Showing papers on "Fourier transform published in 2004"

Extensions of compressed sensing

Abstract: We study the notion of compressed sensing (CS) as put forward by Donoho, Candes, Tao and others. The notion proposes a signal or image, unknown but supposed to be compressible by a known transform, (e.g. wavelet or Fourier), can be subjected to fewer measurements than the nominal number of data points, and yet be accurately reconstructed. The samples are nonadaptive and measure 'random' linear combinations of the transform coefficients. Approximate reconstruction is obtained by solving for the transform coefficients consistent with measured data and having the smallest possible l1 norm.We present initial 'proof-of-concept' examples in the favorable case where the vast majority of the transform coefficients are zero. We continue with a series of numerical experiments, for the setting of lp-sparsity, in which the object has all coefficients nonzero, but the coefficients obey an lp bound, for some p ∈ (0, 1]. The reconstruction errors obey the inequalities paralleling the theory, seemingly with well-behaved constants.We report that several workable families of 'random' linear combinations all behave equivalently, including random spherical, random signs, partial Fourier and partial Hadamard.We next consider how these ideas can be used to model problems in spectroscopy and image processing, and in synthetic examples see that the reconstructions from CS are often visually "noisy". To suppress this noise we postprocess using translation-invariant denoising, and find the visual appearance considerably improved.We also consider a multiscale deployment of compressed sensing, in which various scales are segregated and CS applied separately to each; this gives much better quality reconstructions than a literal deployment of the CS methodology.These results show that, when appropriately deployed in a favorable setting, the CS framework is able to save significantly over traditional sampling, and there are many useful extensions of the basic idea.

Harmonic analysis

Abstract: This paper describes a method of calculating the transforms, currently obtained via Fourier and reverse Fourier transforms. The method allows calculating efficiently the transforms of a signal having an arbitrary dimension of the digital representation by reducing the transform to a vector-to-circulant matrix multiplying. There is a connection between harmonic equations in rectangular and polar coordinate systems. The connection established here and used to create a very robust recursive algorithm for a conformal mapping calculation. There is also suggested a new ratio (and an efficient way of computing it) of two oscillative signal.

Accelerating the Nonuniform Fast Fourier Transform

Abstract: The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain. When the sampling is uniform, the fast Fourier transform (FFT) allows this calculation to be computed in O(N log N ) operations rather than O(N 2 ) operations. Unfortunately, when the sampling is nonuniform, the FFT does not apply. Over the last few years, a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform FFTs (NUFFTs). These rely on a mixture of interpolation and the judicious use of the FFT on an oversampled grid (A. Dutt and V. Rokhlin, SIAM J. Sci. Comput., 14 (1993), pp. 1368-1383). In this paper, we observe that one of the standard interpolation or "gridding" schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights. This is of particular value in two- and three- dimensional settings, saving either 10 d N in storage in d dimensions or a factor of about 5-10 in CPUtime (independent of dimension).

Lensless imaging of magnetic nanostructures by X-ray spectro-holography

Abstract: Our knowledge of the structure of matter is largely based on X-ray diffraction studies of periodic structures and the successful transformation (inversion) of the diffraction patterns into real-space atomic maps. But the determination of non-periodic nanoscale structures by X-rays is much more difficult. Inversion of the measured diffuse X-ray intensity patterns suffers from the intrinsic loss of phase information, and direct imaging methods are limited in resolution by the available X-ray optics. Here we demonstrate a versatile technique for imaging nanostructures, based on the use of resonantly tuned soft X-rays for scattering contrast and the direct Fourier inversion of a holographically formed interference pattern. Our implementation places the sample behind a lithographically manufactured mask with a micrometre-sized sample aperture and a nanometre-sized hole that defines a reference beam. As an example, we have used the resonant X-ray magnetic circular dichroism effect to image the random magnetic domain structure in a Co/Pt multilayer film with a spatial resolution of 50 nm. Our technique, which is a form of Fourier transform holography, is transferable to a wide variety of specimens, appears scalable to diffraction-limited resolution, and is well suited for ultrafast single-shot imaging with coherent X-ray free-electron laser sources.

On the diffraction theory of optical images

Abstract: The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object. There results a four-fold integral involving these functions, and the complex conjugate functions of the latter two. This integral is evaluated in terms of the Fourier transforms of the coherence function, the diffraction distribution function and its complex conjugate. In fact, these transforms are respectively the distribution of intensity in an 'effective source', and the complex transmission of the optical system-they are the data initially known and are generally of simple form. A generalized 'transmission factor' is found which reduces to the known results in the simple cases of perfect coherence and complete incoherence. The procedure may be varied in a manner more suited to non-periodic objects. The theory is applied to study inter alia the influence of the method of illumination on the images of simple periodic structures and of an isolated line.

Handbook of Holographic Interferometry: Optical and Digital Methods

Abstract: Preface.1 Introduction.1.1 Scope of the Book.1.2 Historical Developments.1.3 Holographic Interferometry as a Measurement Tool.2 Optical Foundations of Holography.2.1 Light Waves.2.2 Interference of Light.2.3 Coherence.2.4 Scalar Diffraction Theory.2.5 Speckles.2.6 Holographic Recording and Optical Reconstruction.2.7 Elements of the Holographic Setup.2.8 CCD- and CMOS-Arrays.3 Digital Recording and Numerical Reconstruction of Wave Fields.3.1 Digital Recording of Holograms.3.2 Numerical Reconstruction by the Fresnel Transform.3.3 Numerical Reconstruction by the Convolution Approach.3.4 Further Numerical Reconstruction Methods.3.5 Wave-Optics Analysis of Digital Holography.3.6 Non-Interferometric Applications of Digital Holography.4 Holographic Interferometry.4.1 Generation of Holographic Interference Patterns.4.2 Variations of the Sensitivity Vectors.4.3 Fringe Localization.4.4 Holographic Interferometric Measurements.5 Quantitative Determination of the Interference Phase.5.1 Roleof Interference Phase.5.2 Disturbances of Holographic Interferograms.5.3 Fringe Skeletonizing.5.4 Temporal Heterodyning.5.5 Phase Sampling Evaluation.5.6 Fourier Transform Evaluation.5.7 Dynamic Evaluation.5.8 Digital Holographic Interferometry.5.9 Interference Phase Demodulation.6 Processing of the Interference Phase.6.1 Displacement Determination.6.2 TheSensitivity Matrix.6.3 Holographic Strain and Stress Analysis.6.4 Hybrid Methods.6.5 Vibration Analysis.6.6 Holographic Contouring.6.7 Contour Measurement by Digital Holography.6.8 Comparative Holographic Interferometry.6.9 Measurement Range Extension.6.10 Refractive Index Fields in Transparent Media.6.11 Defect Detection by Holographic Non-Destructive Testing.7 Speckle Metrology.7.1 Speckle Photography.7.2 Electronic and Digital Speckle Interferometry.7.3 Electro-optic Holography.7.4 Speckle Shearography.Appendix.A Signal Processing Fundamentals.A.1 Overview.A.2 Definition of the Fourier Transform.A.3 Interpretation of the Fourier Transform.A.4 Properties of the Fourier Transform.A.5 Linear Systems.A.6 Fourier Analysis of Sampled Functions.A.7 The Sampling Theorem and Data Truncation Effects.A.8 Interpolation and Resampling.A.9 Two-Dimensional Image Processing.A.10 The Fast Fourier Transform.A.11 Fast Fourier Transform for N!= 2n.A.12 Cosine and Hartley Transform.A.13 The Chirp Function and the Fresnel Transform.B Computer Aided Tomography.B.1 Mathematical Preliminaries.B.2 The Generalized Projection Theorem.B.3 Reconstruction by Filtered Backprojection.B.4 Practical Implementation of Filtered Backprojection .B.5 Algebraic Reconstruction Techniques.C Bessel FunctionsBibliographyAuthor IndexSubject Index.

Windowed Fourier transform for fringe pattern analysis

Abstract: Fringe patterns in optical metrology systems need to be demodulated to yield the desired parameters. Time-frequency analysis is a useful concept for fringe demodulation, and a windowed Fourier transform is chosen for the determination of phase and phase derivative. Two approaches are developed: the first is based on the concept of filtering the fringe patterns, and the second is based on the best match between the fringe pattern and computer-generated windowed exponential elements. I focus on the extraction of phase and phase derivatives from either phase-shifted fringe patterns or a single carrier fringe pattern. Principles as well as examples are given to show the effectiveness of the proposed methods.

ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems

Abstract: We propose an efficient, hybrid Fourier-wavelet regularized deconvolution (ForWaRD) algorithm that performs noise regularization via scalar shrinkage in both the Fourier and wavelet domains. The Fourier shrinkage exploits the Fourier transform's economical representation of the colored noise inherent in deconvolution, whereas the wavelet shrinkage exploits the wavelet domain's economical representation of piecewise smooth signals and images. We derive the optimal balance between the amount of Fourier and wavelet regularization by optimizing an approximate mean-squared error (MSE) metric and find that signals with more economical wavelet representations require less Fourier shrinkage. ForWaRD is applicable to all ill-conditioned deconvolution problems, unlike the purely wavelet-based wavelet-vaguelette deconvolution (WVD); moreover, its estimate features minimal ringing, unlike the purely Fourier-based Wiener deconvolution. Even in problems for which the WVD was designed, we prove that ForWaRD's MSE decays with the optimal WVD rate as the number of samples increases. Further, we demonstrate that over a wide range of practical sample-lengths, ForWaRD improves on WVD's performance.

Measurement and Data Analysis for Engineering and Science

Abstract: Fundamentals of Experimentation Introduction Experiments Chapter Overview Experimental Approach Role of Experiments The Experiment Classification of Experiments Plan for Successful Experimentation Hypothesis Testing* Design of Experiments* Factorial Design* Problems Bibliography Fundamental Electronics Chapter Overview Concepts and Definitions Circuit Elements RLC Combinations Elementary DC Circuit Analysis Elementary AC Circuit Analysis Equivalent Circuits* Meters* Impedance Matching and Loading Error* Electrical Noise* Problems Bibliography Measurement Systems: Sensors and Transducers Chapter Overview Measurement System Overview Sensor Domains Sensor Characteristics Physical Principles of Sensors Electric Piezoelectric Fluid Mechanic Optic Photoelastic Thermoelectric Electrochemical Sensor Scaling* Problems Bibliography Measurement Systems: Other Components Chapter Overview Signal Conditioning, Processing, and Recording Amplifiers Filters Analog-to-Digital Converters Smart Measurement Systems Other Example Measurement Systems Problems Bibliography Measurement Systems: Calibration and Response Chapter Overview Static Response Characterization by Calibration Dynamic Response Characterization Zero-Order System Dynamic Response First-Order System Dynamic Response Second-Order System Dynamic Response Measurement System Dynamic Response Problems Bibliography Measurement Systems: Design-Stage Uncertainty Chapter Overview Design-Stage Uncertainty Analysis Design-Stage Uncertainty Estimate of a Measurand Design-Stage Uncertainty Estimate of a Result Problems Bibliography Signal Characteristics Chapter Overview Signal Classification Signal Variables Signal Statistical Parameters Problems Bibliography The Fourier Transform Chapter Overview Fourier Series of a Periodic Signal Complex Numbers and Waves Exponential Fourier Series Spectral Representations Continuous Fourier Transform Continuous Fourier Transform Properties* Discrete Fourier Transform Fast Fourier Transform Problems Bibliography Digital Signal Analysis Chapter Overview Digital Sampling Digital Sampling Errors Windowing* Determining a Sample Period Problems Bibliography Probability Chapter Overview Relation to Measurements Basic Probability Concepts Sample versus Population Plotting Statistical Information Probability Density Function Various Probability Density Functions Central Moments Probability Distribution Function Problems Bibliography Statistics Chapter Overview Normal Distribution Normalized Variables Student's t Distribution Rejection of Data Standard Deviation of the Means Chi-Square Distribution Pooling Samples* Problems Bibliography Uncertainty Analysis Chapter Overview Modeling and Experimental Uncertainties Probabilistic Basis of Uncertainty Identifying Sources of Error Systematic and Random Errors Quantifying Systematic and Random Errors Measurement Uncertainty Analysis Uncertainty Analysis of a Multiple-Measurement Result Uncertainty Analyses for Other Measurement Situations Uncertainty Analysis Summary Finite-Difference Uncertainties* Uncertainty Based upon Interval Statistics* Problems Bibliography Regression and Correlation Chapter Overview Least-Squares Approach Least-Squares Regression Analysis Linear Analysis Higher-Order Analysis* Multi-Variable Linear Analysis* Determining the Appropriate Fit Regression Confidence Intervals Regression Parameters Linear Correlation Analysis Signal Correlations in Time* Problems Bibliography Units and Significant Figures Chapter Overview English and Metric Systems Systems of Units SI Standards Technical English and SI Conversion Factors Prefixes Significant Figures Problems Bibliography Technical Communication Chapter Overview Guidelines for Writing Technical Memo Technical Report Oral Technical Presentation Problems Bibliography A Glossary B Symbols C Review Problem Answers Index

Time-fractional telegraph equations and telegraph processes with brownian time

Abstract: We study the fundamental solutions to time-fractional telegraph equations of order 2α. We are able to obtain the Fourier transform of the solutions for any α and to give a representation of their inverse, in terms of stable densities. For the special case α=1/2, we can show that the fundamental solution is the distribution of a telegraph process with Brownian time. In a special case, this becomes the density of the iterated Brownian motion, which is therefore the fundamental solution to a fractional diffusion equation of order 1/2 with respect to time.

Linear phase imaging using differential interference contrast microscopy.

Abstract: We propose an extension to Nomarski differential interference contrast microscopy that enables isotropic linear phase imaging. The method combines phase shifting, two directions of shear and Fourier-space integration using a modified spiral phase transform. We simulated the method using a phantom object with spatially varying amplitude and phase. Simulated results show good agreement between the final phase image and the object phase, and demonstrate resistance to imaging noise.

Full range polarization-sensitive Fourier domain optical coherence tomography

Abstract: A swept source based polarization-sensitive Fourier domain optical coherence tomography (FDOCT) system was developed that can acquire the Stokes vectors, polarization diversity intensity and birefringence images in biological tissue by reconstruction of both the amplitude and phase terms of the interference signal. The Stokes vectors of the reflected and backscattered light from the sample were determined by processing the analytical complex fringe signals from two perpendicular polarizationdetection channels. Conventional time domain OCT (TDOCT) and spectrometer based FDOCT systems are limited by the fact that the input polarization states are wavelength dependent. The swept source based FDOCT system overcomes this limitation and allows accurate setting of the input polarization states. From the Stokes vectors for two different input polarization states, the polarization diversity intensity and birefringence images were obtained.

Multiresolution S-transform-based fuzzy recognition system for power quality events

Abstract: The paper proposes a novel fuzzy pattern recognition system for power quality disturbances. It is a two-stage system in which a mulitersolution S-transform is used to generate a set of optimal feature vectors in the first stage. The multiresolution S-transform is based on a variable width analysis window, which changes with frequency according to a user-defined function. Thus, the resolution in time or the related resolution in frequency is a general function of the frequency and two parameters, which can be chosen according to signal characteristics. The multiresolution S-transform can be seen either as a phase-corrected version of the wavelet transform or a variable window short time Fourier transform that simultaneously localizes both real and imaginary spectra of the signal. The features obtained from S-transform analysis of the power quality disturbance signals are much more amenable for pattern recognition purposes unlike the currently available wavelet transform techniques. In stage two, a fuzzy logic-based pattern recognition system is used to classify the various disturbance waveforms generated due to power quality violations. The fuzzy approach is found to be very simple and classification accuracy is more than 98% in most cases of power quality disturbances.

Detection and parameter estimation of multicomponent LFM signal based on the fractional Fourier transform

Abstract: This paper presents a new method for the detection and parameter estimation of multicomponent LFM signals based on the fractional Fourier transform. For the optimization in the fractional Fourier domain, an algorithm based on Quasi-Newton method is proposed which consists of two steps of searching, leading to a reduction in computation without loss of accuracy. And for multicomponent signals, we further propose a signal separation technique in the fractional Fourier domain which can effectively suppress the interferences on the detection of the weak components brought by the stronger components. The statistical analysis of the estimate errors is also performed which perfects the method theoretically, and finally, simulation results are provided to show the validity of our method.

Covariance nuclear magnetic resonance spectroscopy.

Abstract: Covariance nuclear magnetic resonance (NMR) spectroscopy is introduced, which is a new scheme for establishing nuclear spin correlations from NMR experiments. In this method correlated spin dynamics is directly displayed in terms of a covariance matrix of a series of one-dimensional (1D) spectra. In contrast to two-dimensional (2D) Fourier transform NMR, in a covariance spectrum the spectral resolution along the indirect dimension is determined by the favorable spectral resolution obtainable along the detection dimension, thereby reducing the time-consuming sampling requirement along the indirect dimension. The covariance method neither involves a second Fourier transformation nor does it require separate phase correction or apodization along the indirect dimension. The new scheme is demonstrated for cross-relaxation (NOESY) and J-coupling based magnetization transfer (TOCSY) experiments.

Projection−Reconstruction Technique for Speeding up Multidimensional NMR Spectroscopy

Abstract: The acquisition of multidimensional NMR spectra can be speeded up by a large factor by a projection-reconstruction method related to a technique used in X-ray scanners. The information from a small number of plane projections is used to recreate the full multidimensional spectrum in the familiar format. Projections at any desired angle of incidence are obtained by Fourier transformation of time-domain signals acquired when two or more evolution intervals are incremented simultaneously at different rates. The new technique relies on an established Fourier transform theorem that relates time-domain sections to frequency-domain projections. Recent developments in NMR instrumentation, such as increased resolution and sensitivity, make fast methods for data gathering much more practical for protein and RNA research. Hypercomplex Fourier transformation generates projections in symmetrically related pairs that provide two independent "views" of the spectrum. A new reconstruction algorithm is proposed, based on the inverse Radon transform. Examples are presented of three- and four-dimensional NMR spectra of nuclease A inhibitor reconstructed by this technique with significant savings in measurement time.

Fourier Analysis and Approximation of Functions

Abstract: 1. Representation Theorems.- 1.1 Theorems on representation at a point.- 1.2 Integral operators. Convergence in Lp-norm and almost everywhere.- 1.3 Multidimensional case.- 1.4 Further problems and theorems.- 1.5 Comments to Chapter 1.- 2. Fourier Series.- 2.1 Convergence and divergence.- 2.2 Two classical summability methods.- 2.3 Harmonic functions and functions analytic in the disk.- 2.4 Multidimensional case.- 2.5 Further problems and theorems.- 2.6 Comments to Chapter 2.- 3. Fourier Integral.- 3.1 L-Theory.- 3.2 L2-Theory.- 3.3 Multidimensional case.- 3.4 Entire functions of exponential type. The Paley-Wiener theorem.- 3.5 Further problems and theorems.- 3.6 Comments to Chapter 3.- 4. Discretization. Direct and Inverse Theorems.- 4.1 Summation formulas of Poisson and Euler-Maclaurin.- 4.2 Entire functions of exponential type and polynomials.- 4.3 Network norms. Inequalities of different metrics.- 4.4 Direct theorems of Approximation Theory.- 4.5 Inverse theorems. Constructive characteristics. Embedding theorems.- 4.6 Moduli of smoothness.- 4.7 Approximation on an interval.- 4.8 Further problems and theorems.- 4.9 Comments to Chapter 4.- 5. Extremal Problems of Approximation Theory.- 5.1 Best approximation.- 5.2 The space Lp. Best approximation.- 5.3 Space C. The Chebyshev alternation.- 5.4 Extremal properties for algebraic polynomials and splines.- 5.5 Best approximation of a set by another set.- 5.6 Further problems and theorems.- 5.7 Comments to Chapter 5.- 6. A Function as the Fourier Transform of A Measure.- 6.1 Algebras A and B. The Wiener Tauberian theorem.- 6.2 Positive definite and completely monotone functions.- 6.3 Positive definite functions depending only on a norm.- 6.4 Sufficient conditions for belonging to Ap and A*.- 6.5 Further problems and theorems.- 6.6 Comments to Chapter 6.- 7. Fourier Multipliers.- 7.1 General properties.- 7.2 Sufficient conditions.- 7.3 Multipliers of power series in the Hardy spaces.- 7.4 Multipliers and comparison of summability methods of orthogonal series.- 7.5 Further problems and theorems.- 7.6 Comments to Chapter 7.- 8. Summability Methods. Moduli of Smoothness.- 8.1 Regularity.- 8.2 Applications of comparison. Two-sided estimates.- 8.3 Moduli of smoothness and K-functionals.- 8.4 Moduli of smoothness and strong summability in Hp(D), 0erences.- Author Index.- Topic Index.

Wavelet deconvolution in a periodic setting

Abstract: In this paper, we present an inverse estimation procedure which combines Fourier analysis with wavelet expansion. In the periodic setting, our method can recover a blurred function observed in white noise. The blurring process is achieved through a convolution operator which can either be smooth (polynomial decay of the Fourier transform) or irregular (such as the convolution with a box-car). The proposal is non-linear and does not require any prior knowledge of the smoothness class; it enjoys fast computation and is spatially adaptive. This contrasts with more traditional ltering methods which demand a certain amount of regularisation and often fail to recover non-homogeneous functions. A ne tuning of our method is derived via asymptotic minimax theory which reveals some key dierences with the direct case of Donoho et al. (1995): (a) band-limited wavelet families have nice theoretical and computing features; (b) the high frequency cut o depends on the spectral characteristics of the convolution kernel; (c) thresholds are level dependent in a geometric fashion. We tested our method using simulated lidar data for underwater remote sensing. Both visual and numerical results show an improvement over existing methods. Finally, the theory behind our estimation paradigm gives a complete characterisation of the ’Maxiset’ of the method i.e. the set of functions where the method attains a near-optimal rate of convergence for a variety of L p loss functions.

Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry

Abstract: We present an analysis of a spatial carrier-fringe pattern in three-dimensional (3-D) shape measurement by using the wavelet transform, a tool excelling for its multiresolution in the time- and space-frequency domains. To overcome the limitation of the Fourier transform, we introduce the Gabor wavelet to analyze the phase distributions of the spatial carrier-fringe pattern. The theory of wavelet transform profilometry, an accuracy check by means of a simulation, and an example of 3-D shape measurement are shown.

Quantum process tomography of the quantum Fourier transform

Abstract: The results of quantum process tomography on a three-qubit nuclear magnetic resonance quantum information processor are presented and shown to be consistent with a detailed model of the system-plus-apparatus used for the experiments. The quantum operation studied was the quantum Fourier transform, which is important in several quantum algorithms and poses a rigorous test for the precision of our recently developed strongly modulating control fields. The results were analyzed in an attempt to decompose the implementation errors into coherent (overall systematic), incoherent (microscopically deterministic), and decoherent (microscopically random) components. This analysis yielded a superoperator consisting of a unitary part that was strongly correlated with the theoretically expected unitary superoperator of the quantum Fourier transform, an overall attenuation consistent with decoherence, and a residual portion that was not completely positive—although complete positivity is required for any quantum operat...

Mode-switching and nonlinear effects in compressible flow over a cavity

Abstract: Multiple distinct peaks of comparable strength in unsteady pressure autospectra often characterize compressible flow-induced cavity oscillations. It is unclear whether these different large-amplitude tones (i.e., Rossiter modes) coexist or are the result of a mode-switching phenomenon. The cause of additional peaks in the spectrum, particularly at low frequency, is also unknown. This article describes the analyses of unsteady pressure data in a cavity using time-frequency methods, namely the short-time Fourier transform (STFT) and the continuous Morlet wavelet transform, and higher-order spectral techniques. The STFT and wavelet analyses clearly show that the dominant mode switches between the primary Rossiter modes. This is verified by instantaneous schlieren images acquired simultaneously with the unsteady pressures. Furthermore, the Rossiter modes experience some degree of low-frequency amplitude modulation. An estimate of the modulation frequency, obtained from the wavelet analysis, matches the low-fr...

Spaces of Test Functions via the STFT

Abstract: We characterize several classes of test functions, among them Bjorck's ultra-rapidly decaying test functions and the Gelfand-Shilov spaces of type S, in terms of the decay of their short-time Fourier transform and in terms of their Gabor coefficients.

Fourier transform imaging of spin vortex eigenmodes.

Abstract: Thin-circular lithographically defined magnetic elements with a spin vortex configuration are excited with a short perpendicular magnetic field pulse. We report the first images of excited magnetic eigenmodes up to third order, obtained by means of a phase sensitive Fourier transform imaging technique. Both axially symmetric and symmetry breaking azimuthal eigenmodes are observed. We observe strong oscillations of the magnetization in the central part of the magnetic elements. The experimental data are in good agreement with micromagnetic simulations.

Multiresolution techniques for the detection of gravitational-wave bursts

Abstract: We present two search algorithms that implement logarithmic tiling of the time–frequency plane in order to efficiently detect astrophysically unmodelled bursts of gravitational radiation. The first is a straightforward application of the dyadic wavelet transform. The second is a modification of the windowed Fourier transform which tiles the time–frequency plane for a specific Q. In addition, we also demonstrate adaptive whitening by linear prediction, which greatly simplifies our statistical analysis. This is a methodology paper that aims to describe the techniques for identifying significant events as well as the necessary pre-processing that is required in order to improve their performance. For this reason we use simulated LIGO noise in order to illustrate the methods and to present their preliminary performance.

Review of iterative Fourier-transform algorithms for beam shaping applications

Abstract: We present a comparison of some of the most used iterative Fourier transform algorithms (IFTA) for the design of continuous and multilevel diffractive optical elements (DOE). Our aim is to provide optical engineers with advice for choosing the most suited algorithm with re- spect to the task. We tackle mainly the beam-shaping and the beam- splitting problems, where the desired light distributions are almost binary. We compare four recent algorithms, together with the historical error- reduction and input-output methods. We conclude that three of these algorithms are interesting for continuous-phase kinoforms, and two, namely the three-step method proposed by Wyrowski and the over- compensation of Prongue ´, still perform well with multilevel- and binary- phase DOE. © 2004 Society of Photo-Optical Instrumentation Engineers.

Practical Fourier Analysis for Multigrid Methods

Abstract: PRACTICAL APPLICATION OF LFA AND xlfa Introduction Some Notation Basic Iterative Schemes A First Discussion of Fourier Components From Residual Correction to Coarse-Grid Correction Multigrid Principle and Components A First Look at the Graphical User Interface Main Features of Local Fourier Analysis for Multigrid The Power of Local Fourier Analysis Basic Ideas Applicability of the Analysis Multigrid and Its Components in LFA Multigrid Cycling Full Multigrid xlfa Functionality-An Overview Implemented Coarse-Grid Correction Components Implemented Relaxations Using the Fourier Analysis Software Case Studies for 2D Scalar Problems Case Studies for 3D Scalar Problems Case Studies for 2D SYSTEMS of Equations Creating New Applications THE THEORY BEHIND LFA Fourier One-Grid or Smoothing Analysis Elements of Local Fourier Analysis High and Low Fourier Frequencies Simple Relaxation Methods Pattern Relaxations Smoothing Analysis for Systems Multistage (MS) Relaxations Further Relaxation Methods The Measure of h-Ellipticity Fourier Two- and Three-Grid Analysis Basic Assumptions Two-Grid Analysis for 2D Scalar Problems Two-Grid Analysis for 3D Scalar Problems Two-Grid Analysis for Systems Three-Grid Analysis Further Applications of Local Fourier Analysis Orders of Transfer Operators Simplified Fourier k-Grid Analysis Cell-Centered Multigrid Fourier Analysis for Multigrid Preconditioned by GMRES APPENDIX Fourier Representation of Relaxation Two-Dimensional Case Three-Dimensional Case

A New Spectral Representation of Earthquake Data: Hilbert Spectral Analysis of Station TCU129, Chi-Chi, Taiwan, 21 September 1999

Abstract: A new method of spectral analysis, using an approach we call the empirical mode decomposition (EMD) and the Hilbert spectrum analysis (HSA), is presented. The EMD method decomposes any data into a finite number of intrinsic mode function (IMF) components with time-variable amplitudes and frequencies. This decomposition is nearly orthogonal and totally adaptive. With the decomposition, a Hilbert, rather than Fourier, transform is applied to each IMF component, which gives each component instantaneous frequency and energy density. This approach is totally new, and it is different from any of the existing methods: it uses differentiation to define the frequency rather than the traditional convolution computation; thus, it gives the instantaneous frequency and energy density. The greatest advantage of the new approach is that it is the only spectral analysis method applicable to nonstationary and nonlinear data. To illustrate the capability of his new method, we have applied it to the earthquake record from station TCU129, at Chi-Chi, Taiwan, collected during the 21 September 1999 earthquake. The same record is also analyzed with Fourier analysis, wavelet transform, and response spectrum analysis. Comparisons among the different analysis methods indicate that the Hilbert spectral analysis gives the most detailed information in a time-frequency-energy presentation. It also emphasizes the potentially damage-causing low-frequency energy in the earthquake signal missed by all the other methods. Manuscript received 19 December 2000.

Extended heat-fluctuation theorems for a system with deterministic and stochastic forces.

Abstract: Heat fluctuations over a time tau in a nonequilibrium stationary state and in a transient state are studied for a simple system with deterministic and stochastic components: a Brownian particle dragged through a fluid by a harmonic potential which is moved with constant velocity. Using a Langevin equation, we find the exact Fourier transform of the distribution of these fluctuations for all tau. By a saddle-point method we obtain analytical results for the inverse Fourier transform, which, for not too small tau, agree very well with numerical results from a sampling method as well as from the fast Fourier transform algorithm. Due to the interaction of the deterministic part of the motion of the particle in the mechanical potential with the stochastic part of the motion caused by the fluid, the conventional heat fluctuation theorem is, for infinite and for finite tau, replaced by an extended fluctuation theorem that differs noticeably and measurably from it. In particular, for large fluctuations, the ratio of the probability for absorption of heat (by the particle from the fluid) to the probability to supply heat (by the particle to the fluid) is much larger here than in the conventional fluctuation theorem.

Super-resolution overlay in multi-projector displays

Abstract: A technique, associated system and computer executable program code, for projecting a superimposed image onto a target display surface under observation of one or more cameras. A projective relationship between each projector being used and the target display surface is determined using a suitable calibration technique. A component image for each projector is then estimated using the information from the calibration, and represented in the frequency domain. Each component image is estimated by: Using the projective relationship, determine a set of sub-sampled, regionally shifted images, represented in the frequency domain; each component image is then composed of a respective set of the sub-sampled, regionally shifted images. In an optimization step, the difference between a sum of the component images and a frequency domain representation of a target image is minimized to produce a second, or subsequent, component image for each projector. Here, a second set of frequency domain coefficients for use in producing a frequency domain representation of the second component image for each projector is identified. Taking the inverse Fourier transform of the frequency domain representation of the second component image, converts the information into a spatial signal that is placed into the framebuffer of each component projector and projected therefrom to produce the superimposed image.