| The fractional Fourier transform (FRFT) is an extension of the classical Fourier transform (FT). The FRFT has not only the excellent characteristics as those of the FT, but also many special features that FT does not have. Therefore, FRFT can be directly applied in the fields where FT has been used, and meanwhile it shows new features. The FRFT has been one of the powerful tools in optical information processing and digital signal processing. Many new fractional transforms have been obtained by extending the conception of the FRFT. These novel mathematical transforms will also be widely used in the fields of digital signal processing, digital image processing and optical information processing. Due to the parallel and high speed processing properties, the optical information processing system has been a new focus in the study of the image encryption. And many feasible image encryption methods have been proposed. Phase encoding and fractional Fourier encoding techniques have received increasing attention at home and abroad for their high security and more keys. In this thesis, three methods of optical image encryption is studied in theory and numerically simulated. They are algorithms based on the anamorphic fractional Fourier Transform (AFRFT), double random phase encoding technique and iterative encryption algorithm, respectively. The main research work is shown as follows:(1) Based on double random phase encoding, a new method for image encryption using AFRFT and linear congruence theory is proposed. Random phase masks are generated using Linear Congruential Generator (LCG). This encryption algorithm is characterized by that the linear function with more than four parameters as keys. AFRFT has two more fractional orders than conventional FRFT. The number of keys increases from two to eight, so the encryption system is more secure. Moreover, the structure of the encryption system is flexible for two auxiliary keys. Therefore, the security of the image has been further improved. The results of simulation show that only when the eight keys are exactly correct, can the original image be reconstructed. Reliability of the technique has been evaluated by calculating the mean square error (MSE) between the decrypted image and the original image. (2) A multiple-image encryption method based on FRFT and phase encoding has been studied. Amplitude encoding and phase encoding are comprehensively considered, and phase masks transformed from the original images are placed in the system to carry out multiple-image encryption. In this algorithm, an original image is encoded into amplitude part and other images are encoded into phase information. extracting the phase informations from the output of FRFT with different orders, the key of the encryption algorithm is phase different. In order to reconstruct the original images, correct orders of FRFT and decryption phase are needed. Numerical simulations have demonstrated the efficiency and the security of this algorithm. Using image information instead of random phase encryption simplifies the complexity of the system. And digital simulation demonstrates the proposed algorithm can be used in a color image encryption.(3) A color image encryption scheme based on the phase retrieval algorithm in FRFT domain is presented. A color image can be considered as three monochromatic images which can be processed as three gray images. After encryption, three components are combined to form a color encrypted image. Using double-image encryption based on the phase retrieval algorithm twice, triple images as phase informations of three complex-value images are encrypted into single image. Multiple-image encryption can be achieved by encrypting images in the fractional Fourier domains with different orders. Numerical simulations have demonstrated the effectiveness of this image encryption scheme. This algorithm can hide both true and false informations into one image. False informations can mislead attackers and hence intensify the security. |
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