Zhao Liang, Healy John J, Sheridan John T
J Opt Soc Am A Opt Image Sci Vis. 2014 Dec 1;31(12):2631-41. doi: 10.1364/JOSAA.31.002631.
The two-dimensional (2D) nonseparable linear canonical transform (NS-LCT) is a unitary, linear integral transform that relates the input and output monochromatic, paraxial scalar wave fields of optical systems characterized by a 4×4 ray tracing matrix. In addition to the obvious generalizations of the 1D LCT (which are referred to as separable), the 2D-NS-LCT can represent a variety of nonaxially symmetric optical systems including the gyrator transform and image rotation. Unlike the 1D LCT, the numerical approximation of the 2D-NS-LCT has not yet received extensive attention in the literature. In this paper, (1) we develop a sampling theorem for the general 2D-NS-LCT which generalizes previously published sampling theorems for the 1D case and (2) we determine which sampling rates may be chosen to ensure that the obvious discrete transform is unitary.
二维（2D）不可分离线性规范变换（NS-LCT）是一种酉线性积分变换，它关联了以4×4光线追迹矩阵为特征的光学系统的输入和输出单色傍轴标量波场。除了一维LCT的明显推广（称为可分离的）之外，二维NS-LCT还可以表示包括旋转变换和图像旋转在内的各种非轴对称光学系统。与一维LCT不同，二维NS-LCT的数值逼近在文献中尚未受到广泛关注。在本文中，（1）我们为一般的二维NS-LCT推导了一个采样定理，该定理推广了先前发表的一维情况下的采样定理；（2）我们确定了可以选择哪些采样率以确保明显的离散变换是酉的。
