基于小波变换的三维面形测量技术研究
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摘要
三维轮廓术在高速在线检测、质量监控、机器视觉以及医疗诊断等领域的应用日益广泛。具有非接触性的光学投影式测量方法由于其高分辨率、无破损、数据获取速度快等优点而被公认为最有前途的三维轮廓测量方法。本文以傅里叶变换轮廓术所存在的缺陷为出发点，提出了一种新的测量方法——小波变换轮廓术，并从理论，以及计算机模拟和实验各方面进行了深入的研究。
首先在傅里叶变换的基础上引入高斯窗函数提出了窗口傅里叶变换条纹分析法，进而在高斯窗函数中引入伸缩变换提出伸缩窗口傅里叶变换分析。并从理论和实验中分析了这两种方法的局限性。
再而提出了小波变换轮廓术，并通过理论分析和实际测量证明了该技术的有效性。另外，还对小波变换的关键技术问题，如母小波函数的选择、小波尺度函数的构造以及小波变换脊的确定进行了详细的研究与分析。
最后，为了解决非连续物体的包裹相位解调的问题，本论文发展了钟金钢老师提出的双频条纹相位查表法，结合小波变换轮廓术，提出了双频光栅+小波变换+查表相位解包技术，实现在一幅变形双频光栅图像中进行小波分析得到两个频率所对应的确定的相位分布。该方法可以用于由具有台阶、陡峭表面和噪声等复杂表面形成的非平稳光栅信号的分析。
The application of 3-D shape measurement is more and more important in the domain of industrial inspection, quality controlling, machine vision, and medical science, etc. And because of its high precision, nondestructive feature san also fastness of data acquisition, the method of non-contact optical measurement becomes very popular. Take the limitation of Fourier transform profilometry as the beginning, wavelet transform profilometry is brought forward as a new measurement method. The theoretical analysis, simulation and experimental results are presented.
Firstly, the Gauss window is introduced to Fourier transform, i.e. Gabor transform, and applied to the fringe analysis. By introducing dilating transformation to the Gauss window function on the space scale, dilating Gabor transform is got. The theoretical analysis, simulation and experimental results are presented that demonstrate the limitation of them.
Furthermore, wavelet transform(WT) is applied to analyze the spatial carrier-fringes, that is, wavelet transform profilometry. The theoretical analysis, simulation and experimental results presented demonstrate the validity of the principle. And the key technique of WT, such as the selection of the mother wavelet, the scale function, and the ridge of the wavelet transform are studied.
Finally, the spatial bi-frequency grating pattern, the ratio of the two frequencies of that is an irrational number, is introduced. By using the wavelet transform profilometry, the wrapped modulated phase distributions corresponding to the two frequencies respectively are got in one bi-frequency grating pattern. And then applying the phase unwrapping by a lookup table method for unwrapping, the determinate phases corresponding to the two frequencies are got. Such technique can be applied to the analysis of the nonstationary and transient signals.
首先在傅里叶变换的基础上引入高斯窗函数提出了窗口傅里叶变换条纹分析法，进而在高斯窗函数中引入伸缩变换提出伸缩窗口傅里叶变换分析。并从理论和实验中分析了这两种方法的局限性。
再而提出了小波变换轮廓术，并通过理论分析和实际测量证明了该技术的有效性。另外，还对小波变换的关键技术问题，如母小波函数的选择、小波尺度函数的构造以及小波变换脊的确定进行了详细的研究与分析。
最后，为了解决非连续物体的包裹相位解调的问题，本论文发展了钟金钢老师提出的双频条纹相位查表法，结合小波变换轮廓术，提出了双频光栅+小波变换+查表相位解包技术，实现在一幅变形双频光栅图像中进行小波分析得到两个频率所对应的确定的相位分布。该方法可以用于由具有台阶、陡峭表面和噪声等复杂表面形成的非平稳光栅信号的分析。
The application of 3-D shape measurement is more and more important in the domain of industrial inspection, quality controlling, machine vision, and medical science, etc. And because of its high precision, nondestructive feature san also fastness of data acquisition, the method of non-contact optical measurement becomes very popular. Take the limitation of Fourier transform profilometry as the beginning, wavelet transform profilometry is brought forward as a new measurement method. The theoretical analysis, simulation and experimental results are presented.
Firstly, the Gauss window is introduced to Fourier transform, i.e. Gabor transform, and applied to the fringe analysis. By introducing dilating transformation to the Gauss window function on the space scale, dilating Gabor transform is got. The theoretical analysis, simulation and experimental results are presented that demonstrate the limitation of them.
Furthermore, wavelet transform(WT) is applied to analyze the spatial carrier-fringes, that is, wavelet transform profilometry. The theoretical analysis, simulation and experimental results presented demonstrate the validity of the principle. And the key technique of WT, such as the selection of the mother wavelet, the scale function, and the ridge of the wavelet transform are studied.
Finally, the spatial bi-frequency grating pattern, the ratio of the two frequencies of that is an irrational number, is introduced. By using the wavelet transform profilometry, the wrapped modulated phase distributions corresponding to the two frequencies respectively are got in one bi-frequency grating pattern. And then applying the phase unwrapping by a lookup table method for unwrapping, the determinate phases corresponding to the two frequencies are got. Such technique can be applied to the analysis of the nonstationary and transient signals.
引文
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[2] M. Takeda. Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview. Indust. Metrol, 1990, 1(1):79～99
[3] Chen Wenyi, Tan Yushan, Zhao Hong. Automatic Analysis Technique of Spatial Carrier-Fringe Patterns. Optics and Lasers in Engineering, 1996, 25:111～120
[4] 王晋疆，杨志文．光学三维传感．半导体光电．2002，23(1)：51～53
[5] 陈晓荣，蔡萍，施文康．光学非接触三维形貌测量技术新进展．光学 精密工程．2002，10(5)：528～531
[6] 苏显渝，李继陶．三维面形测量技术的新进展．物理．1996，25(10)：614～620
[7] 田丰，赵宏，陈文艺等．三维面形的光栅投影合成测量技术．光子学报．1996，25(12)：1120～1124
[8] 牛小兵，林玉池，赵美蓉等．光栅投影三维测量的原理及关键技术分析．光电子 激光，2002，13(9)：983～986
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[10] 王学礼，桑波，赵宏等．投影莫尔技术应用于静态物面微小变形测试的研究．仪器仪表学报．2001，22(6)：622～625
[11] Mitsuo Takeda, Kazuhiro Mutoh. Fourier Transform Profilometry for The Automatic Measurement of 3-D Object Shapes. Appl. Opt, 1983, 22(24): 3977～3982
[12] Su Xianyu, Chen Wenjing. Fourier transform profilometry: a review. Optics and Laser Engineering, 2001, 35(5):263～284
[13] 苏显渝，谭松新，向立群等．基于傅里叶变换轮廓术方法的复杂物体三维面形测量．光学学报．1998，18(9)：1228-1233
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[15] Schwider J, Falkenstorfer O, Schreiber H, Zoller A, Streibl N. New compensating four-phase algorithm for phase-shift interferometry. Opt Eng, 1993, 32(8): 1883～1885
[16] Kreis Thomas M. Computer Aided Evaluation of Fringe Patterns. Optics and Lasers in Engineering, 1993, 19(4):221～240
[17] Zhong Jingang, Zhao Jin. Three-dimensional Shape measurement system with digital light projector. Proc SPIE, 2002, 4778:95～104
[18] 翁嘉文，钟金钢．加窗傅里叶变换在三维形貌测量中的应用．光子学报．2003，32(8)：993-996
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[20] 杨力华，戴道清，黄文良等(译)．信号处理的小波导引[M]．北京：机械工业出版社．2002，58-59
[21] 韩世勤，张玉芬．小波变换及其在信号分析中的应用．中国民族学院学报(自然科学版)．1996，15(4)：86～91
[22] 冉启文，王建赜．小波变换及其在时-频分析中的应用．数理统计与管理．1999，18(3)：56～59
[23] 冉启文，小波分析方法及其应用．数理统计与管理．1999，18(1)：52～55
[24] 成琼，于德介．基于复小波变换相位功率谱的齿轮故障诊断．湖南大学学报(自然科学版)．2001，28(3)：23～26
[25] 何晓霞，沈玉娣，张西宁．连续小波变换在滚动轴承故障诊断中的应用．机械科学与技术．2001，20(4)：571～574
[26] 王洪刚，赵继，许树新等．小波变换在瞬变信号检测中的计算机仿真及应用．机械设计与制造工程．2001，30(4)：54～55，20
[27] 陈祥训．正交紧支复小波的生成及其在电力系统的应用．中国电机工程学报．2000，20(7)：83～88
[28] 马晶，谭立英，冉启文．小波分析在光学信息处理中的应用．物理学报．1999，48(7)：1223～1229
[29] Sanaoni G, Biancardi L, Minoni U et al., A novel, adaptive system for 3-D optical profilometry using a liquid crystal light projector. IEEE Trans. Instrum. Meas., 1994, 43(4):558～565
[30] Li J, Su H, Su X. Two-frequency grating used in phase-measuring profilometry. Appl. Opt., 1997, 36(1):277～280
[31] Zhao H, Chen W, Tan Y. Phase-unwrapping algorithm for the measurement of
three-dimensional object shapes. Appl. Opt., 1994, 33(20):4497～1500
[32] Hao Y, Zhao Y, Li D. Multifrequency grating projection profilometry based on the nonlinear excess fraction method. Appl. Opt., 1999,38(19): 4106～4110
[33] Gushov V I, Solodkin Y N. Automatic processing of fringe patterns in integer interferometers. Opt. Lasers Engng., 1991, 14(4,5):311～324
[34] Takeda M, Gu Q, Kinoshita M et al.. Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations. Appl. Opt., 1997, 36(22):5347～5354
[35] 钟金钢，张永林，李丰丽等．空间载频条纹相位分析法中的相位不确定性．光学学报．2001，21(5)：609～614
[36] T. C. Stand. Optical three-dimensional sensing for machine vision. Opt. Eng., 1985,24(1):33
[37] J. A. Jarikio, R.C. Kim, S. K. Case. Three-dimensional inspection using multistripe structured light. Opt. Eng., 1985, 24(6):996
[38] Frank Chen, G. M. Brown, Mumin Song. Overview of three-dimensional shape measurement using optical methods. Opt. Eng., 2000,39(1): 10
[39] J. Villa, M. Servin. Robust profilometer for the measurement of 3-D object shapes based on a regularized phase tracker. Optics and Lasers in Engineering, 1999,31:279～288
[40] Chen F, Brown G,Song M. Overview of three-dimensional shape measurement using optical methods. Opt Eng, 2000, 39(1): 10～22
[41] 郝煜栋，赵洋，李达成．光学投影式三维轮廓测量技术综述．光学技术．1998，5：57～64．
[42] 苏显渝，李继陶．信息光学[M]．北京：科学出版社，1999
[43] Kuznetsova TI. Gabor-transform inversion and superresolution in optical measurements. Optics Communications, 1998, 153(1):5～8
[44] Aydin Akan, Chaparro Luis F. Evolutionary chirp representation of non-stationary signals via Gabor transform. Signal Processing, 2001, 81(11):2429～2436
[45] Rutuparna Panda, Chatterji BN. Unsupervised texture segmentation using tuned filters in Gaborian space. Pattern Recognition Letters, 1997, 18(5):445～453
[46] 郑建明，李言，肖继明等．伸缩窗口短时Fourier分析．振动、测试与诊断．2000，20(4)：254～258
[47] David P. Casasent, John-Scott Smokelin, Anqi Ye. Wavelet and Gabor transforms for detection. Optical Engineering, 1992, 31 (9): 1893～1898
[48] Rong-Seng Chang, Jin-Yi Sheu, Ching-Huang Lin, He-Chiang Liu. Analysis of CCD moir(?) pattern for micro-range measurements using the wavelet transform. Optics and Laser Technology. 2003, 35:43～47
[49] H. Jeong. Analysis of plate wave propagation in anisotropic laminates using a wavelet transform. NDT&E International. 2001, 34:185～190
[50] Brian Teller, Harold H. Szu. New wavelet transform normalization to remove frequency bias. Optical Engineering, 1992, 31(9):1830～1834
[51] FIELD D J. Relations between the statistics of natural images and the response properties of cortical cells. Journal of The Optical Society of America A, 1987, 4(12):2379-2394
[52] Nathalie Delprat, Bernard Escudi(?), Philippe Guillemain et al.. Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies. IEEE Transactions on Information Theory, 1992, 38(2):644～664
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[54] Paolo Tomassini, Antonio Giulietti, Leonida A. et al.. Analyzing laser plasma interferngrams with a continuous wavelet transform ridge extraction technique: the method. Applied Optics, 2001, 40(35):6561～6568
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[59] Zhong Jingang, Weng Jiawen. Dilating Gabor transform for the fringe analysis of 3-D shape measurement. Optical Engineering. 2004, 43(4):895～899
[2] M. Takeda. Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview. Indust. Metrol, 1990, 1(1):79～99
[3] Chen Wenyi, Tan Yushan, Zhao Hong. Automatic Analysis Technique of Spatial Carrier-Fringe Patterns. Optics and Lasers in Engineering, 1996, 25:111～120
[4] 王晋疆，杨志文．光学三维传感．半导体光电．2002，23(1)：51～53
[5] 陈晓荣，蔡萍，施文康．光学非接触三维形貌测量技术新进展．光学 精密工程．2002，10(5)：528～531
[6] 苏显渝，李继陶．三维面形测量技术的新进展．物理．1996，25(10)：614～620
[7] 田丰，赵宏，陈文艺等．三维面形的光栅投影合成测量技术．光子学报．1996，25(12)：1120～1124
[8] 牛小兵，林玉池，赵美蓉等．光栅投影三维测量的原理及关键技术分析．光电子 激光，2002，13(9)：983～986
[9] 赵宏，陈文艺，谭玉山．利用莫尔条纹的准正弦特性的三维轮廓术．光学学报．1994，14(8)：834～837
[10] 王学礼，桑波，赵宏等．投影莫尔技术应用于静态物面微小变形测试的研究．仪器仪表学报．2001，22(6)：622～625
[11] Mitsuo Takeda, Kazuhiro Mutoh. Fourier Transform Profilometry for The Automatic Measurement of 3-D Object Shapes. Appl. Opt, 1983, 22(24): 3977～3982
[12] Su Xianyu, Chen Wenjing. Fourier transform profilometry: a review. Optics and Laser Engineering, 2001, 35(5):263～284
[13] 苏显渝，谭松新，向立群等．基于傅里叶变换轮廓术方法的复杂物体三维面形测量．光学学报．1998，18(9)：1228-1233
[14] 陈文静，苏显渝，谭松新．傅里叶变换中频谱泄漏的讨论．光学学报．2000．20(10)：1429～1434
[15] Schwider J, Falkenstorfer O, Schreiber H, Zoller A, Streibl N. New compensating four-phase algorithm for phase-shift interferometry. Opt Eng, 1993, 32(8): 1883～1885
[16] Kreis Thomas M. Computer Aided Evaluation of Fringe Patterns. Optics and Lasers in Engineering, 1993, 19(4):221～240
[17] Zhong Jingang, Zhao Jin. Three-dimensional Shape measurement system with digital light projector. Proc SPIE, 2002, 4778:95～104
[18] 翁嘉文，钟金钢．加窗傅里叶变换在三维形貌测量中的应用．光子学报．2003，32(8)：993-996
[19] 杨福生．小波变换的工程分析与应用[M]．北京：科学出版社．1999
[20] 杨力华，戴道清，黄文良等(译)．信号处理的小波导引[M]．北京：机械工业出版社．2002，58-59
[21] 韩世勤，张玉芬．小波变换及其在信号分析中的应用．中国民族学院学报(自然科学版)．1996，15(4)：86～91
[22] 冉启文，王建赜．小波变换及其在时-频分析中的应用．数理统计与管理．1999，18(3)：56～59
[23] 冉启文，小波分析方法及其应用．数理统计与管理．1999，18(1)：52～55
[24] 成琼，于德介．基于复小波变换相位功率谱的齿轮故障诊断．湖南大学学报(自然科学版)．2001，28(3)：23～26
[25] 何晓霞，沈玉娣，张西宁．连续小波变换在滚动轴承故障诊断中的应用．机械科学与技术．2001，20(4)：571～574
[26] 王洪刚，赵继，许树新等．小波变换在瞬变信号检测中的计算机仿真及应用．机械设计与制造工程．2001，30(4)：54～55，20
[27] 陈祥训．正交紧支复小波的生成及其在电力系统的应用．中国电机工程学报．2000，20(7)：83～88
[28] 马晶，谭立英，冉启文．小波分析在光学信息处理中的应用．物理学报．1999，48(7)：1223～1229
[29] Sanaoni G, Biancardi L, Minoni U et al., A novel, adaptive system for 3-D optical profilometry using a liquid crystal light projector. IEEE Trans. Instrum. Meas., 1994, 43(4):558～565
[30] Li J, Su H, Su X. Two-frequency grating used in phase-measuring profilometry. Appl. Opt., 1997, 36(1):277～280
[31] Zhao H, Chen W, Tan Y. Phase-unwrapping algorithm for the measurement of
three-dimensional object shapes. Appl. Opt., 1994, 33(20):4497～1500
[32] Hao Y, Zhao Y, Li D. Multifrequency grating projection profilometry based on the nonlinear excess fraction method. Appl. Opt., 1999,38(19): 4106～4110
[33] Gushov V I, Solodkin Y N. Automatic processing of fringe patterns in integer interferometers. Opt. Lasers Engng., 1991, 14(4,5):311～324
[34] Takeda M, Gu Q, Kinoshita M et al.. Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations. Appl. Opt., 1997, 36(22):5347～5354
[35] 钟金钢，张永林，李丰丽等．空间载频条纹相位分析法中的相位不确定性．光学学报．2001，21(5)：609～614
[36] T. C. Stand. Optical three-dimensional sensing for machine vision. Opt. Eng., 1985,24(1):33
[37] J. A. Jarikio, R.C. Kim, S. K. Case. Three-dimensional inspection using multistripe structured light. Opt. Eng., 1985, 24(6):996
[38] Frank Chen, G. M. Brown, Mumin Song. Overview of three-dimensional shape measurement using optical methods. Opt. Eng., 2000,39(1): 10
[39] J. Villa, M. Servin. Robust profilometer for the measurement of 3-D object shapes based on a regularized phase tracker. Optics and Lasers in Engineering, 1999,31:279～288
[40] Chen F, Brown G,Song M. Overview of three-dimensional shape measurement using optical methods. Opt Eng, 2000, 39(1): 10～22
[41] 郝煜栋，赵洋，李达成．光学投影式三维轮廓测量技术综述．光学技术．1998，5：57～64．
[42] 苏显渝，李继陶．信息光学[M]．北京：科学出版社，1999
[43] Kuznetsova TI. Gabor-transform inversion and superresolution in optical measurements. Optics Communications, 1998, 153(1):5～8
[44] Aydin Akan, Chaparro Luis F. Evolutionary chirp representation of non-stationary signals via Gabor transform. Signal Processing, 2001, 81(11):2429～2436
[45] Rutuparna Panda, Chatterji BN. Unsupervised texture segmentation using tuned filters in Gaborian space. Pattern Recognition Letters, 1997, 18(5):445～453
[46] 郑建明，李言，肖继明等．伸缩窗口短时Fourier分析．振动、测试与诊断．2000，20(4)：254～258
[47] David P. Casasent, John-Scott Smokelin, Anqi Ye. Wavelet and Gabor transforms for detection. Optical Engineering, 1992, 31 (9): 1893～1898
[48] Rong-Seng Chang, Jin-Yi Sheu, Ching-Huang Lin, He-Chiang Liu. Analysis of CCD moir(?) pattern for micro-range measurements using the wavelet transform. Optics and Laser Technology. 2003, 35:43～47
[49] H. Jeong. Analysis of plate wave propagation in anisotropic laminates using a wavelet transform. NDT&E International. 2001, 34:185～190
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