Zernike多项式的条纹反射三维面形重建算法研究
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摘要
针对光学镜面测量中的面形重构问题,提出了一种基于Zernike正交多项式的面形重建方法。通过理论推导建立了面形重建数学模型;利用Matlab仿真模拟,分析了Zernike项数与采样点数的关系,研究了不同项数及采样点数对重建结果的影响;以双曲面为例进行了算法的有效性验证,给出了重建后面形的36项Zernike系数;最后对算法进行了噪声等级变化分析。研究结果表明:对于一个初始面形峰谷值(PV)为0.007 5 mm,均方根值(RMS)为0.001 1 mm的双曲面,选择Zernike前36项系数、采样点数为300时,拟合面形残差PV=2.664 5e-15 mm,RMS=4.661 6e-16 mm,拟合效果较好,且当噪声在15%以内,算法适用性较好。研究结果为后续复杂镜面条纹反射测量提供参考。
A new method based on Zernike orthogonal polynomial is proposed for the shape reconstruction in the optical mirror measurement.The mathematical model is established through theoretical derivation,and Matlab simulation is used to analyze the relationship between Zernike terms and sampling points.The effects of different items and sampling points on the reconstruction results are discussed.Taking the hyperboloid as an example,the validity of the algorithm is tested,with the 36 Zernike coefficients of the reconstructed shape given.Finally,the change of the noise level is analyzed.The results show that for a hyperboloid with an initial surface peak-valley value(PV) of 0.007 5 mm and a root mean square value(RMS) of 0.001 1 mm,the fitting surface residual PV=2.664 5 e-15 mm,RMS=4.661 6 e-16 mm when the first 36 items of Zernike polynomial and 300 sampling points are involved.The algorithm has a good fitting effect,and when the noise is within 15%,it has good applicability.The results provide a reference for the further research on complex mirror surface fringe reflection measurement.
A new method based on Zernike orthogonal polynomial is proposed for the shape reconstruction in the optical mirror measurement.The mathematical model is established through theoretical derivation,and Matlab simulation is used to analyze the relationship between Zernike terms and sampling points.The effects of different items and sampling points on the reconstruction results are discussed.Taking the hyperboloid as an example,the validity of the algorithm is tested,with the 36 Zernike coefficients of the reconstructed shape given.Finally,the change of the noise level is analyzed.The results show that for a hyperboloid with an initial surface peak-valley value(PV) of 0.007 5 mm and a root mean square value(RMS) of 0.001 1 mm,the fitting surface residual PV=2.664 5 e-15 mm,RMS=4.661 6 e-16 mm when the first 36 items of Zernike polynomial and 300 sampling points are involved.The algorithm has a good fitting effect,and when the noise is within 15%,it has good applicability.The results provide a reference for the further research on complex mirror surface fringe reflection measurement.
引文
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[17] 冯婕,白瑜,邢廷文.Zernike多项式波面拟合精度研究[J].光电技术应用,2011,26(2):31. FENG Jie,BAI Yu,XING Tingwen.Fitting Accuracy of Wavefront Using Zernike Polynomials[J].Elector-Optic Technology Application,2011,26(2):31.(in Chinese)
[18] 古德温,怀亚特.光学干涉检测[M].苏俊宏,田爱玲,译.杭州:浙江大学出版社,2014. GOODWIN E P,WYANT J C.Feild Guide to Interferometric Optical Testing[M].SU Junhong,TIAN Ailing,Translated.Hangzhou:Zhejiang University Press,2014.(in Chinese)
[19] ZHAO C Y,BURGE J H.Orthonormal Vector Polynomials in a Unit Circle,Part I: Basis Set Derived from Gradients of Zernike Polynomials[J].Optics Express,2007,15(26):18014.
[2] 宋雷,岳慧敏.基于点阵投影的彩色复合光栅傅里叶变换轮廓术[J].光学学报,2010,30(5):1368. SONG Lei,YUE Huimin.Fourier Transform Profilometry of Colorful Composite Grating Based on Point Array Projection[J].Acta Optica Sinica,2010,30(5):1368.(in Chinese)
[3] 袁婷,张峰,陶小平,等.条纹反射法检测光学反射镜面形[J].光子学报,2015,44(9):0912004. YUAN Ting,ZHANG Feng,TAO Xiaoping,et al.Test of Optical Mirror Surface Using Fringe Reflection System[J].Acta Photonica Sinica,2015,44(9):0912004.(in Chinese)
[4] 刘元坤,苏显渝,吴庆阳.基于条纹反射的类镜面三维面形测量方法[J].光学学报,2006,26(11):1636. LIU Yuankun,SU Xianyu,WU Qingyang.Three-Dimensional Shape Measurement for Specular Surface Based on Fringe Reflection[J].Acta Optica Sinica,2006,26(11):1636.(in Chinese)
[5] 邵山川,陶小平,王孝坤.基于条纹反射的超精密车削反射镜的在位面形检测[J].激光与光电子学进展,2018,55(7):071203-1. SHAO Shanchuan,TAO Xiaoping,WANG Xiaokun.On-Machine Surface Shape Measurement of Reflective Mirrors by Ultra-Precision Turning Based on Fringe Reflection[J].Laser & Optoelectronics Progress,2018,55(7):071203-1.(in Chinese)
[6] ZHANG H W,HAN S J,LIU S G,et al.3D Shape Reconstruction of Large Specular Surface[J].Applied Optics,2012,51(31):7616.
[7] RODDIER F,RODDIER C.Wavefront Reconstruction Using Iterative Fourier Transforms[J].Applied Optics,1991,30(11):1325.
[8] SOUTHWELL W H.Wave-front Estimation from Wave-front Slope Measurements[J].Journal of the Optical Society of America,1980,70(8):998.
[9] 荆海龙,苏显渝,刘元坤,等.基于条纹反射的镜面测量及三维重建算法分析[J].光电工程,2008,35(10):37. JING Hailong,SU Xianyu,LIU Yuankun,et al.Specular Surface Measurement Based on Fringe Reflection and Analysis of 3D Shape Reconstruction Technique[J].Opto-Electronic Engineering,2008,35(10):37.(in Chinese)
[10] 荆海龙,苏显渝,刘元坤.基于梯度数据的三维面形重建方法[J].激光杂志,2007,28(6):41. JING Hailong,SU Xianyu,LIU Yuankun.Shape Reconstruction Methods from Gradient Field[J].Laser Journal,2007,28(6):41.(in Chinese)
[11] WANG J F,LIU K,LI Y Q,et al.Comparison of Wavefront Reconstruction with Modal Method and Zonal Method for the Inspection of Catadioptric Projection Optics Using Hartmann Wavefront Sensor[C]//2011 International Conference on Optical Instruments and Technology:Optical Systems and Modern Optoelectronic Instruments.Beijing:Proceedings of SPIE,2011:8.
[12] LI G,LI Y,LIU K,et al.Improving Wavefront Reconstruction Accuracy by Using Integration Equations with Higher-order Truncation Errors in the Southwell Geometry[J].Journal of the Optical Society of America,2013,30(7):1448.
[13] HUANG L,ASUNDI A.Improvement of Least Squares Integration Method with Iterative Compensation for Shape Reconstruction from Gradient[J].Proceedings of SPIE,2012,8430(14):25.
[14] ZHOU T,CHEN K,WEI H,et al.Improved Method for Rapid Shape Recovery of Large Specular Surfaces Based on Phase Measuring Deflectometry[J].Applied Optics,2016,55(10):2760.
[15] 张伟,刘剑峰,龙夫年,等.基于Zernike多项式进行波面拟合研究[J].光学技术,2005,31(5):675. ZHANG Wei,LIU Jianfeng,LONG Funian,et al.Study on Wavefront Fitting Using Zernike Polynomials[J].Optical Technique,2005,31(5):675.(in Chinese)
[16] 谢苏隆.Zernike多项式拟合曲面中拟合精度与采样点数目研究[J].应用光学,2010,31(6):943. XIE Sulong.Sampling Point Number in Curved Surface Fitting with Zernike Polynomials[J].Journal of Applied Optics,2010,31(6):943.(in Chinese)
[17] 冯婕,白瑜,邢廷文.Zernike多项式波面拟合精度研究[J].光电技术应用,2011,26(2):31. FENG Jie,BAI Yu,XING Tingwen.Fitting Accuracy of Wavefront Using Zernike Polynomials[J].Elector-Optic Technology Application,2011,26(2):31.(in Chinese)
[18] 古德温,怀亚特.光学干涉检测[M].苏俊宏,田爱玲,译.杭州:浙江大学出版社,2014. GOODWIN E P,WYANT J C.Feild Guide to Interferometric Optical Testing[M].SU Junhong,TIAN Ailing,Translated.Hangzhou:Zhejiang University Press,2014.(in Chinese)
[19] ZHAO C Y,BURGE J H.Orthonormal Vector Polynomials in a Unit Circle,Part I: Basis Set Derived from Gradients of Zernike Polynomials[J].Optics Express,2007,15(26):18014.