Study And Applications Of Fractional Fourier Transform

With the rapid development of computer, network and communicationtechnology, especially the popularization of Internet, the multimediainformation exchange provides a convenient way for using and spreadingdigital produces. Therefore, the characteristic that digital producesare easily copied and tampered, which makes digital produces securityprotection become an urgently resolved problem, information securityand data encryption are becoming more and more important. By analyzingand discussing the techniques of Fourier transform, fractional Fouriertransform and image scrambling, methods of an image encryption and ablind watermarking algorithm based on chaotic sequences and FractionalFourier Transform have been proposed and numerically simulated. Theprinciple work is as follows:(1)Properties of the fractional Fourier transform are investigateddetailedly in this thesis, the numerical simulation system of thefractional Fourier written by the MATLAB language is set up on the basisof the Namias's fractional transform theory, experiment result for thismethod is given. Chaotic sequences have several good propertiesincluding the ease of their generation, the sensitive dependence ontheir initial condition and the statistics property of noise. Wediscussed an image encryption arithmetic base on chaotic sequences.Preliminary results are satisfactory, which can protect secret keyeffectively and insure the security of the image.(2)By using the two parameters of dissymmetrical discrete fractionalFourier transform as secret keys, a new encrypted transform method ofdigital image is presented. The fractional Fourier transform of theinput image was performed by two one-dimensional FRFT (FractionalFourier Transform) with different orders in the x and y directions in cascade. The two orders are considered to keys, only when the transformorder in each direction is respectively matched with that of theasymmetry fractional Fourier transform, the encoded image can bereconstructed exactly. The experiment show that we can see some of theoriginal image information when the ecryption is close to the key.Thispaper presents an image encryption technology based on chaotic sequencesand fractional Fourier transform. At first, the real number valuechaotic sequences are generated by using the key value, then encryptthe image combining the method of image scrambling in space. Combingit and fractional Fourier transform, multi2-channel image encryptionis implemented. The experimental results show that the technology isvalid and has good security.(3)A digital image watermark algorithm in FRFTdomain with chaoticencryption was presented, we design a binary image watermarking signal.In order to increase the security of the system,the real-value chaoticsequences are used to modulate the watermark. First, the original imageis fractional Fourier transformed with an order. Then, through thecorrespondence between the secret key and the original watermarkimage, the watermark is embedded in fractional Fourier spectral domain.Final, the watermarked image is obtained by inversely transforming thejoint spectra. Experimental results demonstrate that the watermark isinvisible and robust against crop, common noise and JPEG compress etc.