Eigenfunctions of the offset Fourier,fractional Fourier,and linear canonical transforms

The offset Fourier transform (offset FT), offset fractional Fourier transform (offset FRFT), and offset linear canonical transform (offset LCT) are the space-shifted and frequency-modulated versions of the original transforms. They are more general and flexible than the original ones. We derive the eigenfunctions and the eigenvalues of the offset FT, FRFT, and LCT. We can use their eigenfunctions to analyze the self-imaging phenomena of the optical system with free spaces and the media with the transfer function exp[j(h2x2 + h1x + h0)] (such as lenses and shifted lenses). Their eigenfunctions are also useful for resonance phenomena analysis, fractal theory development, and phase retrieval.     

Uncertainty principles in linear canonical transform domains and some of their implications in optics

The linear canonical transform (LCT) is the name of a parameterized continuum of transforms that include, as particular cases, many widely used transforms in optics such as the Fourier transform, fractional Fourier transform, and Fresnel transform. It provides a generalized mathematical tool for representing the response of any first-order optical system in a simple and insightful way. In this work we present four uncertainty relations between LCT pairs and discuss their implications in some common optical systems.     

Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm

A method for the calculation of the fractional Fourier transform (FRT) by means of the fast Fourier transform (FFT) algorithm is presented. The process involves mainly two FFT's in cascade; thus the process has the same complexity as this algorithm. The method is valid for fractional orders varying from -1 to 1. Scaling factors for the FRT and Fresnel diffraction when calculated through the FFT are discussed.     

Speckle orientation in paraxial optical systems

The statistical properties of speckles in paraxial optical systems depend on the system parameters. In particular, the speckle orientation and the lateral dependence (x and y) of the longitudinal speckle size can vary significantly. For example, the off-axis longitudinal correlation length remains equal to the on-axis size for speckles in a Fourier transform system, while it decreases dramatically as the observation position moves off axis in a Fresnel system. In this paper, we review the speckle correlation function in general linear canonical transform (LCT) systems, clearly demonstrating that speckle properties can be controlled by introducing different optical components, i.e., lenses and sections of free space. Using a series of numerical simulations, we examine how the correlation function changes for some typical LCT systems. The integrating effect of the camera pixel and the impact this has on the measured first- and second-order statistics of the speckle intensities is also examined theoretically. A series of experimental results are then presented to confirm several of these predictions. First, the effect the pixel size has on the measured first-order speckle statistics is demonstrated, and second, the orientation of speckles in a Fourier transform system is measured, showing that the speckles lie parallel to the optical axis.     

Fast spectral-domain method for acoustic scattering problems

This paper presents the application of the conjugate-gradient (CG) fast Fourier transform (FFT) (CG-FFT) method and the CG nonuniform FFT (CG-NUFFT) method for the integral equation arising from acoustic scattering problems. In the conventional method of moments (MoM) for integral equations, the CPU and memory requirements are O(N³) and O(N²), respectively, where N is the number of unknowns in the problem. The CG-FFT method, which combines the iterative conjugate-gradient method with FFT, reduces these requirements to O(KN log₂N) and O(N), respectively, where K is the number of CG iterations. The CG-NUFFT method differs from the CG-FFT method in that it makes use of nonuniform FFT algorithms instead of FFT to allow a nonuniform discretization. Therefore, the CG-NUFFT method can solve the integral equation with both uniform and nonuniform grid while retaining the efficiency of the CG-FFT method. These two methods are applied to solve for two-dimensional constant density acoustic scattering problems. Numerical. results demonstrate that they can solve much larger problems than the MoM     

Precise and fast phase wraps reduction in fringe projection profilometry

Phase wraps in a 2D wrapped phase map can be completely eliminated or greatly reduced by frequency shifting. But it usually cannot be optimally reduced using conventional fast Fourier transform (FFT) because the spectrum can be shifted only by a integer number in the frequency domain. In order to achieve a significant phase wrap reduction, we propose a fast and precise two-step method for phase wraps reduction in this paper, which is based on the iterative local discrete Fourier transform (DFT). Firstly, initial estimate of the frequency peak is obtained by FFT. Then sub-pixel spectral peak with high resolution is determined by iteratively upsampling the local DFT around the initial peak location. Finally, frequency shifting algorithm that operates in the spatial domain is used to eliminate phase wraps. Simulations and experiments are conducted to demonstrate the superb computing efficiency and overall performance of the proposed method.     

Eigenfunctions and self-imaging phenomena of the two-dimensional nonseparable linear canonical transform

The two-dimensional (2D) nonseparable linear canonical transform (NSLCT) is a generalization of the fractional Fourier transform (FRFT) and the LCT. It is useful in signal analysis and optics. The eigenfunctions of both the FRFT and the LCT have been derived. In this paper, we extend the previous work and derive the eigenfunctions of the 2D NSLCT. Although the 2D NSLCT is very complicated and has 16 parameters, with the proposed methods, we can successfully find the eigenfunctions of the 2D NSLCT in all cases. Since many optical systems can be represented by the 2D NSLCT, our results are useful for analyzing the self-imaging phenomena of optical systems.     

Bayesian versus Fourier spectral analysis of ion cyclotron resonance time-domain signals

The frequency-domain spectrum obtained by Fourier transformation (FT) of a time-domain signal is accurate only for a continuous noiseless time-domain signal of infinite duration. For discrete noisy truncated time-domain signals, non-FT (e.g., Bayesian analysis) methods may provide more accurate spectral estimates of time-domain signal frequencies, relaxation time(s), and relative abundances. In this paper, we show that Bayesian analysis of simulated and experimental ion cyclotron resonance (ICR) time-domain noisy signals can produce a spectrum with mass accuracy improved by a factor of 10 or more over that obtained from a magnitude-mode discrete fast Fourier transform (FFT) spectrum. Moreover, Bayesian analysis offers the useful advantage that it automatically estimates the precision of its iteratively determined spectral parameters. The main disadvantage of Bayesian analysis is its lengthy computation time compared to that of FFT (hours vs seconds on the same hardware for approximately 4K time-domain data points); the Bayesian computation time increases rapidly with the number of spectral peaks and (less rapidly) with the number of time-domain data points. Bayesian analysis should thus prove useful for those FT/ICR applications involving relatively few data points and/or requiring high mass accuracy.     

Fast Fourier Transform for Step-Like Functions: The Synthesis of Three Apparently Different Methods

In 1965 Cooley and Tukey published an algorithm for rapid calculation of the discrete Fourier transform (DFT), a particularly convenient calculating technique, which can well be applied to impulse-like functions whose beginning and end lie at the same level. Independently, various propositions were made to overcome the truncation error which arises, if a step-like function, i.e. one whose end level differs from its starting level, is treated in the same way. It was argued that they behave differently under the influence of noise, band-limited violation, and other experimental inconveniences. The aim of this paper is to show that the three widely and satisfactorily used techniques of Samulon, Nicolson, and Gans, which originate from apparently different ideas, are exactly the same. An extended DFT and fast Fourier transform (FFT) formula is deduced which is adapted as well to impulse-like as to step-like functions.     

Pure absorption-mode spectra from Bayesian maximum entropy analysis of ion cyclotron resonance time-domain signals

Fourier transform ion cyclotron resonance (FT/ICR) mass spectra are normally reported in the (phase-independent) magnitude-mode format. In principle, the absorption-mode format offers spectral resolution enhanced by a factor ranging from square root of 3 to 2 over the corresponding magnitude-mode spectrum obtained by discrete FT of the same unapodized time-domain data. However, an absorption-mode display is generally unsuitable in practice because of the auxiliary spectral peaks (Gibbs oscillations) resulting from the relatively long time delay between excitation and detection. Although the resulting large phase variation (up to 100 pi rad or more across the Nyquist spectral bandwidth) can be corrected exactly for a continuous time-domain signal, phase correction of a discrete time-domain signal results in Gibbs oscillations even for a perfectly phased absorption-mode spectrum. In this paper, we show that a Bayesian "maximum-entropy" analysis of simulated and experimental ion cyclotron resonance time-domain noisy signals can recover a precisely phased absorption-mode frequency-domain spectrum that is devoid of Gibbs oscillations, is less sensitive to noise, and offers improved mass accuracy over that obtained from a conventional magnitude-mode discrete fast Fourier transform (FFT) spectrum.     

3种频域数字水印算法的分析和比较

阐述了离散傅里叶算法(DFT)、离散余弦算法(DCT)、离散小波变换算法(DWT)这3种常用的频域数字水印算法的原理,并分别采用不同算法对数字图像进行了水印嵌入,检测其峰值信噪比(PSNR)和相关系数(NC)的值,分析并比较了这3种算法的优缺点,证明了离散小波变换在频域算法中能够得到最好的效果。     

LIDFT-the DFT linear interpolation method

The method of linear interpolation of the discrete Fourier transform (LIDFT) to estimate parameters of a signal consisting of many sinusoidal oscillations, has been presented in the paper. The LIDFT method combines beneficial properties from known procedures of nonlinear interpolation of a spectrum, obtained as a result of DFT and from parametric methods based on the Prony method. The LIDFT algorithm with beneficial numerical properties has been obtained after formulating the assumptions of the LIDFT method, providing a linear matrix equation and following symbolic transformations of this equation. The basic operations involved are the FFT algorithm and linear matrix equation solving procedure. The parametric, linearizing data window together with the method-developed to choose the window parameter allow for effective application of the LIDFT method depending on the examined signal character     

A fast algorithm for three-dimensional electrostatics analysis: fast Fourier transform on multipoles (FFTM)

In this paper, we propose a new fast algorithm for solving large problems using the boundary element method (BEM). Like the fast multipole method (FMM), the speed-up in the solution of the BEM arises from the rapid evaluations of the dense matrix–vector products required in iterative solution methods. This fast algorithm, which we refer to as fast Fourier transform on multipoles (FFTM), uses the fast Fourier transform (FFT) to rapidly evaluate the discrete convolutions in potential calculations via multipole expansions. It is demonstrated that FFTM is an accurate method, and is generally more accurate than FMM for a given order of multipole expansion (up to the second order). It is also shown that the algorithm has approximately linear growth in the computational complexity, implying that FFTM is as efficient as FMM. Copyright © 2004 John Wiley & Sons, Ltd.     

Hartley transform ion cyclotron resonance mass spectrometry

The Hartley transform offers a useful alternative to the Fourier transform for the conversion of a time-domain ion cyclotron resonance (ICR) signal into its corresponding frequency-domain mass spectrum. The Hartley transform has the advantage that it eliminates the need for complex variables, when (as for linearly polarized signals) the time-domain signal can be represented by a mathematically real function. Moreover, the Hartley transform produces the same spectra (absorption mode, dispersion mode, magnitude mode) as does the Fourier transform. In particular, the discrete fast Hartley transform (FHT) produces the same spectrum at twice the speed of a complex fast Fourier transform (FFT), making the FHT equivalent in speed to a "real" FFT. Hartley and Fourier transform methods in ICR mass spectrometry are compared and demonstrated experimentally. Essentially the same advantages and computational methods should apply to the use of the Hartley transform in place of the Fourier transform in other forms of spectrometry (e.g., nuclear magnetic resonance, infrared, etc.).     

Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms

By use of matrix-based techniques it is shown how the space-bandwidth product (SBP) of a signal, as indicated by the location of the signal energy in the Wigner distribution function, can be tracked through any quadratic-phase optical system whose operation is described by the linear canonical transform. Then, applying the regular uniform sampling criteria imposed by the SBP and linking the criteria explicitly to a decomposition of the optical matrix of the system, it is shown how numerical algorithms (employing interpolation and decimation), which exhibit both invertibility and additivity, can be implemented. Algorithms appearing in the literature for a variety of transforms (Fresnel, fractional Fourier) are shown to be special cases of our general approach. The method is shown to allow the existing algorithms to be optimized and is also shown to permit the invention of many new algorithms.     

半功率点法估计阻尼的一种改进

为了提高半功率点法估计阻尼的精度，探讨了快速傅立叶变换（FFT）离散谱线之间函数值的意义，基于这种连续性理解，构造了半功率点估计阻尼的新算法，采用理论分析与数值仿真相结合的方法研究了半功率点法的主要误差因素和各自特性，特别是其中窗阻尼变化规律，给出了保证阻尼误差小于10％，5％和1％对窗长的要求。     

快速Fourier变换的信号流图自动生成

快速Ｆｏｕｒｉｅｒ变换信号流图对于快速Ｆｏｕｒｉｅｒ变换的研究是很有意义的，本文系统地了快速Ｆｏｕｉｅｒ变换信号流图自动生成的算法，包括一般快速Ｆｏｕｒｉｅｒ变换和广义滑动快速Ｆｏｕｉｅｒ变换的信号流图的自动生成算法，根据算法提出在向对象的实现方法并开发了相应的应用程序。     

Paraxial speckle-based metrology systems with an aperture

Digital speckle photography can be used in the analysis of surface motion in combination with an optical linear canonical transform (LCT). Previously [D. P. Kelly et al. Appl. Opt.44, 2720 (2005)] it has been shown that optical fractional Fourier transforms (OFRTs) can be used to vary the range and sensitivity of speckle-based metrology systems, allowing the measurement of both the magnitude and direction of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. This requires two bulk optical systems. We extend the OFRT analysis to more general LCT systems with a single limiting aperture. The effect of a limiting aperture in LCT systems is examined in more detail by deriving a generalized Yamaguchi correlation factor. We demonstrate the benefits of using an LCT approach to metrology design. Using this technique, we show that by varying the curvature of the illuminating field, we can effectively change the output domain. From a practical perspective this means that estimation of the motion of a target can be achieved by using one bulk optical system and different illuminating conditions. Experimental results are provided to support our theoretical analysis.     

FFT calculation of magnetic fields in air coils

The magnetostatic field produced by an air coil that possesses one-dimensional periodicity can be expressed as a one-dimensional discrete convolution of two functions. The first function expresses the field produced by a single turn of the coil. The second is a shape function; it expresses the spatial position and strength of current of each turn of the coil. The discrete convolution of these two functions gives the magnetostatic field produced by the coil. This paper presents one application of linear system theory to an air coil calculation, the use of the fast Fourier transform (FFT) in computing magnetostatic magnetic fields from air coils. A program is described which uses FFT convolution to perform this calculation.     

Interpolation by fast Fourier and Chebyshev transforms

Transform methods for the interpolation of regularly spaced data are described, based on fast evaluation using discrete Fourier transforms. For periodic data adequately sampled, the fast Fourier transform (FFT) is used directly. With undersampled or aperiodic data, a Chebyshev interpolating polynomial is evaluated by means of the FFT to provide minimum deviation and distributed ripple. The merits of two kinds of Chebyshev series are compared. All the methods described produce an interpolation passing directly through the given values and are applied easily to the multi-dimensional case.