A fast numerical algorithm for the 2D non-separable linear canonical transform based on a decomposition of the ABCD matrix

The two-dimensional non-separable linear canonical transform (2D-NS-LCT) can model a wide range of paraxial optical systems. Digital algorithms to calculate the 2D-NS-LCTs are of great interested in both light propagation modeling and digital signal processing. We have previously reported that the transform of a 2D image with rectangular sampling grid generally results in a parallelogram output sampling grid, thus complicating further calculations. One possible solution is to use interpolation techniques. However, it usually leads to poor calculation speed and reduced accuracy. To alleviate this problem, we previously proposed a unitary algorithm by choosing an advantageous sampling rate related to the system parameters. In this paper, a fast algorithm is further proposed based on a novel matrix decomposition, which can significantly improve the efficiency of the numerical approximations.