基于奇异值分解的数字波前拟合算法

提出了基于奇异值分解、采用泽尼克多项式拟合干涉波前的算法,该算法直接从线性方程组入手,对矩阵进行奇异值分解分解,在求解逆矩阵的过程中,采用阈值法对奇异值的倒数进行非常规的置换(∞→0),可直接得到系数向量。理论分析和实验证明,相对于传统的格拉姆施密特正交法,该算法可首先通过求解条件数判断线性方程矩阵是否奇异,对于解决病态方程组或奇异矩阵的最小二乘问题,有很好的稳定性,避免了由最小二乘构造的法方程组出现病态而引入的计算误差,且易于编程。

Algorithm for Digital Wavefront Fitting Based on Singular Value Decomposition

An algorithm based on singular value decomposition(SVD) is presented to fit the wavefront with Zernike polynomials.In the current algorithm,SVD is applied to decompose the matrix of the linear equations directly.In the process of computing the inverse matrix,a threshold value is employed to modify the reciprocals of singular values which are zero or too small.Then the Zernike coefficients can be worked out immediately.Compared with the typical Gram-Schmidt orthogonalization,SVD has good stabilization when solving the least square problem of the ill-conditioned equations or singular matrix. And also because of avoiding construction of normal equation group,the computational error is eliminated.Moreover,it is quite easy to be programmed.