用Zernike多项式进行波面拟合的一种新算法

数学物理学报 ›› 2000, Vol. 20 ›› Issue (3): 378-385.

用Zernike多项式进行波面拟合的一种新算法

（1中国科学院南京天文仪器研制中心 |南京 210042）

（2南京建筑工程学院南京 210009）

出版日期:2000-05-12 发布日期:2000-05-12

A New Algorithm for Wavefront Fitting Using Zernike Polynomial

(1 Nanjing Astronomical Instruments Research Center, Chinese Academy of Sciences, Nanjing 210042)
(2 Nanjing Architectural &|Civil Engineering Institute, Nanjing 210009)

Online:2000-05-12 Published:2000-05-12

摘要/Abstract

摘要：

该文提出了一种用于计算全息数字波面干涉仪中实现波面Zernike多项式拟合的精确算法. 该算法不同于传统的直接构造法方程和Gram-Schmidt正交化方法，而是用Householder变换对矛盾方程的广义增广矩阵进行正交三角化，直接求解拟合系数.它避免了构造法方程组，从而避免了以前的方法因构造的法方程组出现严重病态而引入的计算误差，并且易于编程，因而是一种比较理想的实现Zernike多项式拟合的算法.

关键词: 波面拟合, Ｚｅｒｎｉｋｅ多项式, ＧｒａｍＳｃｈｍｉｄｔ正交法, Ｈｏｕｓｅｈｏｌｄｅｒ变换

Abstract:

A precise algorithm for wavefront fitting using Zernike polynomial, which is applied in the computer-generated hologram digital wavefront interferometer, is studied. Differing from the conventional method of direct constructing normal equation group or the GramSchmidt orthogonalization method, the new algorithm orthogonalizes and triangulates the matrix of inconsistent equation group by using Householder transformation, and then the Zernike coefficients are worked out using a backsubstitution technique. It is proposed for the first time. By avoiding constructing normal equation group, the computational error introduced by constructing normal equation group is eliminated. It is also quite easy to be programmed and has proved to be an efficacious algorithm.

Key words: Ｗａｖｅｆｒｏｎｔｆｉｔｔｉｎｇ, Ｚｅｒｎｉｋｅｐｏｌｙｎｏｍｉａｌ, ＧｒａｍＳｃｈｍｉｄｔｏｒｔｈｏｇｏｎａｌｉｚａｔｉｏｎｍｅｔｈｏｄ, Ｈｏｕｓｅｈｏｌｄｅｒｔｒａｎｓｆｏｒｍａｔｉｏｎ

中图分类号:

６５Ｄ１５

鄢静舟1 孙厚环2 高志强1 周必方1. 用Zernike多项式进行波面拟合的一种新算法[J]. 数学物理学报, 2000, 20(3): 378-385.

Yan Jingzhou1 Sun Houhuan2 Gao Zhiqiang1 Zhou Bifang1. A New Algorithm for Wavefront Fitting Using Zernike Polynomial[J]. Acta mathematica scientia,Series A, 2000, 20(3): 378-385.