Qiwen Ran

Optical image encryption based on multifractional Fourier transforms

We propose a new image encryption algorithm based on a generalized fractional Fourier transform, to which we refer as a multifractional Fourier transform. We encrypt the input image simply by performing the multifractional Fourier transform with two keys. Numerical simulation results are given to verify the algorithm, and an optical implementation setup is also suggested.

Generalized Sampling Expansion for Bandlimited Signals Associated With the Fractional Fourier Transform

The aim of the generalized sampling expansion (GSE) is the reconstruction of an unknown continuously defined function f(t), from the samples of the responses of M linear time invariant (LTI) systems, each sampled by the 1/M th Nyquist rate. In this letter, we investigate the GSE in the fractional Fourier transform (FRFT) domain. Firstly, the GSE for fractional bandlimited signals with FRFT is proposed based on new linear fractional systems, which is the generalization of classical generalized Papoulis sampling expansion. Then, by designing fractional Fourier filters, we obtain reconstruction method for sampling from the signal and its derivative based on the derived GSE and the property of FRFT. Last, the potential application of the GSE is presented to show the advantage of the theory.

A Convolution and Correlation Theorem for the Linear Canonical Transform and Its Application

As a generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) plays an important role in many fields of optics and signal processing. Many properties for this transform are already known, but the correlation theorem, similar to the version of the Fourier transform (FT), is still to be determined. In this paper, firstly, we introduce a new convolution structure for the LCT, which is expressed by a one dimensional integral and easy to implement in filter design. The convolution theorem in FT domain is shown to be a special case of our achieved results. Then, based on the new convolution structure, the correlation theorem is derived, which is also a one dimensional integral expression. Last, as an application, utilizing the new convolution theorem, we investigate the sampling theorem for the band limited signal in the LCT domain. In particular, the formulas of uniform sampling and low pass reconstruction are obtained.

On Bandlimited Signals Associated With Linear Canonical Transform

We first show that the bandlimited signals associated with linear canonical transform (LCT) form a reproducing kernel Hilbert space. An orthogonal basis for the class of bandlimited signals associated with LCT is then proposed by use of the reproducing kernel, with respect to which the coordinates of signal are actually values taken by the signal at certain instants of time. Finally, a nonuniform sampling theorem for bandlimited signals associated with LCT is presented.

Reconstruction of Bandlimited Signals in Linear Canonical Transform Domain From Finite Nonuniformly Spaced Samples

We investigate the reconstruction of bandlimited signals in the linear canonical transform (LCT) domain from a finite set of nonuniformly spaced samples. Based on the reproducing property of the reproducing kernel belonging to the class of bandlimited signals in LCT domain, we derive an interpolating formula with minimum mean-squared error that interpolates the finite set of nonuniformly spaced samples, and show that it is identical to the minimum energy bandlimited in LCT domain interpolator. Singular value decomposition is also used to set up a reconstruction algorithm which guarantees that the reconstruction result also achieves the minimum energy reconstruction.

Deficiencies of the cryptography based on multiple-parameter fractional Fourier transform

Methods of image encryption based on fractional Fourier transform have an incipient flaw in security. We show that the schemes have the deficiency that one group of encryption keys has many groups of keys to decrypt the encrypted image correctly for several reasons. In some schemes, many factors result in the deficiencies, such as the encryption scheme based on multiple-parameter fractional Fourier transform [Opt. Lett.33, 581 (2008)]. A modified method is proposed to avoid all the deficiencies. Security and reliability are greatly improved without increasing the complexity of the encryption process.

Generalized Prolate Spheroidal Wave Functions Associated With Linear Canonical Transform

Time-limited and (a, b, c, d)-band-limited signals are of great interest not only in theory but also in real applications. In this paper, we use the sampling theorem associated with linear canonical transform to investigate an operator whose effect on a signal is to produce its first time-limited then (b, b, c, d)-band-limited version. First, the eigenvalue problem for the operator is shown to be equivalent to a discrete eigenvalue problem for an infinite matrix. Then the eigenfunctions of the operator, which are referred to as generalized prolate spheroidal wave functions (GPSWFs), are shown to be first, orthogonal over finite as well as infinite intervals, and second, complete over L2(-L, L) and the class of (a, b, c, d)-band-limited signals. A simple method based on sampling theorem for computing GPSWFs is presented and the definite parity of GPSWFs is also given. Finally, based on the dual orthogonality and completeness of GPSWFs, several applications of GPSWFs to the representation of time-limited and (a, b, c, d)-band-limited signals are presented.

Sampling of bandlimited signals in the linear canonical transform domain

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. This paper investigates the sampling of bandlimited signals in LCT domain. First, we propose the linear canonical series (LCS) based on the LCT, which is a generalized pattern of Fourier series. Moreover, the LCS inherits all the nice properties from the LCT. Especially, the Parseval’s relation is presented for the LCS, which is used to derive the sampling theorem of LCT. Then, utilizing the generalized form of Parseval’s relation for the complex LCS, we obtain the sampling expansion for bandlimited signals in LCT domain. The advantage of this reconstruction method is that the sampling expansion can be deduced directly not based on the Shannon theorem.

Sampling of Bandlimited Signals in Fractional Fourier Transform Domain

Fractional Fourier transformed bandlimited signals are shown to form a reproducing kernel Hilbert space. Basic properties of the kernel function are applied to the study of a sampling problem in the fractional Fourier transform (FRFT) domain. An orthogonal sampling basis for the class of bandlimited signals in the FRFT domain is then given. A nonuniform sampling theorem for bandlimited signals in the FRFT domain is also presented. Numerical experiments are given to demonstrate the effectiveness of the proposed nonuniform sampling theorem.

Multiplicative filtering in the fractional Fourier domain

In this article, we investigate the multiplicative filtering in the fractional Fourier transform (FRFT) domain based on the generalized convolution theorem which states that the convolution of two signals in time domain results in simple multiplication of their FRFTs in the FRFT domain. In order to efficiently implement multiplicative filtering, we express the generalized convolution structure by the conventional convolution operation. Utilizing the generalized convolution structure, we convert the multiplicative filtering in the FRFT domain easily to the time domain. Based on the model of multiplicative filtering in the FRFT domain, a practical method is proposed to achieve the multiplicative filtering through convolution in the time domain. This method can be realized by classical Fast Fourier transform (FFT) and has the same capability compared with the method achieved in the FRFT domain. As convolution can be performed by FFT, this method is more useful from practical engineering perspective.