一阶光学系统分数傅里叶变换的相空间分析

在维格纳相空间中,通过将一阶光学系统的传输矩阵分解为坐标旋转、比例缩放和啁啾矩阵的组合,得到了一阶光学系统在空域的分数傅里叶表示.结果表明:任意一阶光学系统均可表示为经过比例缩放和二次相位调制的分数傅里叶变换.通过将输入输出光场在相空间中作π/2角旋转,得到了一阶光学系统在频域的传输矩阵和衍射积分公式,进而得到了一阶光学系统在频域的分数傅里叶表示.比较空域和频域一阶光学系统的相空间变换矩阵,说明2个系统本质上属同一变换在不同基坐标下的表示,并推导出了光学系统在空域和频域具有相同分数傅里叶变换的条件.     

Symplectic Radon Transform and the Metaplectic Representation

We study the symplectic Radon transform from the point of view of the metaplectic representation of the symplectic group and its action on the Lagrangian Grassmannian. We give rigorous proofs in the general setting of multi-dimensional quantum systems. We interpret the Radon transform of a quantum state as a generalized marginal distribution for its Wigner transform; the inverse Radon transform thus appears as a “demarginalization process” for the Wigner distribution.     

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.     

Image displacement based on phase-encoded reference joint fractional transform correlator

In order to utilize the space of an input plane efficiently and make the optical structure more flexible, an image displacement measurement based on phase-encoded reference joint fractional transform correlator (PER-JFrTC) is proposed. We use a random phase mask to encode the reference image and overlay it with the target image forming the input image. Joint power spectrum (JPS) of the input image is obtained by Fourier transform and the resultant is encoded by the same phase mask. Then a fractional Fourier transform with an order p is applied to the phase-encoded JPS (PJPS), resulting in a correlation output with a sharp cross-correlation peak, which includes the displacement information between the reference and the target image. Contrast to displacement measurement based on traditional joint transform correlator (JTC), PER-JFrTC can use the space of the input plane efficiently and reduces the influence of the auto-correlation. Also the position of cross-correlation peak can be fixed arbitrarily according to the fractional order p as well as the optical set-up can be more flexible and easier to implement. Results based on digital computation show that PER-JFrTC could detect the displacement accurately and verify our proposal. A possible optical set-up is suggested.     

量子光学态的Ridgelet变换

利用有序算符内积分技术推导出一个有用的双模算符正规乘积公式. 然后在量子力学框架下, 计算出相干态、特殊压缩相干态、中介纠缠态表象的Radon变换. 在此基础上, 通过选取“墨西哥帽”母小波函数, 分别分析了以上三种量子光学态的Ridgelet变换. 关键词： 有序算符内积分技术 Radon变换 Ridgelet变换     

光学可分离小波变换的研究

为改进光学小波变换,提出了光学可分离小波变换的概念。通过对光学小波变换的必要条件进行分析,采用层叠算法和二维可分离小波变换构造方法计算出了尺度和小波函数的二维离散近似序列。将新的母小波引入到光学小波变换中,获得了光学可分离小波变换。通过与现有光学小波变换及二维可分离离散小波变换进行比较说明了其优点。     

Characterizing Young's fringes’ orientation and spacing by Fourier transform and Radon transform

Orientation and spacing are important features of fringe patterns, especially in speckle photography when the displacement information of the object is determined by pointwisely filtering the double-exposure specklegram. The Fourier transform and Radon transform are both simple and effective tools for solving this problem, which extract the information in spectral domain and spatial domain, respectively. A hybrid method combining Radon transform and Fourier transform is also possible. The results can be further improved by enhanced Fourier transform and enhanced Radon transform. The theories, as well as real applications, are given in this paper.     

On the Radon transformation of Wigner function altered with various optical processes

We discuss what happens to the Radon transformation of signal's Wigner functions (i.e., signal's Wigner transformation (WT)) if the signal function undergoes various optical processes, such as Fraunhofer diffraction, lens transformation and Fresnel diffraction, etc. Because the usual Wigner transforms can be studied via their corresponding transforms of the Wigner operator, we use the Weyl ordered form of the Wigner operator and the Weyl ordering invariance under similar transformations to derive the result, we find that the alteration of Radon transformation of signal's Wigner function (or named the variation of tomogram function), through these optical processes, can be ascribed to the variation of Radon transformation parameters once the parameter of WT is given.     

The representation of waist-to-waist transformation of Gaussian beams by the scaled fractional Fourier transform

The beam waist-to-waist transformation of Gaussian beams between input and output reference planes described by the scaled fractional Fourier transform is analyzed in this paper. We obtain the transfer matrix of ABCD optical system that corresponds to the scaled fractional Fourier transform. The results show that the beam waist-to-waist transformation of Gaussian beams can be described by the scaled fractional Fourier transform when the ABCD optical system has a suitable transfer matrix. The relationship between the input and output waist planes and some particular cases when a Gaussian beam passes through a thin lens is also discussed.     

Radon transforms of the Wigner operator on hyperplanes

The generalization of tomographic maps to hyperplanes is considered. We find that the Radon transform of the Wigner operator in multi-dimensional phase space leads to a normally ordered operator in binomial distribution---a mixed-state density operator. Reconstruction of the Wigner operator is also feasible. The normally ordered form and the Weyl ordered form of the Wigner operator are used in our derivation. The operator quantum tomography theory is expressed in terms of some operator identities, with the merit of revealing the essence of the theory in a simple and concise way.     

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。     

Collins formula interpreted as wavelet transform

Based on the manipulation of input and output complex amplitude definitions of a light field in space domain, one direct relation between the Collins formula and the wavelet transform has been derived, namely, the Collins formula has been interpreted as a generalization of the wavelet transform. Then some elementary optical systems’ behavior has been described according to this derivation for example. This observation would be useful in various optical applications by virtue of wavelet transform's character.     

三能级钾原子气体三维傅里叶变换频谱的解析解

利用投影切片定理、傅里叶位移定理和误差函数给出三能级钾原子气体三维傅里叶变换频谱在T=0界面的解析解.固定均匀线宽,非均匀展宽和对角线相关系数可以定量地识别,通过在适当方向上拟合三维傅里叶变换频谱谱峰的切片来确定.结果表明,非均匀展宽增大,频谱图沿着对角线方向延伸,对角线相关系数增大,频谱图逐渐变圆,振幅也逐渐变小.     

Polarization tomography of the tokamak plasma magnetic field

Optical polarization tomography of the tokamak plasma magnetic field is considered in the context accepted for large magnetic fields. Two models of the interaction between the magnetic field and radiation are studied: (1) the Faraday rotation (FR) of the polarization plane is imposed on birefringence caused by the toroidal component and (2) the Cotton-Mouton effect (CME) is imposed on the FR. Previously, these problems were solved with allowance for only one of the optical effects. New integrals caused by the amplification of the magnetic field are added to ray integrals used earlier. The retrieval of the magnetic field is based on the vector Radon transform (VRT) of imaginary absorption. Formulas of the inverse VRT are derived using generalized conditions from R. Novikov on the Radon transform with absorption.     

Wavelet-modified fringe-adjusted joint transform correlator

In this paper, we implement a wavelet-modified fringe-adjusted joint transform correlator (JTC) for real-time target recognition applications. In real-time situation the input scene is captured using a charge-coupled device (CCD) camera. The obtained joint power spectrum is multiplied by a pre-synthesized fringe-adjusted filter and the resultant function is processed with an appropriately scaled wavelet filter. Three performance measure parameters: correlation peak intensity, peak-to-sidelobe ratio, and signal-to-clutter ratio have been calculated for fringe-adjusted joint transform correlator (FJTC) and wavelet-modified fringe-adjusted joint transform correlator (WFJTC). The WFJTC has been found to yield better results in comparison to conventional FJTC. To suppress the undesired strong dc, the resultant function is differentiated. Differential processing wavelet-modified fringe-adjusted joint power spectrum removes the zero-order spectra and hence improves the detection efficiency. To focus the correlation terms in different planes in order to capture one of the desired autocorrelation peaks and discard the strong dc and another autocorrelation peak, chirp-encoding technique has also been applied. Targets with Gaussian and speckle noise have also been used to check the correlation outputs. Computer simulation and experimental results are presented.     

Schlieren成像和声场可视化

Schlieren技术是利用声场引起透明媒质光折射率的变化而实现声场可视化的光学成像技术。它具有对声场无干扰、快速、瞬态成像的特点。本文利用二维光学Fourier变换分析了Schlieren技术的成像原理,在采用连续激光和高速ICCD的Schlieren成像系统中,实验研究了平面波声场和线聚焦声场中换能器光学校准方法和声压的定量检测技术。发展声场瞬态和动态成像技术,观测了声波的聚焦过程和固-液界面的声场分布和变化。这些结果表明Schlieren技术是一种有效的声场可视化和定量检测的光学成像技术。     

基于自适应正交有限脊波变换的图像编码算法

在有限脊波变换的基础上,根据对图像有限Radon变换系数的分析,提出了一种自适应的正交有限脊波变换。此变换根据特别设计的代价函数,可自适应地为每列有限Radon变换系数选择最优的正交变换形式,从而可高效地分析图像中的直线奇异。讨论了此变换在图像编码领域中的应用,通过对零树编码算法SPIHT进行改进,实现了图像自适应正交有限脊波变换系数的渐进编码。实验结果表明,对富含直线边缘的图像来说,与小波图像编码算法相比,该算法获得了较高的边缘重建质量。     

Relations between chirp transform and Fresnel diffraction, Wigner distribution function and a fast algorithm for chirp transform

Two physical interpretations of chirp transform related to Fresnel diffraction and Wigner distribution function are given.The chirp transform can be regarded as a Fresnel diffraction observed on a spherical tangent to the diffraction plane,or a rotation and stretching transformation of the Wigner distribution function space.A general fast algorithm for the numerical calculation of chirp transform is developed by employing two fast Fourier transform algorithms.The algorithm,by which a good evaluation can be achieved,unifies the calculations of Fresnel diffraction,arbitrary fractionalorder Fourier transforms and other scalar diffraction systems.The algorithm is used to calculate the Fourier transform of a Gaussian function and the Fourier transform,the Fresnel transform,the Fractional-order Fourier transforms of a rectangle function to evaluate the performance of this algorithm.The calculated results are in good agreement with the analytical results,both in the amplitude and phase.     

Target recognition in clutter scene based on wavelet transform

Edge extraction based on wavelet for optical correlation detection is presented. Optical experiments with joint transform correlator (JTC) show that there is a bright application prospect in the field of optical correlation detection by extracting the edge features of input image with the method of wavelet transform. In the course of processing, the multi-scale character of wavelet is used sufficiently. The energy of correlation peaks and the detection ratio of various targets are greatly enhanced by the approach. To demonstrate the feasibility of edges extraction based on WT, small targets and targets in clutter scene are successfully detected.