Sarver and Associates Inc, Carbondale, Ill, USA.
J Refract Surg. 2010 Jan;26(1):61-5. doi: 10.3928/1081597X-20101215-10.
To develop a method to quickly calculate equal-spaced Zernike polynomial expansion samples on a rectangular or polar grid for analysis or display.
It is well known that a Zernike polynomial expansion can be converted into an equivalent rectangular or polar two-dimensional Taylor polynomial expansion. It is also known how to quickly calculate equal-spaced polynomial samples using difference equations. Using these two techniques, a software class was developed that provides fast evaluation of Zernike polynomial expansion samples on a rectangular or polar grid. To test the method, the time for the direct calculation of 10th order Zernike polynomial expansion was compared to the difference equation approach for a 1000x1000 sample grid.
The direct calculation of the 10th order Zernike polynomial expansion required over 400 times more processing time than the difference equation technique for a 1000x1000 sample grid. The largest difference in calculated values between the two techniques was negligible, indicating 11 digits of accuracy when using double precision variables.
The difference equation approach proves to be a fast and accurate method to calculate equal-spaced Zernike polynomial expansion samples on a rectangular or polar grid. This algorithm has application in both the analysis of optical systems and display of results.
开发一种在矩形或极坐标网格上快速计算等距 Zernike 多项式展开样本的方法，用于分析或显示。
众所周知，Zernike 多项式展开可以转换为等效的矩形或极坐标二维泰勒多项式展开。也知道如何使用差分方程快速计算等距多项式样本。利用这两种技术，开发了一个软件类，提供了在矩形或极坐标网格上快速评估 Zernike 多项式展开样本的功能。为了测试该方法，比较了直接计算 10 阶 Zernike 多项式展开与差分方程方法在 1000x1000 样本网格上的时间。
对于 1000x1000 样本网格，直接计算 10 阶 Zernike 多项式展开所需的处理时间比差分方程技术多 400 多倍。两种技术计算值的最大差异可以忽略不计，表明使用双精度变量时具有 11 位精度。
差分方程方法被证明是一种在矩形或极坐标网格上快速准确地计算等距 Zernike 多项式展开样本的方法。该算法在光学系统分析和结果显示中都有应用。
