The discrete multiple-parameter fractional Fourier transform

As a generalization of the Fourier transform (FT), the fractional Fourier transform (FRFT) has many applications in the areas of optics, signal processing, information security, etc. Therefore, the efficient discrete computational method is the vital fundament for the application of the fractional Fourier transform. The multiple-parameter fractional Fourier transform (MPFRFT) is a generalized fractional Fourier transform, which not only includes FRFT as special cases, but also provides a unified framework for the study of FRFT. In this paper, we present in detail the discretization method of the MPFRFT and define the discrete multiple-parameter fractional Fourier transform (DMPFRFT). Then, we utilize the tensor product to define two-dimensional multiple-parameter fractional Fourier transform (2D-MPFRFT) and the corresponding two-dimensional discrete multiple-parameter fractional Fourier transform (2D-DMPFRFT). Finally, as an application, a novel image encryption method based on 2D-DMPFRFT is proposed. Numerical simulations are performed to demonstrate that the proposed method is reliable and more robust to blind decryption than several existing methods.