Navarro Rafael, Arines Justo, Rivera Ricardo
ICMA, Universidad de Zaragoza and Consejo Superior de Investigaciones Científicas, Facultad de Ciencias, Pedro Cerbuna 12, 50009 Zaragoza, Spain.
Opt Express. 2009 Dec 21;17(26):24269-81. doi: 10.1364/OE.17.024269.
An invertible discrete Zernike transform, DZT is proposed and implemented. Three types of non-redundant samplings, random, hybrid (perturbed deterministic) and deterministic (spiral) are shown to provide completeness of the resulting sampled Zernike polynomial expansion. When completeness is guaranteed, then we can obtain an orthonormal basis, and hence the inversion only requires transposition of the matrix formed by the basis vectors (modes). The discrete Zernike modes are given for different sampling patterns and number of samples. The DZT has been implemented showing better performance, numerical stability and robustness than the standard Zernike expansion in numerical simulations. Non-redundant (critical) sampling along with an invertible transformation can be useful in a wide variety of applications.
提出并实现了一种可逆离散泽尼克变换（DZT）。研究表明，随机、混合（扰动确定性）和确定性（螺旋）这三种非冗余采样方式能够使所得的采样泽尼克多项式展开具有完备性。当保证完备性时，我们就能得到一个正交归一基，因此求逆仅需对由基向量（模式）构成的矩阵进行转置。给出了不同采样模式和样本数量下的离散泽尼克模式。在数值模拟中，DZT的实现表现出比标准泽尼克展开更好的性能、数值稳定性和鲁棒性。非冗余（临界）采样与可逆变换在多种应用中可能会很有用。
