Wavelet--fractional Fourier transforms

This paper extends the definition of fractional Fourier transform (FRFT) proposed by Namias V by using other orthonormal bases for $L^{2}\left( R \right)$ instead of Hermite--Gaussian functions. The new orthonormal basis is gained indirectly from multiresolution analysis and orthonormal wavelets. The so defined FRFT is called wavelets-fractional Fourier transform.     

Wavelet-fractional Fourier transforms

This paper extends the definition of fractional Fourier transform （FRFT） proposed by Namias V by using other orthonormal bases for L^2（R） instead of Hermite-Gaussian functions. The new orthonormal basis is gained indirectly from multiresolution analysis and orthonormal wavelets. The so defined FRFT is called wavelets-fractional Fourier transform.     

Finite systems,fractional Fourier transforms and their finite phase spaces

Various harmonic oscillator models define — in a sense to be explained here — fractional Fourier transforms (up to a phase). The fractionalization of the Fourier integral transform is well understood; the finite case is less. There are several discrete and finite oscillator models that contract to the continuous, integral model. The Ankara model can be thought as a ring of point masses joined by springs to their equilibrium positions and to each other; the Cuernavaca model uses the su(2) algebra with a distinct physical interpretation. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

菲涅耳衍射和分数傅里叶变换

介绍了衍射孔的菲涅耳衍射和分数傅里叶变换的对应关系，使得可以用分数傅里叶变换来描述光电原始光场经过菲涅耳衍射区一直到无穷远处夫琅禾费衍射区的自由空间标量衍射传播全过程。     

The asymmetrical fractional Fourier transforms and its optical implement

Compared with the symmetrical lens system, the structure which contains double cylindrical lenses is also the basic units of a complex system and implement practical asymmetrical fractional Fourier transforms. Based on the theory of scalar diffraction and on the Fourier optics, we implement several kinds of asymmetrical fractional Fourier transforms and make the different mathematical manipulation at the corresponding coordinates, while the symmetrical system cannot. Meanwhile, the simulation experiment shows that such system not only makes the two perpendicular coordinates to be independently operated, but also increases the flexibility of the optical setup. All these characteristics make it flexible to application domain in optics.     

Comments on the complete generalized fractional Fourier transform

In this research, we thoroughly investigate the complete generalized fractional Fourier transform (CGFRFT) and draw the following conclusions that are different from the original literature: (1) The CGFRFT is not a generalized version, but a special case that ignores the marginal postulate; (2) If the period parameter is not a multiple of four, the CGFRFT can never perform a Fourier transform regardless of the value of the transform order. The simulation results of a rectangular signal support the above conclusions.     

Signal separation using linear canonical and fractional Fourier transforms

Signal separation is an important operation in many areas such as communications, geophysics, and optics including optical signal processing. In this article, the issue of signal separation based on the linear canonical transform (LCT) and the fractional Fourier transform (FRFT) is considered. An iterative algorithm for signal separation in the 2-ray ground reflected model using the FRFT is also presented along with the simulation results.     

Joint extended fractional Fourier transform correlator

Based on the conventional correlation and fractional correlation, the extended fractional correlation (EFC) is presented. And based on the configuration of the nonconventional joint transform correlator, we propose the joint extended fractional Fourier transform correlator (JEFRTC). The properties of the extended fractional cross correlation peak (EFCCP) in theory are analyzed. A sound conclusion is drawn that the width of EFCCP is narrower than that of fractional correlation peak under some conditions. This JEFRTC can permit lower precision of the systemic parameters when implemented with optical configuration. That will improve correlator’s character discriminability.     

Free-space Fresnel diffraction for the approximation of fractional Fourier transform

Fractional Fourier transformation of an object can be approximated by the object's free-space Fresnel diffraction pattern under some restricted conditions and plane wave illumination according to Hua's method. A better approximation is achieved under least-squared conditions developed in this paper. Simulation results verify that our theoretical development works for any fractional order a compared with the previous approach.     

基于分数阶Fourier变换的数字图像实值加密方法

构造了一种新的保实化的分数阶Fourier变换,提出了一种基于该变换的数字图像实值加密方法。利用保实分数阶Fourier变换的保实特性和阶数可加性等完成了数字图像的加密与解密,明文和密文分别位于空域和由密钥决定的保实分数阶Fourier变换域中,具有较强的抗统计破译能力。密图是一个实值图像,便于显示和存储。仿真实验结果表明,该加密方法密钥简单,无数据膨胀,对参数敏感度高,具有一定的鲁棒性和安全性。在信息安全领域具有良好的研究前景和实用价值。     

不对称离散分数傅里叶变换实现数字图像的加密变换

利用不对称分数傅里叶变换的特性,提出了一种图像加密变换的新方法。对图像的x,y方向分别实施不同级次的一维分数傅里叶变换,得到加密图像。解密方法就是对变换后的图像实施对应级次的分数傅里叶逆变换,只有当x,y方向的逆变换级次分别与原变换级次都相同或者满足周期条件时,才能恢复原图像。加密变换有效地提高了图像加密和防伪力度。数值计算验证了方法的正确性和可行性。     

Displacement measurement based on joint fractional Fourier transform correlator

A displacement measurement technology based on joint fractional Fourier transform is firstly proposed. Contrast to conventional displacement measurement based on joint Fourier transform correlator, the position of cross correlation peak in the proposed technology could be fixed arbitrarily according to the order of fractional Fourier transform. The optical setup in the proposed technology is more flexible and easier to implement. Simulation and experiment results are given out to verify the analysis.     

基于分数阶傅里叶变换的光学加密影片

随着通信与网络的快速发展,能够包含大容量信息的影片得到了广泛应用。利用分数阶傅里叶变换的特点,提出了一种全光学加密解密影片的方法。将现有的影片加密密钥从二重变为了四重,显著增加了影片的安全性。通过改变振幅型正弦光栅的作用位置,将影片的加密解密次数由现有的每帧各一次减小为总共一次,优化了算法。仿真结果验证了该方法的有效性。     

强非局域非线性介质中的互诱导分数傅里叶变换

在强非局域非线性介质中,用强抽运光可以诱导另一弱信号光实现互诱导分数傅里叶变换效应.信号光的分数傅里叶变换阶数与传输距离和抽运光功率有关,当传输距离不变时与抽运光功率的开方成正比.互诱导分数傅里叶变换是实现光控光的又一方法,其特性有助于研发新型的分数傅里叶变换器件,并在光信息处理、光学成像等多个领域有潜在的应用. 关键词： 分数傅里叶变换 非局域介质 互诱导 光孤子     

Experimental observation of lensless coincidence fractional Fourier transform with a Gaussian Schell-model beam

A modified lensless optical system for implementing coincidence fractional Fourier transform (FRT) is proposed, and the conditions for the optical system to implement the coincidence FRT with incoherent or partially coherent light are discussed. Furthermore, we report the experimental observation of lensless coincidence FRT of an object (double slits) with a typical partially coherent beam - Gaussian Schell-model (GSM) beam. The experimental results are analyzed and agree reasonably well with the theoretical results.     

一种基于混沌和分数阶傅里叶变换的数字水印算法

基于混沌序列及离散分数阶傅里叶变换,提出了一种基于混沌和分数阶傅里叶变换的数字水印算法,并对算法及抗攻击性能进行了仿真分析.结果表明,该算法简单有效,对JPEG压缩、噪声、滤波等攻击具有良好的鲁棒性. 关键词： 数字水印 混沌 分数阶傅里叶变换 鲁棒性     

Coincidence fractional Fourier transform with a stochastic electromagnetic Gaussian Schell-model beam

Based on classical theory of optical coherence and polarization, coincidence fractional Fourier transform (FRT) with a stochastic electromagnetic Gaussian Schell-model (EGSM) beam is investigated. Dependences of the quality and visibility of the coincidence FRT pattern of an object on the degree of polarization and correlation coefficients of the EGSM beam are studied numerically. It is shown that the quality and visibility of the coincidence FRT pattern of an object are determined by the polarization and coherence of the EGSM beam together.     

Image recovery from double amplitudes in fractional Fourier domain

The classical Gerchberg--Saxton algorithm is introduced into the image recovery in fractional Fourier domain after adaptation. When this algorithm is applied directly, its performance is good for smoothed image, but bad for unsmoothed image. Based on the diversity of fractional Fourier transform on its orders, this paper suggests a novel iterative algorithm, which extracts the information of the original image from amplitudes of its fractional Fourier transform at two orders. This new algorithm consists of two independent Gerchberg--Saxton procedures and an averaging operation in each circle. Numerical simulations are carried out to show its validity for both smoothed and unsmoothed images with most pairs of orders in the interval [0, 1].