Efficient and robust recurrence relations for the Zernike circle polynomials and their derivatives in Cartesian coordinates.
Efficient and robust recurrence relations for the Zernike circle polynomials and their derivatives in Cartesian coordinates.
复制标题
笛卡尔坐标中泽尼克圆多项式及其导数的高效且稳健的递推关系。
DOI：
--
复制
发表时间：
2018
期刊：
影响因子：
3.8
通讯作者：
Torben Brender Andersen
中科院分区：
文献类型：
--
作者：
Torben Brender Andersen
一段时间以来，已知并推荐的是，由于越来越大的抵消误差，应该使用递归关系而不是显式表达式来执行对径向阶数高于8至10的泽尼克多项式的计算。本文给出了一组简单的递推关系式，它既可用于极坐标系下的单位规格化Zernike多项式，也可用于直角坐标系下。递归关系也很适合于计算的Zernike多项式的笛卡尔导数。递推关系很容易推广到任意高阶。对标准64位浮点运算可达到的精度的评估表明，可以在单位盘上计算径向阶数高达30的泽尼克多项式，误差不超过5E-14，并且径向阶数高达50的误差不超过1.2E-13。与OpticStudio（Zemax）中的Zernike能力的比较表明，递归关系在性能（速度和精度）上优于软件中实现的现有算法。给出了计算泽尼克多项式及其导数的通用伪代码。
For some time it has been known and recommended that the calculation of Zernike polynomials to radial orders higher than 8 to 10 should be performed using recurrence relations rather than explicit expressions due increasingly large cancellation errors. This paper presents a set of simple recurrence relations that can be used for the unit-normalized Zernike polynomials in polar coordinates and easily adapted to Cartesian coordinates as well. The recurrence relations are also well suited for the calculation of the Cartesian derivatives of the Zernike polynomials. The recurrence relations are easily extended to arbitrarily high orders. Assessments of the precision achievable with standard 64-bit floating point arithmetic show that Zernike polynomials up to radial order 30 can be calculated over the unit disc with errors not exceeding 5E-14, and up to radial order 50 with errors not exceeding 1.2E-13. Comparison with the Zernike capability in OpticStudio (Zemax) shows that the recurrence relations are superior in performance (both speed and precision) over the existing algorithm implemented in the software. General pseudo-code for the calculation of Zernike polynomials and their derivatives is also presented.
影响因子：
3.8
作者：
Kaye, Elena A.;Hertzberg, Yoni;Pauly, Kim Butts
通讯作者：
Pauly, Kim Butts