Pei Soo-Chang, Huang Shih-Gu
J Opt Soc Am A Opt Image Sci Vis. 2016 Feb 1;33(2):214-27. doi: 10.1364/JOSAA.33.000214.
As a generalization of the 2D Fourier transform (2D FT) and 2D fractional Fourier transform, the 2D nonseparable linear canonical transform (2D NsLCT) is useful in optics and signal and image processing. To reduce the digital implementation complexity of the 2D NsLCT, some previous works decomposed the 2D NsLCT into several low-complexity operations, including 2D FT, 2D chirp multiplication (2D CM), and 2D affine transformations. However, 2D affine transformations will introduce interpolation error. In this paper, we propose a new decomposition called CM-CC-CM-CC decomposition, which decomposes the 2D NsLCT into two 2D CMs and two 2D chirp convolutions. No 2D affine transforms are involved. Simulation results show that the proposed methods have higher accuracy, lower computational complexity, and smaller error in the additivity property compared with the previous works. Plus, the proposed methods have a perfect reversibility property, meaning that one can reconstruct the input signal/image losslessly from the output.
作为二维傅里叶变换（2D FT）和二维分数傅里叶变换的推广，二维非可分线性规范变换（2D NsLCT）在光学以及信号与图像处理中很有用。为了降低二维非可分线性规范变换的数字实现复杂度，之前的一些工作将二维非可分线性规范变换分解为几个低复杂度运算，包括二维傅里叶变换、二维线性调频乘法（2D CM）和二维仿射变换。然而，二维仿射变换会引入插值误差。在本文中，我们提出一种名为CM - CC - CM - CC分解的新分解方法，该方法将二维非可分线性规范变换分解为两个二维线性调频乘法和两个二维线性调频卷积。不涉及二维仿射变换。仿真结果表明，与之前的工作相比，所提方法具有更高的精度、更低的计算复杂度以及在可加性属性方面更小的误差。此外，所提方法具有完美的可逆性，这意味着可以从输出无损地重建输入信号/图像。
