Fractional Fourier transform based image multiplexing and encryption technique for four-color images using input images as keys

A digital technique for multiplexing and encryption of four RGB images has been proposed using the fractional Fourier transform (FRT). The four input RGB images are first converted into their indexed image formats and subsequently multiplexed into a single image through elementary mathematical steps prior to the encryption. The encryption algorithm uses two random phase masks in the input- and the FRT domain, respectively. These random phase masks are especially designed using the input images. As the encryption is carried out through a single channel, the technique is more compact and faster as compared to the multichannel techniques. Different fractional orders, the random masks in input-, and FRT domain are the keys for decryption as well as de-multiplexing. The algorithms to implement the proposed multiplexing-, and encryption scheme are discussed, and results of digital simulation are presented. Simulation results show that the technique is free from cross-talk. The performance of the proposed technique has also been analyzed against occlusion, noise, and attacks using partial windows of the correct random phase keys. The robustness of the technique against known-, and chosen plain-text attacks has also been explained.     

分数傅里叶域数字水印算法

基于离散分数傅里叶变换(DFRFT)快速算法发展了一种分数傅里叶谱域图像水印算法.该算法根据分数傅里叶变换谱具有空域和频域双域信息表达能力,分别对原始图像和所加水印信息进行不同级次的分数傅里叶变换,提取水印分数傅里叶谱的低频成分并直接将其叠加到原始图像的分数傅里叶谱中的对角像元上,然后再进行逆变换得到水印图像.在JPEG压缩、图像旋转及剪切等攻击方式下,对该水印算法进行了鲁棒性分析,数值实验表明该水印算法具有良好的抗攻击性和安全性.     

联合分数变换相关器的分数相关特性分析

从分数傅里叶变换（FRFT）的定义出发，理论分析了联合分数变换相关器（JFRTC）的分数相关特性.从所得JFRTC的数学表达式中可以看出，将FRFT应用到联合变换相关器（JTC）中得到的JFRTC具有与传统JTC不同的性质.对于传统JTC，一旦输入平面上参考图像与目标图像之间的距离给定，相关输出峰的位置即确定，而JFRTC的相关输出峰的位置则可以由分数级次p₁和p₂来自由调节，这个特性在实际模式识别中非常有用.另一方面，JFRTC的相关输出峰值在大多数情况下低于传统JTC的相关峰值，却是JFRTC的一大缺点.最后，从FRFT的比例性质出发，给出了FRFT谱畸变不变的实现条件，并由此预言了JFRTC畸变不变模式识别的功能.     

Fractional Fourier transform of Lorentz beams

This paper introduces Lorentz beams to describe certain laser sources that produce highly divergent fields. The fractional Fourier transform (FRFT) is applied to treat the propagation of Lorentz beams. Based on the definition of convolution and the convolution theorem of the Fourier transform, an analytical expression for a Lorentz beam passing through a FRFT system has been derived. By using the derived formula, the properties of a Lorentz beam in the FRFT plane are illustrated numerically.     

Fractional Fourier transform of Airy beams

An analytical expression of an Airy beam passing through a fractional Fourier transform (FRFT) system is presented. The effective beam size of the Airy beam in the FRFT plane is also derived. The influences of the order of FRFT, the modulation parameter, and the transverse scale on the normalized intensity distribution and the effective beam size of an Airy beam in the FRFT plane are examined. The order of FRFT controls the effective beam size and the orientation of the beam spot. The modulation parameter affects the effective beam size and the number of lateral side lobes. The normalized intensity distribution and the effective beam size are both proportional to the transverse scale.     

分数傅立叶变换的进展与展望

在综合国内外有关文献的基础上，评述分数傅立叶变换在信息光学中的新进展。概述分数傅立叶变换的发展过程，并对其研究现状进行了简略总结，同时对其未来的发展作出展望。     

Fractional Fourier Transform of Cantor Sets

A new kind of multifractal is constructed by fractional Fourier transform of Cantor sets. The wavelet transform modulus maxima method is applied to calculate the singularity spectrum under an operational definition of multifractal. In particular, an analysing procedure to determine the spectrum is suggested for practice. Nonanalyticities of singularity spectra or phase transitions are discovered, which are interpreted as some indications on the range of Boltzmann temperature q, on which the scaling relation of partition function holds.     

From Complex Fractional Fourier Transform to Complex Fractional Radon Transform

We show that for n-dimensional complex fractional Fourier transform the corresponding complex fractional Radon transform can also be derived, however, it is different from the direct product of two n-dimensional real fractional Radon transforms. The complex fractional Radon transform of two-mode Wigner operator is calculated.     

From Complex Fractional Fourier Transform to Complex Fractional Radon Transform

We show that for n-dimensional complex fractional Fourier transform the corresponding complex fractional Radon transform can also be derived, however, it is different from the direct product of two n-dimensional real fractional Radon transforms. The complex fractional Radon transform of two-mode Wigner operator is calculated.     

Fractional Fourier transform and geometric quantization

Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of the phase-space: no linear structure is necessary. It is shown that the “fractional Fourier transform” provides a simple example of this construction. As an application of this technique we show that for any linear Hamiltonian system, its quantum dynamics can be obtained exactly as the lift of the corresponding classical dynamics by means of the above transformation. Moreover, it can be deduced from the free quantum evolution. This way new, unknown symmetries of the Schrödinger equation can be constructed. It is also argued that the above construction defines in a natural way a connection in the bundle of quantum states, with the base space describing all their possible representations. The non-flatness of this connection would be responsible for the non-existence of a quantum representation of the complete algebra of classical observables.     

黑白胶片光栅编码彩色摄影术

用空间光栅将彩色景物记录到一张黑白胶片上（这一过程称为将彩色景物编码，所得的黑白透明片称为编码片）并通过再现（称为解码）装置再现彩色景物的思想和技术可以追溯到１８９９年，已有近百年的历史，开始时，类似于电子学中用时间信号调制载波方法，用彩色图像来调制光栅，十多年后便出现第一个萧条时期，持续将近半个世纪，期间比较重要的成果是有三个光栅进行三次编码记录的方法，到本世纪六十年代，又兴旺了一段时间，其中最     

Fractional Fourier transforms and optical systems

Fractional Fourier transforms of an arbitrary degree can be implemented by refractive lenses. Perfect imaging systems and correlators may be implemented by cascading fractional Fourier transformer units of the same family and fractional degree.     

Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams

A new kind of laser beams named the elliptical Hermite-sinusoidal-Gaussian beams (EHSGBs) is introduced and defined by use of tensor method, and the elliptical Hermite-cosh-Gaussian beam (EHChGB) can be regarded as special case of EHSGBs. The fractional Fourier transform (FRFT) is applied to treat the propagation of EHChGBs. An analytical expression for an EHChGB passing through an FRFT system is derived by using vector integration. Some numerical simulations are illustrated for the propagation properties of EHChGBs through FRFT systems.     

Fourier images of fractal objects

Fourier images corresponding to diffraction patterns in the Fraunhofer zone were obtained for a set of fractal objects by means of computer simulation with the use of net functions. The fractal dimensions of the objects being investigated were determined on the basis of the intensity distribution in the diffraction patterns.     

Fractional Fourier transform of Lorentz-Gauss vortex beams

An analytical expression for a Lorentz-Gauss vortex beam passing through a fractional Fourier transform (FRFT) system is derived. The influences of the order of the FRFT and the topological charge on the normalized intensity distribution, the phase distribution, and the orbital angular momentum density of a Lorentz-Gauss vortex beam in the FRFT plane are examined. The order of the FRFT controls the beam spot size, the orientation of the beam spot, the spiral direction of the phase distribution, the spatial orientation of the two peaks in the orbital angular momentum density distribution, and the magnitude of the orbital angular momentum density. The increase of the topological charge not only results in the dark-hollow region becoming large, but also brings about detail changes in the beam profile. The spatial orientation of the two peaks in the orbital angular momentum density distribution and the phase distribution also depend on the topological charge.     

On the Scaled Fractional Fourier Transformation Operator

Based on our previous study [Chin. Phys. Lett. 24 （2007） 2238] in which the Fresnel operator corresponding to classical Fresnel transform was introduced, we derive the fractional Fourier transformation operator, and the optical operator method is then enriched.     

Fractional Fourier transform for off-axis dark-hollow beams

Based on the definition of fractional Fourier transform (FRT) in the rectangular coordinate system, the FRT for off-axis circular and elliptical dark-hollow beams (DHB) is studied in detail, and analytical formulae are derived for the FRT for off-axis circular and elliptical DHB. The properties of off-axis DHB in the FRT plane are illustrated numerically by use of the derived formulae. The results show that the properties of the off-axis DHB in the FRT plane are closely related with the parameters of the beam and fractional order. The derived formula provides a convenient way for analyzing and calculating the FRT of off-axis DHBs.     

Convolution and Filtering in Fractional Fourier Domains

Fractional Fourier transforms, which are related to chirp and wavelet transforms, lead to the notion of fractional Fourier domains. The concept of filtering of signals in fractional domains is developed, revealing that under certain conditions one can improve upon the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.     

A New Expression of the Fractional Fourier Transformation

With the help of su(2) algebra technique, a new equivalent form of the fractional Fourier transformation is given. Two examples are illustrated for their physical application in quantum optics.``