Fractional Transforms and Their Applications in Image Encryption and Filtering 

Author  LiuZhengJun 
Tutor  LiuShuTian 
School  Harbin Institute of Technology 
Course  Optics 
Keywords  Fractional random transform Fractional Fourier transform Complex order Fourier transform Image encryption Image sharing Multiple image encryption Phase retrieval Hollow Gaussian beam 
CLC  O438 
Type  PhD thesis 
Year  2007 
Downloads  600 
Quotes  4 
Optical information security is a new research topic in both of the optical information processing and the information science. Up to now most of the opticalinformation security techniques are based on optical transforms together with therandom key generation algorithms. Therefore exploring new transforms and imageencryption algorithms are of great significance for optical information security techniques in both theory and practical applications. In recent years, the fractional Fouriertransforms and their applications in the field of optical information processing haveattracted much attention in optical community. The fractional Fourier transforms havenot only been one of the powerful tools in optical information and digital signal processing, but also its conceptional extensions in fractional orders have derived manynew fractional transforms. These novel mathematical transforms will also have valuable applications in the fields of digital signal processing, digital image processingand optical information processing. In this dissertation, we propose various kindsof novel fractional order transforms and image encryption algorithms based on thefractional Fourier transforms. We also give a physical interpretation of the complexorder Fourier transform and propose a hollow Gaussian beam generation scheme. Theresearch work has been summarized in detail as follows.Propose the concept of the discrete fractional random transform. We haveanalyzed the kernel matrix of the discrete fractional Fourier transform and found thatthe eigenvector matrix and the eigenvalue matrix are relatively independent, and hencecan be changeable. We therefore defined a discrete fractional random transform byrandomizing the eigenvector matrix in the discrete fractional Fourier transform. Thediscrete fractional random transform can result random data, however, maintains mostof the excellent mathematical properties as the fractional Fourier transforms have,such as linearity, index additivity, unitary, periodicity and energy conservation, etc.Due to its randomness, this transform can be directly used in the image encryptionsand decryptions. Based on the discrete fractional random transform, we also definedthe discrete fractional random Sine/Cosine transforms, which have the same mathematical properties as well as keeping the symmetries for even and odd signals. The discrete fractional random transform can be reconstructed by the discrete fractionalrandom Sine and Cosine transforms. The computational load of such reconstructedfractional random transform can be reduced to a half of the original. As a matter offact, the eigenvector matrix of the discrete fractional Fourier transform can be arbitrarily changed. We then defined a new transform, referred as the discrete fractionalangular transform, by redefining a set of eigenvectors via an angle parameter withrecursion method. Such transform has a very small computational load and thereforecan be used in fast signal processing.Propose the random Fourier transform and the random fractional Fouriertransform. According to the multiplicity of the eigenvalues of the fractional Fouriertransforms, the eigenvalues of the fractional Fourier transforms can be directly randomly chosen. Hence we defined a random Fourier transform and give its mathematical representation. We also proposed an optical implementation of the randomFourier transform by using a 4f system with random phase filtering. Based on theoptical physical process of the fractional Fourier transform, we proposed a randomfractional Fourier transform, in which the transform kernel function of the fractionalFourier has been randomized. This random fractional Fourier transform can be implemented optically and hence used directly as an optical image encryption scheme.Further more, the conventional double random phase encoding algorithm can be modeled by cascading the random fractional Fourier transforms.Propose various kinds of novel image encryption schemes. We have analyzedthe image encryption scheme based on optical Hartley transform and found its securitywas low and difficult for optical implementation. From this point of view, we proposeda novel image encryption scheme based on random Hartley transform and doublerandom amplitude encoding, which has high security under attacks of naked and blinddecryption. We have proposed an image encryption scheme based on the commutationand anticommutation rules that uses the onedimensional Fourier of fractional Fouriertransforms. Its optical implementation has been given and its security has been provedby numerical simulations. We proposed a double image encryption scheme based onthe phase retrieval algorithm in the fractional Fourier domain. Two images can beencrypted to the fractional Fourier domain with two different fractional orders. Thisalgorithm can be extended to a multiple image encryption scheme with the fractionalorder multiplexing. Also we proposed a multiple image encryption scheme by shifting the multiple central parts of spectra together with the double random phase encodingtechnique.Propose two simple and practical image sharing scheme. Based on the mathematical combination theory, we proposed a simple image sharing scheme which caneffectively implement the (t,n) threshold sharing with high security. Based on thediscrete fractional random transform and linear equation theory, we proposed an image sharing scheme, in which the image can be shared with a set of linear matrixequations.Study the physical nature of the complexorder Fourier transform. We haveanalyzed the mathematical properties of the complexorder Fourier transform. Wehave proved that the energy nonconservation property of the complexorder Fouriertransform by using the notion of the Wigner distribution function. Hence we pointedout that the complexorder Fourier transform can not be realized in the first orderlossless optical system and clarified some misunderstandings of the optical implementation of the complexorder Fourier transform in optical community. Based onour analysis, we gave a physical interpretation of the imaginaryorder Fourier transform with an incoherent light propagation model in a homogeneous medium withconstant intensity absorption.Propose a hollow Gaussian beam generation method. According to the differential property of the Fourier transform, we proved that the hollow Gaussian beamcan be generated via a spatial amplitude filtering in the Fourier domain. The filtercan be represented by a binomial combination of the Hermite polynomials. We haveproved our proposal with numerical simulations and designed two 4f optical systemsthat can generate the hollow Gaussian beams as well.