Use Of Fourier Methods Research Articles

Using the Fourier Methods for Cycle Counting of Bimodal Stress Histories with Variable in Time Amplitudes of Components

The application of the Fourier methods to perform cycle identification and description for different cases of bimodal stress histories are presented and discussed in the paper. The direct spectral method and the modified direct spectral method, previously co-proposed by the author, together with the aspects of the use of Fourier methods discussed in this article, offer a unique alternative to the methods known in the literature for identifying and counting various types of bimodal stress variations in time (with constant or variable amplitudes; completely reversed, reversal, or pulsed type). The application of the Fast Fourier Transform (FFT) and the Short-Time Fourier Transform (STFT) is discussed. The method is useful, especially in cases when it is known that due to the form of work of the mechanical system the two existing components of vibrations with various frequencies can be identified, of which the one with higher frequency has a vibration amplitude lower than a component with a lower frequency and, above all, a variable in time.

Impact of iterative reconstruction algorithms on the applicability of Fourier-based detectability index for x-ray CT imaging.

The increasing application of iterative reconstruction algorithms in clinical computed tomography to improve image quality and reduce radiation dose, elicits strong interest, and needs model observers to optimize CT scanning protocols objectively and efficiently. The current paradigm for evaluating imaging system performance relies on Fourier methods, which presuppose a linear, wide-sense stationary system. Long-range correlations introduced by iterative reconstruction algorithms may narrow the applicability of Fourier techniques. Differences in the implementation of reconstruction algorithms between manufacturers add further complexity. The present work set out to quantify the errors entailed by the use of Fourier methods, which can lead to design decisions that do not correlate with detectability. To address this question, we evaluated the noise properties and the detectability index of the ideal linear observer using the spatial approach and the Fourier-based approach. For this purpose, a homogeneous phantom was imaged on two scanners: the Revolution CT (GE Healthcare) and the Somatom Definition AS+ (Siemens Healthcare) at different exposure levels. Images were reconstructed using different strength levels of IR algorithms available on the systems considered: Adaptative Statistical Iterative Reconstruction (ASIR-V) and Sinogram Affirmed Iterative Reconstruction (SAFIRE). Our findings highlight that the spatial domain estimate of the detectability index is higher than the Fourier domain estimate. This trend is found to be dependent on the specific regularization used by IR algorithms as well as the signal to be detected. The eigenanalysis of the noise covariance matrix and of its circulant approximation yields explanation about the evoked trends. In particular, this analysis suggests that the predictive power of the Fourier-based ideal linear observer depends on the ability of each basis analyzed to be relevant to the signal to be detected. The applicability of Fourier techniques is dependent on the specific regularization used by IR algorithms. These results argue for verifying the assumptions made when using Fourier methods since Fourier-task-based detectability index does not always correlate with signal detectability.

Iterative wave-front reconstruction in the Fourier domain

The use of Fourier methods in wave-front reconstruction can significantly reduce the computation time for large telescopes with a high number of degrees of freedom. However, Fourier algorithms for discrete data require a rectangular data set which conform to specific boundary requirements, whereas wave-front sensor data is typically defined over a circular domain (the telescope pupil). Here we present an iterative Gerchberg routine modified for the purposes of discrete wave-front reconstruction which adapts the measurement data (wave-front sensor slopes) for Fourier analysis, fulfilling the requirements of the fast Fourier transform (FFT) and providing accurate reconstruction. The routine is used in the adaptation step only and can be coupled to any other Wiener-like or least-squares method. We compare simulations using this method with previous Fourier methods and show an increase in performance in terms of Strehl ratio and a reduction in noise propagation for a 40×40 SPHERE-like adaptive optics system. For closed loop operation with minimal iterations the Gerchberg method provides an improvement in Strehl, from 95.4% to 96.9% in K-band. This corresponds to ~ 40 nm improvement in rms, and avoids the high spatial frequency errors present in other methods, providing an increase in contrast towards the edge of the correctable band.

A Limit Theorem for a Weiss Epidemic Process

For a Markov two-dimensional death-process of a special class we consider the use of Fourier methods to obtain an exact solution of the Kolmogorov equations for the exponential (double) generating function of the transition probabilities. Using special functions, we obtain an integral representation for the generating function of the transition probabilities. We state the expression of the expectation and variance of the stochastic process and establish a limit theorem.

A Limit Theorem for a Weiss Epidemic Process

For a Markov two-dimensional death-process of a special class we consider the use of Fourier methods to obtain an exact solution of the Kolmogorov equations for the exponential (double) generating function of the transition probabilities. Using special functions, we obtain an integral representation for the generating function of the transition probabilities. We state the expression of the expectation and variance of the stochastic process and establish a limit theorem.

One shot profilometry using phase partitions

Shape measurement using structured light systems involves the difficulty of detecting sharp discontinuities higher than one period of the projected fringe pattern; moreover, phase unwrapping becomes a problem. In this paper, a method to retrieve surface topography trough the projection of a single fringe pattern in gray levels is proposed. The correct phase is unwrapped through the use of Fourier methods and partition functions obtained from the phase. Experimental results show that the method can deal with the projection of high frequency fringes, being limited mainly by the resolution of the projector–detector system.

Optimal tuning of the tunable translational-rotational dynamic absorbers in global vibration control problems in beams

The paper deals with vibration of the beam subjected to the concentrated and distributed harmonic forces, with the translational-rotational dynamic vibration absorbers attached. The linear Euler-Bernoulli beam model is assumed, the solution of the equation of motion is found with the use of Fourier method. The timeLaplace transformation allows to obtain the displacement amplitude in the frequency domain. Numerical examples present a problem of minimization of the kinetic energy of the whole or the part of the beam by optimal tuning the translational-rotational dynamic vibration absorber. The results of calculation confirm the effectiveness of using the translational-rotational dynamic absorbers in beams. In the examples studied the optimal tuning strategies minimizing the kinetic energy are determined.

Fast and Accurate Evaluation of Nonlocal Coulomb and Dipole-Dipole Interactions via the Nonuniform FFT

We present a fast and accurate algorithm for the evaluation of nonlocal (long-range) Coulomb and dipole-dipole interactions in free space. The governing potential is simply the convolution of an interaction kernel $U(\mathbf{x})$ and a density function $\rho(\mathbf{x})=|\psi(\mathbf{x})|^2$ for some complex-valued wave function $\psi(\mathbf{x})$, permitting the formal use of Fourier methods. These are hampered by the fact that the Fourier transform of the interaction kernel $\widehat{U}(\mathbf{k})$ has a singularity and/or $\widehat{\rho}(\mathbf{k})\ne0$ at the origin $\mathbf{k}={\bf 0}$ in Fourier (phase) space. Thus, accuracy is lost when using a uniform Cartesian grid in $\mathbf{k}$ which would otherwise permit the use of the FFT for evaluating the convolution. Here, we make use of a high-order discretization of the Fourier integral, accelerated by the nonuniform fast Fourier transform (NUFFT). By adopting spherical and polar phase-space discretizations in three and two dimensions, respectively, the singularity in $\hat{U}(\mathbf{k})$ at the origin is canceled so that only a modest number of degrees of freedom are required to evaluate the Fourier integral, assuming that the density function $(\rho\mathbf{x})$ is smooth and decays sufficiently quickly as $|\mathbf{x}| \rightarrow \infty$. More precisely, the calculation requires $O(N\log N)$ operations, where $N$ is the total number of discretization points in the computational domain. Numerical examples are presented to demonstrate the performance of the algorithm.

Detecting Change in Biological Rhythms: A Multivariate Permutation Test Approach to Fourier‐Transformed Data

Treatment‐related changes in neurobiological rhythms are of increasing interest to psychologists, psychiatrists, and biological rhythms researchers. New methods for analyzing change in rhythms are needed, as most common methods disregard the rich complexity of biological processes. Large time series data sets reflect the intricacies of underlying neurobiological processes, but can be difficult to analyze. We propose the use of Fourier methods with multivariate permutation test (MPT) methods for analyzing change in rhythms from time series data. To validate the use of MPT for Fourier‐transformed data, we performed Monte Carlo simulations and compared statistical power and family‐wise error for MPT to Bonferroni‐corrected and uncorrected methods. Results show that MPT provides greater statistical power than Bonferroni‐corrected tests, while appropriately controlling family‐wise error. We applied this method to human, pre‐ and post‐treatment, serially‐sampled neurotransmitter data to confirm the utility of this method using real data. Together, Fourier with MPT methods provides a statistically powerful approach for detecting change in biological rhythms from time series data.

Figures of merit for detectors in digital radiography. II. Finite number of secondaries and structured backgrounds.

The current paradigm for evaluating detectors in digital radiography relies on Fourier methods. Fourier methods rely on a shift-invariant and statistically stationary description of the imaging system. The theoretical justification for the use of Fourier methods is based on a uniform background fluence and an infinite detector. In practice, the background fluence is not uniform and detector size is finite. We study the effect of stochastic blurring and structured backgrounds on the correlation between Fourier-based figures of merit and Hotelling detectability. A stochastic model of the blurring leads to behavior similar to what is observed by adding electronic noise to the deterministic blurring model. Background structure does away with the shift invariance. Anatomical variation makes the covariance matrix of the data less amenable to Fourier methods by introducing long-range correlations. It is desirable to have figures of merit that can account for all the sources of variation, some of which are not stationary. For such cases, we show that the commonly used figures of merit based on the discrete Fourier transform can provide an inaccurate estimate of Hotelling detectability.

The Fringe Region Technique and the Fourier Method Used in the Direct Numerical Simulation of Spatially Evolving Viscous Flows

To eliminate the problem with artificial boundary conditions and facilitate the use of Fourier methods, the fringe region (or filter, damping layer, absorbing layer, sponge layer) technique has been used in direct simulations of transitional and turbulent boundary layers. Despite the fact that good computational results have been obtained with this technique, it is not fully understood. The analysis in this paper indicates that the primary importance of the fringe region technique is to damp out the deviation associated with large scales in the direction normal to the wall. The lack of boundary conditions is compensated by the knowledge of an exact solution in the fringe region of the computational domain. The upstream influence from the fringe region is small. Numerical experiments verifying the theoretical predictions are presented.

4] Artifacts of fourier series analysis

Publisher Summary This chapter reviews some of the artifacts of Fourier series analysis. In its simplest form, a Fourier series analysis consists of a three-step process: (1) the experimental data are approximated by a Fourier series; (2) the Fourier series is transformed into a power spectrum; and (3) the power spectrum is interpreted in terms of one, or more, frequencies with significant amplitudes. The basic idea of representing the experimental data by a Fourier series is to find an equation that is the sum of sine and cosine terms that passes through every data point. The Fourier series equations are usually represented as an infinite series of terms to describe an infinite number of data points. The chapter discusses the infinite series form and then describes the ways it can be modified to work with a finite number of discrete data points. Fourier series methods are both useful and powerful. However, problems might exist with any particular use of Fourier methods.

On the use of Fourier methods in the analysis of composite structures

Different types of Fourier synthesis for composite structures are discussed. They include the two-dimensional projection based on the reflections common to both sublattices, the three-dimensional projection, which shows the average structure of one sublattice and the projection down a main (external) axis of the second sublattice, and the full four-dimensional Fourier synthesis. The three-dimensional projection includes the internal coordinate of one of the sublattices and therefore gives direct information on the nature of the modulation. The use of Fourier methods in the solution of the average structure and the analysis of the modulations is discussed. Series-termination effects are shown to affect the height of the atom strings along the internal t coordinate.

The geodetic approximations in the conversion of geoid height to gravity anomaly by Fourier transform

Six sources of error in the use of Fourier methods for the conversion of geoid heights to gravity anomalies are considered. The errors due to spherical approximation are unimportant. The errors due to approximations in Stokes' integral may be eliminated by use of the gravity coating rather than the gravity anomaly. The chord-to-arc error and the truncation error may be reduced by using a reference field. Tapering of the edges of the measurement window reduces the truncation error. The long-wavelength components of the high degree spherical harmonics cause small offsets in the resulting gravity anomalies. The errors due to the plane approximation can be reduced by appropriate choice of map projection and area of integration.

Theoretical Determination of Exchange Integrals in Salts

The exchange integrals of the crystal salts were investigated making use of Fourier Method. The mathematical relations between the exchange integrals were found out by determining the stability conditions for the magnetic modes, following the requirement for which the quadratic form must be definite positive. The magnetic modes and the exchange integrals which are the solutions of an eigen value problem, were studied theoretically in case of magnetite crystal.

Structure Analysis by Electron Diffraction

B. K. Vainshtein Translated and Edited by E. Feigl and J. A. Spink Oxford: Pergamon Press. 1964. Pp. vi + 420. Price 100s. The analysis of crystal structure by electron diffraction, particularly by the use of Fourier methods, is a subject which was originally developed in Russia, notably by Vainshtein, Pinsker and Zvyagin, and quickly followed up in Australia, by Cowley, Rees, and Spink. It is fitting then, that the present book should be a Russo—Australian effort, written by one pioneer, and jointly translated by another.

Improved Resolution of X-Ray Diffraction Patterns of Protein Crystals at Low Temperature

POSSIBILITIES for X-ray crystal structure analysis of proteins are limited by the minimum spacings observed in diffraction. For many proteins, the minimum spacing is about 2.5 A.—a limit which does not permit the resolution of individual atoms in Fourier maps prepared from the data, even if these are accurately measured, and the problem of the determination of the phases of the X-ray diffraction spectra can be solved successfully. This lack of resolution puts a severe restriction on the use of Fourier methods for structure determination of proteins. For a few proteins, such as bovine pancreatic ribonuclease, data with a minimum spacing of 1.5 A. are available, so that resolution of atoms is possible in principle. Thus the study of protein structure is highly dependent on fortune in obtaining crystals permitting atomic resolution in their diffraction effects.

Some aspects of neutron single crystal analysis.

Abstract : A comparison is made of applications of neutron and x-ray scattering to single crystal analysis. The statistical convergence of neutron scattering factors in more or less random structures, such as KH2PO4 and NH4Cl, was demonstrated. The coupling of this convergence with the effects of natural thermal oscillation or artificial convergence factors will permit a rather general use of Fourier methods in neutron crystallography. Extinction effects in single crystals were negligible. Neutron scattering data were considered more amenable to algebraic phase-determining methods than x-ray data. Effects on structure analysis of variations of the neutron scattering power for various nuclei are discussed.