Prata A, Rusch W V
Appl Opt. 1989 Feb 15;28(4):749-54. doi: 10.1364/AO.28.000749.
A numerically efficient algorithm for expanding a function in a series of Zernike polynomials is presented. The algorithm evaluates the expansion coefficients through the standard 2-D integration formula derived from the Zernike polynomials' orthogonal properties. Quadratic approximations are used along with the function to be expanded to eliminate the computational problems associated with integrating the oscillatory behavior of the Zernike polynomials. This yields a procedure that is both fast and numerically accurate. Comparisons are made between the proposed scheme and a procedure using a nested 2-D Simpson's integration rule. The results show that typically at least a fourfold improvement in computational speed can be expected in practical use.
提出了一种在一系列泽尼克多项式中展开函数的数值高效算法。该算法通过从泽尼克多项式的正交特性导出的标准二维积分公式来计算展开系数。二次近似与待展开函数一起使用，以消除与泽尼克多项式振荡行为积分相关的计算问题。这产生了一种既快速又数值精确的方法。将所提出的方案与使用嵌套二维辛普森积分规则的方法进行了比较。结果表明，在实际应用中，通常可以预期计算速度至少提高四倍。
