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.

[1]  V. Namias.  The Fractional Order Fourier Transform and its Application to Quantum Mechanics , 1980 .

[2]  Ran Tao,et al.  Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function , 2010 .

[3]  Cagatay Candan,et al.  The discrete fractional Fourier transform , 2000, IEEE Trans. Signal Process..

[4]  John T. Sheridan,et al.  Optical image encryption by random shifting in fractional Fourier domains. , 2003, Optics letters.

[5]  Ran Tao,et al.  The multiple-parameter discrete fractional Hadamard transform , 2009 .

[6]  C. Shih.  Fractionalization of Fourier transform , 1995 .

[7]  M. Yeary,et al.  Orthogonal Projections and Discrete Fractional Fourier Transforms , 2006, 2006 IEEE 12th Digital Signal Processing Workshop & 4th IEEE Signal Processing Education Workshop.

[8]  Gozde Bozdagi Akar,et al.  Digital computation of the fractional Fourier transform , 1996, IEEE Trans. Signal Process..

[9]  Luís B. Almeida,et al.  The fractional Fourier transform and time-frequency representations , 1994, IEEE Trans. Signal Process..

[10]  Ran Tao,et al.  Research progress on discretization of fractional Fourier transform , 2008, Science in China Series F: Information Sciences.

[11]  G. Unnikrishnan,et al.  Optical encryption by double-random phase encoding in the fractional Fourier domain. , 2000, Optics letters.

[12]  Nicola Laurenti,et al.  Multiplicity of fractional Fourier transforms and their relationships , 2000, IEEE Trans. Signal Process..

[13]  S Liu,et al.  Optical image encryption based on multifractional Fourier transforms. , 2000, Optics letters.

[14]  Ran Tao,et al.  The Generalized Weighted Fractional Fourier Transform and its Application to Image Encryption , 2009, 2009 2nd International Congress on Image and Signal Processing.

[15]  Ran Tao,et al.  Optical image encryption based on the multiple-parameter fractional Fourier transform. , 2008, Optics letters.

[16]  Qian Yang,et al.  MIMO-OFDM system based on fractional Fourier transform and selecting algorithm for optimal order , 2008, Science in China Series F: Information Sciences.

[17]  Ran Tao,et al.  The multiple-parameter fractional Fourier transform , 2008, Science in China Series F: Information Sciences.