Showing papers on "Fourier transform published in 2010"

Electrochemical Impedance Spectroscopy

Abstract: This review describes recent advances in electrochemical impedance spectroscopy (EIS) with an emphasis on its novel applications to various electrochemistry-related problems. Section 1 discusses the development of new EIS techniques to reduce measurement time. For this purpose, various forms of multisine EIS techniques were first developed via a noise signal synthesized by mixing ac waves of various frequencies, followed by fast Fourier transform of the signal and the resulting current. Subsequently, an entirely new concept was introduced in which true white noise was used as an excitation source, followed by Fourier transform of both excitation and response signals. Section 2 describes novel applications of the newly developed techniques to time-resolved impedance measurements as well as to impedance imaging. Section 3 is devoted to recent applications of EIS techniques, specifically traditional measurements in various fields with a special emphasis on biosensor detections.

Numerical Simulation of Optical Wave Propagation With Examples in MATLAB

Abstract: Foundations of Scalar Diffraction Theory Digital Fourier Transforms Simple Computations Using Fourier Transforms Fraunhofer Diffraction and Lenses Imaging Systems and Aberrations Fresnel Diffraction in Vacuum Sampling Requirements for Fresnel Diffraction Relaxed Sampling Constraints with Partial Propagations Propagation Through Atmospheric Turbulence Appendix A: Function Definitions Appendix B: MATLAB Code Listings References Index

Asymmetric cryptosystem based on phase-truncated Fourier transforms.

Abstract: We propose an asymmetric cryptosystem based on a phase-truncated Fourier transform. With phase truncation in Fourier transform, one is able to produce an asymmetric ciphertext as real-valued and stationary white noise by using two random phase keys as public keys, while a legal user can retrieve the plaintext using another two different private phase keys in the decryption process. Owing to the nonlinear operation of phase truncation, high robustness against existing attacks could be achieved. A set of simulation results shows the validity of proposed asymmetric cryptosystem.

Nonlinear Ocean Waves and the Inverse Scattering Transform

Abstract: For more than 200 years, the Fourier Transform has been one of the most important mathematical tools for understanding the dynamics of linear wave trains. "Nonlinear Ocean Waves and the Inverse Scattering Transform" presents the development of the nonlinear Fourier analysis of measured space and time series, which can be found in a wide variety of physical settings including surface water waves, internal waves, and equatorial Rossby waves. This revolutionary development will allow hyperfast numerical modelling of nonlinear waves, greatly advancing our understanding of oceanic surface and internal waves. Nonlinear Fourier analysis is based upon a generalization of linear Fourier analysis referred to as the inverse scattering transform, the fundamental building block of which is a generalized Fourier series called the Riemann theta function. Elucidating the art and science of implementing these functions in the context of physical and time series analysis is the goal of this book. It presents techniques and methods of the inverse scattering transform for data analysis. Geared toward both the introductory and advanced reader venturing further into mathematical and numerical analysis, this book is suitable for classroom teaching as well as research.

Poisson-Gap Sampling and Forward Maximum Entropy Reconstruction for Enhancing the Resolution and Sensitivity of Protein NMR Data

Abstract: The Fourier transform has been the gold standard for transforming data from the time domain to the frequency domain in many spectroscopic methods, including NMR spectroscopy. While reliable, it has the drawback that it requires a grid of uniformely sampled data points, which is not efficient for decaying signals, and it also suffers from artifacts when dealing with nondecaying signals. Over several decades, many alternative sampling and transformation schemes have been proposed. Their common problem is that relative signal amplitudes are not well-preserved. Here we demonstrate the superior performance of a sine-weighted Poisson-gap distribution sparse-sampling scheme combined with forward maximum entropy (FM) reconstruction. While the relative signal amplitudes are well-preserved, we also find that the signal-to-noise ratio is enhanced up to 4-fold per unit of data acquisition time relative to traditional linear sampling.

Mining periodic behaviors for moving objects

Abstract: Periodicity is a frequently happening phenomenon for moving objects. Finding periodic behaviors is essential to understanding object movements. However, periodic behaviors could be complicated, involving multiple interleaving periods, partial time span, and spatiotemporal noises and outliers. In this paper, we address the problem of mining periodic behaviors for moving objects. It involves two sub-problems: how to detect the periods in complex movement, and how to mine periodic movement behaviors. Our main assumption is that the observed movement is generated from multiple interleaved periodic behaviors associated with certain reference locations. Based on this assumption, we propose a two-stage algorithm, Periodica, to solve the problem. At the first stage, the notion of observation spot is proposed to capture the reference locations. Through observation spots, multiple periods in the movement can be retrieved using a method that combines Fourier transform and autocorrelation. At the second stage, a probabilistic model is proposed to characterize the periodic behaviors. For a specific period, periodic behaviors are statistically generalized from partial movement sequences through hierarchical clustering. Empirical studies on both synthetic and real data sets demonstrate the effectiveness of our method.

Integral Geometry and Radon Transforms

Abstract: 1.- The Radon Transformon on Rn 1.1- Introduction 1.2- The Radon Transform: The Support Theorem 1.3- The Inversion Formula: Injectivity Questions 1.4- The Plancherel Formula 1.5- Radon Transform of Distribution 1.6- Integration over d-planes: X-Ray Transforms 1.7- Applications 2.- A Duality in Integral Geometry 2.1- Homogeneous Spaces in Duality 2.2- The Radon Transform for the Double Fibration: Principal Problems 2.3- Orbital Integrals 2.4- Examples of Radon Transforms for Homogeneous Spaces in Duality 3.- The Radon Transform on Two-Point Homogeneous Spaces 3.1- Spaces of Constant Curvature: Inversion and Support Theorems 3.2- Compact Two-Point Homogeneous Spaces: Applications 3.3- Noncompact Two-Point Homogeneous Spaces 3.4- Support Theorems Relative to Horocycles 4.- The X-Ray Transform on a Symmetric Space 4.1- Compact Symmetric Spaces: Injectivity and Local Inversion: Support Theorem 4.2- Noncompact Symmetric Spaces: Global Inversion and General Support Theorem 4.3- Maximal Tori and Minimal Spheres in Compact Symmetric Spaces 5.- Orbital Integrals 5.1- Isotropic Spaces 5.2- Orbital Integrals 5.3- Generalized Riesz Potentials 5.4- Determination of a Function from its Integral over Lorentzian Spheres 5.5- Orbital Integrands and Huygens' Principle 6.- The Mean-Value Operator 6.1- An Injectivity Result 6.2- Asgeirsson's Mean-Value Theorem Generalized 6.3- John's Indentities 7.- Fourier Transforms and Distribution: A Rapid Course 7.1-The Topology of Spaces D(Rn), E(Rn) and S(Rn) 7.2- Distribution 7.3- Convolutions 7.4- The Fourier Transform 7.5- Differential Operators with Constant Coefficients 7.6- Riesz Potentials 8.- Lie Transformation Groups and Differential Operators 8.1- Manifolds and Lie Groups 8.2- Lie Transformation Groups and Radon Transforms 9.- Bibliography 10.- Notational Conventions 11.- Index.

Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian

Abstract: The efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics. Previous wave equations which use time-domain fractional operators require the storage of the complete pressure field at previous time steps (such operators are convolution based). This makes them unsuitable for many three-dimensional problems of interest. Here, a wave equation that utilizes two lossy derivative operators based on the fractional Laplacian is derived. These operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into Fourier based pseudospectral and k-space methods without the increase in memory required by their time-domain fractional counterparts. A framework for encoding the developed wave equation using three coupled first-order constitutive equations is discussed, and the model is demonstrated through several one-, two-, and three-dimensional simulations.

Susceptibility mapping in the human brain using threshold-based k-space division.

Abstract: A method for calculating quantitative three-dimensional susceptibility maps from field measurements acquired using gradient echo imaging at high field is presented. This method is based on division of the three-dimensional Fourier transforms of high-pass-filtered field maps by a simple function that is the Fourier transform of the convolution kernel linking field and susceptibility, and uses k-space masking to avoid noise enhancement in regions where this function is small. Simulations were used to show that the method can be applied to data acquired from objects that are oriented at one angle or multiple angles with respect to the applied field and that the use of multiple orientations improves the quality of the calculated susceptibility maps. As part of this work, we developed an improved approach for high-pass filtering of field maps, based on using an arrangement of dipoles to model the fields generated by external structures. This approach was tested on simulated field maps from the substantia nigra and red nuclei. Susceptibility mapping was successfully applied to experimental measurements on a structured phantom and then used to make measurements of the susceptibility of the red nuclei and substantia nigra in healthy subjects at 3 and 7 T. Magn Reson Med 63:1292–1304, 2010. © 2010 Wiley-Liss, Inc.

Programmable unitary spatial mode manipulation

Abstract: Free space propagation and conventional optical systems such as lenses and mirrors all perform spatial unitary transforms. However, the subset of transforms available through these conventional systems is limited in scope. We present here a unitary programmable mode converter (UPMC) capable of performing any spatial unitary transform of the light field. It is based on a succession of reflections on programmable deformable mirrors and free space propagation. We first show theoretically that a UPMC without limitations on resources can perform perfectly any transform. We then build an experimental implementation of the UPMC and show that, even when limited to three reflections on an array of 12 pixels, the UPMC is capable of performing single mode tranforms with an efficiency greater than 80% for the first four modes of the transverse electromagnetic basis.

Synchrosqueezing-based Recovery of Instantaneous Frequency from Nonuniform Samples

Abstract: We propose a new approach for studying the notion of the instantaneous frequency of a signal. We build on ideas from the Synchrosqueezing theory of Daubechies, Lu and Wu and consider a variant of Synchrosqueezing, based on the short-time Fourier transform, to precisely define the instantaneous frequencies of a multi-component AM-FM signal. We describe an algorithm to recover these instantaneous frequencies from the uniform or nonuniform samples of the signal and show that our method is robust to noise. We also consider an alternative approach based on the conventional, Hilbert transform-based notion of instantaneous frequency to compare to our new method. We use these methods on several test cases and apply our results to a signal analysis problem in electrocardiography.

Short-Time Fractional Fourier Transform and Its Applications

Abstract: The fractional Fourier transform (FRFT) is a potent tool to analyze the chirp signal. However, it fails in locating the fractional Fourier domain (FRFD)-frequency contents which is required in some applications. The short-time fractional Fourier transform (STFRFT) is proposed to solve this problem. It displays the time and FRFD-frequency information jointly in the short-time fractional Fourier domain (STFRFD). Two aspects of its performance are considered: the 2-D resolution and the STFRFD support. The time-FRFD-bandwidth product (TFBP) is defined to measure the resolvable area and the STFRFD support. The optimal STFRFT is obtained with the criteria that maximize the 2-D resolution and minimize the STFRFD support. Its inverse transform, properties and computational complexity are presented. Two applications are discussed: the estimations of the time-of-arrival (TOA) and pulsewidth (PW) of chirp signals, and the STFRFD filtering. Simulations verify the validity of the proposed algorithms.

A Fourier Transform Method for Spread Option Pricing

Abstract: Spread options are a fundamental class of derivative contracts written on multiple assets and are widely traded in a range of financial markets. There is a long history of approximation methods for computing such products, but as yet there is no preferred approach that is accurate, efficient, and flexible enough to apply in general asset models. The present paper introduces a new formula for general spread option pricing based on Fourier analysis of the payoff function. Our detailed investigation, including a flexible and general error analysis, proves the effectiveness of a fast Fourier transform implementation of this formula for the computation of spread option prices. It is found to be easy to implement, stable, efficient, and applicable in a wide variety of asset pricing models.

Analysis of Fourier Transform Valuation Formulas and Applications

Abstract: The aim of this article is to provide a systematic analysis of the conditions such that Fourier transform valuation formulas are valid in a general framework; i.e. when the option has an arbitrary payoff function and depends on the path of the asset price process. An interplay between the conditions on the payoff function and the process arises naturally. We also extend these results to the multi-dimensional case and discuss the calculation of Greeks by Fourier transform methods. As an application, we price options on the minimum of two assets in Levy and stochastic volatility models.

Compressive Fresnel Holography

Abstract: Compressive sensing is a relatively new measurement paradigm which seeks to capture the “essential” aspects of a high-dimensional object using as few measurements as possible. In this work we demonstrate successful application of compressive sensing framework to digital Fresnel holography. It is shown that when applying compressive sensing approach to Fresnel fields a special sampling scheme should be adopted for improved results.

Time-Frequency Analysis of Power-Quality Disturbances via the Gabor–Wigner Transform

Abstract: Recently, many signal-processing techniques, such as fast Fourier transform, short-time Fourier transform, wavelet transform (WT), and wavelet packet transform (WPT), have been applied to detect, identify, and classify power-quality (PQ) disturbances. For research on PQ analysis, it is critical to apply the appropriate signal-processing techniques to solve PQ problems. In this paper, a new time-frequency analysis method, namely, the Gabor-Wigner transform (GWT), is introduced and applied to detect and identify PQ disturbances. Since GWT is an operational combination of the Gabor transform (GT) and the Wigner distribution function (WDF), it can overcome the disadvantages of both. GWT has two advantages which are that it has fewer cross-term problems than the WDF and higher clarity than the GT. Studies are presented which verify that the merits of GWT make it adequate for PQ analysis. In the case studies considered here, the various PQ disturbances, including voltage swell, voltage sag, harmonics, interharmonics, transients, voltage changes with multiple frequencies and voltage fluctuation, or flicker, will be thoroughly investigated by using this new time-frequency analysis method.

Heavy Petroleum Composition. 1. Exhaustive Compositional Analysis of Athabasca Bitumen HVGO Distillates by Fourier Transform Ion Cyclotron Resonance Mass Spectrometry: A Definitive Test of the Boduszynski Model

Abstract: Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR MS) allows detailed characterization of complex petroleum samples at the level of elemental composition assignment. Ultrahigh-resolution (450 000−650 000 at m/z 500) enables identification of isobaric species that differ in mass by 3 mDa or less, and high mass accuracy (mass error of better than 300 ppb), combined with Kendrick mass sorting, allows for unambiguous molecular formula assignment to each of more than 10 000−20 000 peaks in each mass spectrum. Thus, it is possible to identify, sort, and monitor thousands of elemental compositions simultaneously, as a function of the boiling point. Here, the detailed FT-ICR MS characterization of an Athabasca bitumen heavy vacuum gas oil (HVGO) distillation series exposes the progression of heteroatom class, type (double bond equivalents (DBE), number of rings plus double bonds to carbon), and carbon number for tens of thousands of crude oil species, as a function of the boiling point. Specific...

Robust FFT-Based Scale-Invariant Image Registration with Image Gradients

Abstract: We present a robust FFT-based approach to scale-invariant image registration. Our method relies on FFT-based correlation twice: once in the log-polar Fourier domain to estimate the scaling and rotation and once in the spatial domain to recover the residual translation. Previous methods based on the same principles are not robust. To equip our scheme with robustness and accuracy, we introduce modifications which tailor the method to the nature of images. First, we derive efficient log-polar Fourier representations by replacing image functions with complex gray-level edge maps. We show that this representation both captures the structure of salient image features and circumvents problems related to the low-pass nature of images, interpolation errors, border effects, and aliasing. Second, to recover the unknown parameters, we introduce the normalized gradient correlation. We show that, using image gradients to perform correlation, the errors induced by outliers are mapped to a uniform distribution for which our normalized gradient correlation features robust performance. Exhaustive experimentation with real images showed that, unlike any other Fourier-based correlation techniques, the proposed method was able to estimate translations, arbitrary rotations, and scale factors up to 6.

Hyperspectral Region Classification Using a Three-Dimensional Gabor Filterbank

Abstract: A 3-D spectral/spatial discrete Fourier transform can be used to represent a hyperspectral image region using a dense sampling in the frequency domain. In many cases, a more compact frequency-domain representation that preserves the 3-D structure of the data can be exploited. For this purpose, we have developed a new model for spectral/spatial information based on 3-D Gabor filters. These filters capture specific orientation, scale, and wavelength-dependent properties of hyperspectral image data and provide an efficient means of sampling a 3-D frequency-domain representation. Since 3-D Gabor filters allow for a large number of spectral/spatial features to be used to represent an image region, the performance and efficiency of algorithms that use this representation can be further improved if methods are available to reduce the size of the model. Thus, we have derived methods for selecting features that emphasize the most significant spectral/spatial differences between the various classes in a scene. We demonstrate the performance of the 3-D Gabor features for the classification of regions in Airborne Visible/Infrared Imaging Spectrometer hyperspectral data. The new features are compared against pure spectral features and multiband generalizations of gray-level co-occurrence matrix features.

On the Fourier Extension of Nonperiodic Functions

Abstract: We obtain exponentially accurate Fourier series for nonperiodic functions on the interval $[-1,1]$ by extending these functions to periodic functions on a larger domain. The series may be evaluated, but not constructed, by means of the FFT. A complete convergence theory is given based on orthogonal polynomials that resemble Chebyshev polynomials of the first and second kinds. We analyze a previously proposed numerical method, which is unstable in theory but stable in practice. We propose a new numerical method that is stable both in theory and in practice.

Antileakage Fourier transform for seismic data regularization in higher dimensions

Abstract: Wide-azimuth seismic data sets are generally acquired more sparsely than narrow-azimuth seismic data sets. This brings new challenges to seismic data regularization algorithms, which aim to reconstruct seismic data for regularly sampled acquisition geometries from seismic data recorded from irregularly sampled acquisition geometries. The Fourier-basedseismicdataregularizationalgorithmfirstestimates the spatial frequency content on an irregularly sampled input grid. Then, it reconstructs the seismic data on any desired grid. Three main difficulties arise in this process: the “spectral leakage” problem, the accurate estimation of Fourier components, and the effective antialiasing scheme used inside the algorithm. The antileakage Fourier transform algorithm can overcome the spectral leakage problem and handles aliased data. To generalize it to higher dimensions, we propose an area weighting scheme to accurately estimate the Fourier components. However, the computational cost dramatically increases with the sampling dimensions. A windowed Fourier transform reduces the computational cost in high-dimension applications but causes undersampling in wavenumber domain and introduces some artifacts, known as Gibbs phenomena.As a solution, we propose a wavenumber domain oversampling inversion scheme. The robustness and effectiveness of the proposed algorithm are demonstrated with some applications to both synthetic and real data examples.

Resonance lineshapes in two-dimensional Fourier transform spectroscopy

Abstract: We derive an analytical form for resonance lineshapes in two-dimensional (2D) Fourier transform spectroscopy. Our starting point is the solution of the optical Bloch equations for a two-level system in the 2D time domain. Application of the projection-slice theorem of 2D Fourier transforms reveals the form of diagonal and cross-diagonal slices in the 2D frequency data for arbitrary inhomogeneity. The results are applied in quantitative measurements of homogeneous and inhomogeneous broadening of multiple resonances in experimental data.

Nonequispaced curvelet transform for seismic data reconstruction: A sparsity-promoting approach

Abstract: We extend our earlier work on the nonequispaced fast discrete curvelet transform (NFDCT) and introduce a second generation of the transform. This new generation differs from the previous one by the approach taken to compute accurate curvelet coefficients from irregularly sampled data. The first generation relies on accurate Fourier coefficients obtained by an l2 -regularized inversion of the nonequispaced fast Fourier transform (FFT) whereas the second is based on a direct l1 -regularized inversion of the operator that links curvelet coefficients to irregular data. Also, by construction the second generation NFDCT is lossless unlike the first generation NFDCT. This property is particularly attractive for processing irregularly sampled seismic data in the curvelet domain and bringing them back to their irregular record-ing locations with high fidelity. Secondly, we combine the second generation NFDCT with the standard fast discrete curvelet transform (FDCT) to form a new curvelet-based method, coined noneq...

Cluster expansion and the configurational theory of alloys

Abstract: A rigorous mathematical foundation for the cluster-expansion method is presented It is shown that the cluster basis developed by Sanchez et al [Physica A 128, 334 (1984)] is a multidimensional discrete Fourier transform while the general formalism of Sanchez [Phys Rev B 48, 14013 (1993)] corresponds to a multidimensional discrete wavelet transform For functions that depend nonlinearly on the concentration, it is shown that the cluster basis corresponding to a multidimensional discrete Fourier transform does not converge, as it is usually assumed, to a finite cluster expansion or to an Ising-type model representation of the energy of formation of alloys The multidimensional wavelet transform, based on a variable basis cluster expansion, is shown to provide a satisfactory solution to the deficiencies of the discrete Fourier-transform approach Several examples aimed at illustrating the main findings and conclusions of this work are given

Constructible exponential functions, motivic Fourier transform and transfer principle

Abstract: We introduce spaces of exponential constructible functions in the motivic setting for which we construct direct image functors in the absolute and relative settings. This allows us to define a motivic Fourier transformation for which we get various inversion statements. We also define spaces of motivic Schwartz-Bruhat functions on which motivic Fourier transformation induces isomorphisms. Our motivic integrals specialize to nonarchimedean integrals. We give a general transfer principle comparing identities between functions defined by exponential integrals over local fields of characteristic zero, resp. of positive characteristic, having the same residue field. We also prove new results about p-adic integrals of exponential functions and stability of this class of functions under p-adic integration.

Classical electrodynamics coupled to quantum mechanics for calculation of molecular optical properties: A RT-TDDFT/FDTD approach

Abstract: A new multiscale computational methodology was developed to effectively incorporate the scattered electric field of a plasmonic nanoparticle into a quantum mechanical (QM) optical property calculation for a nearby dye molecule. For a given location of the dye molecule with respect to the nanoparticle, a frequency-dependent scattering response function was first determined by the classical electrodynamics (ED) finite-difference time-domain (FDTD) approach. Subsequently, the time-dependent scattered electric field at the dye molecule was calculated using the FDTD scattering response function through a multidimensional Fourier transform to reflect the effect of polarization of the nanoparticle on the local field at the molecule. Finally, a real-time time-dependent density function theory (RT-TDDFT) approach was employed to obtain a desired optical property (such as absorption cross section) of the dye molecule in the presence of the nanoparticle’s scattered electric field. Our hybrid QM/ED methodology was de...

Drawing an elephant with four complex parameters

Abstract: We define four complex numbers representing the parameters needed to specify an elephantine shape. The real and imaginary parts of these complex numbers are the coefficients of a Fourier coordinate expansion, a powerful tool for reducing the data required to define shapes.