Zernike多项式的条纹反射三维面形重建算法研究

Algorithm for Three-Dimensional Surface Reconstruction of Fringe Reflection Using Zernike Polynomial

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 5e-15 mm,RMS=4.661 6e-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.