Zhao Chunyu, Burge James H
Opt Express. 2013 Dec 16;21(25):31430-43. doi: 10.1364/OE.21.031430.
Zernike polynomials are an orthonormal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an orthonormal set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a set of orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials.
泽尼克多项式是圆域上一组正交归一的标量函数，常用于表示波前相位或表面不规则性。在光学测试中，有时会测量表面或波前的斜率或曲率，进而得到表面或波前图。之前我们推导了一组正交归一的向量多项式，它们适用于斜率测量数据，并能生成由泽尼克多项式表示的表面或波前图。这里我们定义了一个用于表示连续表面二阶导数的三元曲率向量，并推导了一组用泽尼克多项式表示的正交归一曲率基函数。我们将新的曲率函数称为C多项式。给出了完整基集的闭式关系，并展示了如何从由C多项式表示的曲率数据中确定泽尼克表面系数。
