DFRFT And DFRHT Theory And Its Applications On Optical Image Encryption

The discrete fractional Fourier transform (DFRFT) and the discrete fractional Hartley transform (DFRHT) is the extension of the classical discrete Fourier transform and the discrete Hartley transform. The DFRFT and the DFRHT provide much more forms of choices than the traditional DFT and DHT. Especially in the study of optical information processing, optical fractional Fourier transform provides the ability of disposing non-focal-plane of signal, which brings optical information processing much convenience.The main work of this paper is firstly we studied the properties of the DFT and DHT's matrices with the eigenvalues and the eigenvectors. Since there's no definite form of the DFT and DHT's matrices, we give a new definition of the DFT and DHT's matrices by introducing some further conditions. Then, we gave the new definition of the DFRFT and the DFRHT. After that, we gave the transform relationship between them and we analysis the properties of the new algorithm. Then we get a now form of weighted extension of the DFRFT, and it would be widely used in the optical area in the future.Finally, as an example, we use them to build an optical image encryption, the numerical analysis show that our method is available and easy to be used. Besides that, our new method greatly enhance the data security by enlarge the encryption key number to match the size of the picture, at the same time, this method do not complex the numerical calculation.