surface Reconstruction from Planar Contours. Communication of the ACM, 20(10),693-702.
[60]唐泽圣.三维数据场可视化[M].北京:清华大学出版社, 1990.
[61]吕林根,许子道.解析几何[M].北京:高等教育出版社, 1997.
[62]江早,王洪成. OpenGL VC/VB图像编程[M].北京:高等教育出版社.
[63]彭群生,鲍虎军,金小刚.计算机真实感图形的算法基础.北京:科学出版社, 1999.
[64]蔡士杰,吴春铭,孙正兴等.计算机图形学.北京:电子工业出版社, 1998.
[65] Shreiner D. OpenGL编程指南.北京:机械工业出版社, 2006.
[66]段瑞玲,李庆祥,李玉和.图像边缘检测方法研究综述[J].光学技术, 2005, 31(3).
[67]车佳斯.零件圆度与圆柱度的图像测量研究[D].吉林:吉林大学, 2007.
[68]陈向伟,王龙山,刘庆民等,基于CCD图像的圆度误差测量的研究[J].半导体光电, 2004, 25(4).
[69]安志勇,李丽娟,石利霞等,非接触圆度误差激光检测及误差分离技术[J].电子测量与仪器学报, 2002年增刊.
[70]于忠党,王龙山.基于图像处理的零件参数检测研究[J].渤海大学学报(自然科学版), 2006, 27(1): 61-65.
[71]朱芳利,葛动元,岳卫宏.轴类零件形位误差的数据处理[J].机械传动, 2003, 31(4): 28~31.
[2]叶云长.计算机层析成像检测[M].北京:机械工业出版社, 2006.
[3] Chye Hwang Yan, T. Robert Whalen, S. Gary Beaupré, Y. Shin Yen, and Sandy Napel. Reconstruction Algorithm for Polychromatic CT Imaging: Application to Beam Hardening Correction [J]. IEEE Transactions on Medical Imaging, 2000, 19(1): 1-11.
[4]庄天戈. CT原理与算法[M].上海:上海交通大学出版社, 1992.
[5]曹玉玲,陈存柱.铸件的工业CT三维检测技术[J]. CT理论与应用研究. 2003, 8(12):36-39.
[6]李俊杰,韩焱,王黎明.基于ICT的石墨密度均匀性检测方法研究[J].无损检测. 2008, 30(3):163-165.
[7]郭履灿,曲韵笙.论CT科技在某些工程项目中的应用[J]. CT理论与应用研究. 1999, 2(8):44-46.
[8]刘丰林,程森林,王珏.工业CT技术[J].现代制造工程. 2003, 3(5): 38-40.
[9]叶海霞.工业CT窄角扇束卷积反投影并行图像重建研究[D].重庆:重庆大学, 2003.
[10]李学军,常智勇等.基于遗传算法的圆柱几何特征信息的测量[J].计算机工程与应用,2006, 22.
[11]邹斌.基于工业CT的几何特征反求与测量算法研究[D].重庆:重庆大学, 2009.
[12]王凯.工业CT切片图像高精度轮廓提取技术研究[D].西安:西北工业大学. 2005.
[13]马睿.工业CT高精度图像测量算法研究[D].重庆:重庆大学, 2008.
[14]王凯,张定华.基于3-D Facet模型的亚体素边缘检测算法研究[J].机械科学与技术, 2005, 24(7).
[15]何斌. Visual C++数字图像处理[M].北京:人民邮电出版社, 2001
[16]张娇娜,郭伟伟,曹衍龙等.圆柱度误差评价方法研究[J].机床与液压, 2008, 36(2).
[17]刘国光.基于Matlab评定圆柱度误差[J].工程设计学报, 2005, 12(4).
[18]倪骁骅.形状误差评定和测量不确定度估计[M].北京:化学工业出版社: 2008.
[19] GB/T 11336-2004,国家标准[S].
[20]陈永鹏.基于MATLAB优化工具箱的机械产品形状误差评定系统研究[D].四川:四川大学, 2003.
[21]张晓光,高顶.射线检测焊接缺陷的提取和自动识别[M].北京:国防工业出版社, 2004.
[22]李弼程,彭天强,彭波.智能图像处理技术[M].北京:电子工业出版社, 2004.
[23]李俊山,李旭辉.数字图像处理[M].北京:清华大学出版社, 2007.
[24]求是科技.数字图像处理典型算法及实现[M].北京:人民邮电出版社, 2006.
[25]朱文忠.一种基于VC实现数字图像的轮廓提取法[J].微计算机信息, 2007, 23(1-3).
[26]杨大地,谈骏渝.实用数值分析[M].重庆:重庆大学出版社, 2000.
[27] A. Marsh, F. Simistria, R. Robb: VR in medicine: Virtual endoscopy, Future Generation Computer Systems.1998, 14(5):253-264.
[28] S. Kourelea, T. Vontetsianos, V. Maniatis, etc: The Application of Virtual Bronchoscopy in the Evaluation of Hemoptysis: Comparative Evaluation with Real Fiberoptic Bronchoscopy. Information Technology Applications in Biomedicine, 2000:246-249.
[29] A. Radetzky, A. Nu. Visualization and simulation techniques for surgical simulators using actualpatient’s data, Artificial Intelligence in Medicine 26(4), 2002:255–279.
[30] Virtual Endoscopy Medical Application. http://www.crd.ge.com/esl/cgsp/fact sheet/virtendo/index.html
[31] S. F. Sheikh, D. S. Paik, C. F. Beaulieu, G. D. Rubin, R. B. Jeffrey, S. Napel. "Wide-Angle Virtual Endoscopyusing Multiple-View Rendering: The Virtual Cockpit." RSNA EJ 1998; 2.
[32] The Visualization Lab of the Computer Science Department at Stony Brook University. http://www.cs.sunvsb.edu/wislab/nroiects/colonosconv/.
[33] T. Nakasatoa, M. Sasakia, S. Ehara, etal. Virtual CT endoscopy of ossicles in the middle ear, Journal of Clinical Imaging, 2001, 25(3):171-177.
[34] Freeffight software. http://www.vec.wfubmc.edu/software/.
[35] The Multidimensional Image Processing Lab at Penn State University. http://cobb.ee.nsu.edu/nroiects/auisk}e/auirksee.htm.
[36] G. C. Kagadis, V. Patrinou, C. P. Kaloger, etal. Virtual endoscopy in the diagnosis of an adult doubletracheal bronchi case, European Journal of Radiology, 2001, 40 (9): 50-53.
[37]刘保权,周明全,赵宏安.一种基于体绘制的交互式实时虚拟内窥镜算法[J].西北大学学报, 2003, 33(2): 155-159.
[38]袁非牛,周荷琴,冯焕清等.虚拟内窥镜系统中的自动导航[J].航天医学与医学工程, 2003, 16(3): 201-205.
[39]彭延军,王元红,石教英等.虚拟内窥镜关键技术研究[J].计算机应用研究, 2005, 5: 37-39.
[40]卢艳平,喻洪麟,王珏.工业虚拟内窥关键技术研究[J].计算机应用研究, 2008, 44(33).
[41] H. BLUM. Biological shapes and visual science[J]. Jour2 nal of TheoreticalBiology, 1973, 38: 205-287.
[42]王怡,周明全,耿国华.虚拟内窥镜中心路径抽取技术[J].西北大学学报(自然科学版) ,2005, 35(6).
[43] M. Wan, F. Dachille, A. Kaufman. Distance-Field Based Skeletons for Virtual Navigation, Visualization 2001, San Diego, 2001: 239 -245.
[44] Son Tran, Liwen Shih. Efficient 3D Binary Image Skeletonization. Proceedings of the 2005 IEEE Computational Systems Bioinformatics Conference Workshops, 2005: 364-372.
[45]袁非牛,周荷琴,冯焕清等.虚拟内窥镜系统中的自动导航[J].航天医学与医学工程,2003, 16(3).
[46] M. Unser, Ten Good Reasons for Using Spline Wavelets[J], Wavelet Applications in Signal and Image Processing , 1997: 422-431.
[47] M. Unser, A. Aldroubi, M. Eden. B-Spline Signal Processing:Part II—Efficient Design and Applications, IEEE Trans. Signal Processing, 1993, 41(2): 834-847.
[48] M. Unser, A. Aldroubi, and M. Eden, The Polynomial Spline Pyramid, IEEE Trans. Pattern Analysis and Machine Intelligence,vol. 15, no. 4: 364-378, Apr. 1993.
[49] M. Unser,“Ten Good Reasons for Using Spline Wavelets,”Proc.SPIE, vol. 3,169, Wavelet Applications in Signal and Image Processing,vol. 5:422-431, 1997.
[50] M. Unser, A. Aldroubi, and M. Eden,“A Family of Polynomial Spline Wavelet Transforms,”Signal Processing, vol. 30, no. 2:141-162, 1993.
[51] Yu-Ping Wang, S. L. Lee. Scale-Space Derived From B-Splines. IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 10: 1040-1055, 1998.
[52] Herman G. T, Liu H. K. Three-dimensional display of human organs from computed tomograms. Computer Graphics Image Processing, 1979,9.
[53] W. E. Lorensen, Cline construction algorithm. H. E. Marching cubes: a high resolution 3D surface Computer Graphics, 1987,V21(4):163-169.
[54]王敏科.基于面绘制的虚拟内窥镜系统[D].浙江:浙江大学, 2006.
[55] G. Nielson, B. Hamann. The asymptotic decider: resolving the ambiguitu in marching cube. Visualization'91, 1991:83-91.
[56] J. Wilhelms, Von fielder A. Topologicalon Volume considerations in isosurface generation. San Diego Workshop Visualization, 1990.
[57]纪凤欣.基于断层图像的几何重建理论与技术研究[D].大连:大连理工大学, 2001.
[58]王世文.基于CT的虚拟内窥系统[D].山东:山东大学, 2005.
[59] H. Fuchs, Z. M. Kedem, S.P. Uselton. 1977. Optimal Surface Reconstruction from Planar Contours. Communication of the ACM, 20(10),693-702.
[60]唐泽圣.三维数据场可视化[M].北京:清华大学出版社, 1990.
[61]吕林根,许子道.解析几何[M].北京:高等教育出版社, 1997.
[62]江早,王洪成. OpenGL VC/VB图像编程[M].北京:高等教育出版社.
[63]彭群生,鲍虎军,金小刚.计算机真实感图形的算法基础.北京:科学出版社, 1999.
[64]蔡士杰,吴春铭,孙正兴等.计算机图形学.北京:电子工业出版社, 1998.
[65] Shreiner D. OpenGL编程指南.北京:机械工业出版社, 2006.
[66]段瑞玲,李庆祥,李玉和.图像边缘检测方法研究综述[J].光学技术, 2005, 31(3).
[67]车佳斯.零件圆度与圆柱度的图像测量研究[D].吉林:吉林大学, 2007.
[68]陈向伟,王龙山,刘庆民等,基于CCD图像的圆度误差测量的研究[J].半导体光电, 2004, 25(4).
[69]安志勇,李丽娟,石利霞等,非接触圆度误差激光检测及误差分离技术[J].电子测量与仪器学报, 2002年增刊.
[70]于忠党,王龙山.基于图像处理的零件参数检测研究[J].渤海大学学报(自然科学版), 2006, 27(1): 61-65.
[71]朱芳利,葛动元,岳卫宏.轴类零件形位误差的数据处理[J].机械传动, 2003, 31(4): 28~31.