基于环扇域正交多项式的频域分析

利用傅里叶变换得到了Zernike多项式和环扇域内正交多项式的功率谱密度（PSD）分布，以及正交多项式每项所对应的峰值径向空间频率和半峰值径向空间频率范围。通过对比发现，正交多项式与相同阶的Zernike多项式PSD分布相似，但是却含有更高的空间频率成分。通过计算机仿真，发现正交多项式中每一项都基本上只代表特定的空间频率范围，根据相位度量的环扇形镜面面形空间频率分布，选择适当的正交多项式的项进行拟合，不仅能够节省运算时间，而且还可以保证拟合精度。

Frequency domain analysis for orthogonal polynomials based on annulus sector area

The power spectrum density (PSD) distribution of Zernike polynomials and the polynomials orthogonalized in an annulus sector area (POAS) were obtained by Fourier transformation. The peak radial spatial frequency and the half-peak radial spatial frequency range corresponding to each item of the orthogonal polynomials were calculated. In comparison with Zernike polynomials, the orthogonal polynomials have similar PSD distribution, but higher spatial frequencies. Computer simulation indicates that each item of POAS represents only a specific frequency range. According to the spatial frequency distribution of the surface figure of a mirror with annulus sector shape, the calculation time is reduced and the fitting accuracy is guaranteed if the proper items of orthogonal polynomials are selected.