A fast two-dimension Discrete Fractional Fourier Transform algorithm and its application on digital watermarking

In order to improve the computing efficiency and accuracy of Discrete Fractional Fourier Transform (DFRFT), a fast algorithm derived from Fast Fourier Transform is introduced on the basis of the study on the periodicity of the eigenvalues of Fractional Fourier Transform (FRFT). Moreover, the algorithm is generalized to implement the two-dimension DFRFT, by which the fractional order spectrum of the digital images can be obtained. In addition, the feasibility of the two-dimension algorithm is proved by its successful application in the domain of digital watermarking. Finally, the properties of the FRFT watermarking are briefly discussed: its sensitivity to attacks in both the spatial domain and the frequency domain indicates that the algorithm can be applied to make fragile watermarkings.