中文摘要：

Zernike多项式系数的求解问题是一个典型的离散不适定问题,最小二乘法、格拉姆-斯密特正交化法和Householder变换法均无法求得稳定的数值解。本文对导致该问题解的不稳定性的原因进行了分析,并采用Tikhonov正则化法对Zernike多项式系数进行求解,利用L曲线准则确定了正则参数。数值仿真结果表明,Tikhonov正则化法有效的保证了解的稳定性,利用该方法得到的拟合面形很好的反映了面形的真实情况。

英文摘要：

To acquire coefficients of Zernike polynomials is a discrete ill-posedness problem.Common methods,such as Least Squares Error method,Cram-Schmidt orthogonalization method and Householder transformation method,can not obtain a stable numerical solution.The instability of ill-posedness problem is analyzed and a Tikhonov regularization method is introduced to solve ill-posed problem.The L curve criterion is used to determine regularization parameter.Simulation result shows that Tikhonov regularization method is stable and effective for solving ill-posedness problem.