Fourier-based Methods Research Articles

Orbitals for classical arbitrary anisotropic colloidal potentials.

Coarse-grained potentials are ubiquitous in mesoscale simulations. While various methods to compute effective interactions for spherically symmetric particles exist, anisotropic interactions are seldom used, due to their complexity. Here we describe a general formulation, based on a spatial decomposition of the density fields around the particles, akin to atomic orbitals. We show that anisotropic potentials can be efficiently computed in numerical simulations using Fourier-based methods. We validate the field formulation and characterize its computational efficiency with a system of colloids that have Gaussian surface charge distributions. We also investigate the phase behavior of charged Janus colloids immersed in screened media, with screening lengths comparable to the colloid size. The system shows rich behaviors, exhibiting vapor, liquid, gel, and crystalline morphologies, depending on temperature and screening length. The crystalline phase only appears for symmetric Janus particles. For very short screening lengths, the system undergoes a direct transition from a vapor to a crystal on cooling; while, for longer screening lengths, a vapor-liquid-crystal transition is observed. The proposed formulation can be extended to model force fields that are time or orientation dependent, such as those in systems of polymer-grafted particles and magnetic colloids.

A multiple-curve Lévy forward rate model in a two-price economy

An advanced Heath–Jarrow–Morton forward rate model driven by time-inhomogeneous Lévy processes is presented which is able to handle the recent development to multiple curves and negative interest rates. It is also able to exploit bid and ask price data. In this approach in order to model spreads between curves for different tenors, credit as well as liquidity risk is taken into account. Deterministic conditions are derived to ensure the positivity of spreads and thus the monotonicity of the curves for the various tenors. Valuation formulas for standard interest rate derivatives such as caps, floors, swaptions and digital options are established. These formulas can be evaluated numerically very fast using Fourier-based valuation methods. In order to exploit bid and ask prices we develop this approach in the context of a two-price economy. Explicit formulas for bid as well as ask prices of the derivatives are stated. A specific model framework based on normal inverse Gaussian and Gamma processes is proposed which allows for calibration to market data. Calibration results are presented based on multiple-curve bootstrapping and cap market quotes. We use data from September 2013 as well as September 2016. The latter is of particular interest since rates were deep in negative territory at that time.

Detection of epileptic dysfunctions in EEG signals using Hilbert vibration decomposition

Abstract Epilepsy is a neurological brain dysfunction that is manifested by recrudescent seizures. Due to high temporal resolution, brain activities recorded by electroencephalography (EEG) are commonly used for localization of seizures and identification of epileptic dysfunctions. However, it is often time consuming and challenging to detect EEG seizures using conventional Fourier-based methods and manual interpretation due to nonlinear and nonstationary dynamics of EEG. In this paper, we propose a new framework based on Hilbert vibration decomposition (HVD) for discriminating normal and epileptic EEG recordings. HVD exploits Hilbert transform presentation of instantaneous frequency and extracts monocomponents that have distinctive time-varying amplitudes and instantaneous frequencies from nonstationary signals. The proposed method employs estimated instantaneous frequencies of largest energy components as features given to least squares support vector machine (LS-SVM) for recognizing epileptic seizures and is shown to be appealing for real time physiological signal processing applications due to its reduced computational complexity. Test results on a benchmark EEG data set achieved 97.66% classification accuracy and area of 0.9914 under the receiver operating characteristics (ROC) curve using the delta, theta and alpha rhythms.

On the wavelet-based SWIFT method for backward stochastic differential equations

We propose a numerical algorithm for backward stochastic differential equations based on time discretization and trigonometric wavelets. This method combines the effectiveness of Fourier-based methods and the simplicity of a wavelet-based formula, resulting in an algorithm that is both accurate and easy to implement. Furthermore, we mitigate the problem of errors near the computation boundaries by means of an antireflective boundary technique, giving an improved approximation. We test our algorithm with different numerical experiments.

New iterative reconstruction methods for fan-beam tomography

In this paper, we present a novel class of iterative reconstruction methods for severely angular undersampled or/and limited-view tomographic problems with fan-beam scanning geometry. The proposed algorithms are based on a new analytical transform which generalizes Fourier-slice theorem to divergent-beam scanning geometries. Using a non-rigid coordinate transform, divergent rays can be reorganized into parallel ones. Therefore, one can employ a simpler parallel-beam projection model instead of more complicated divergent-beam geometries. Various existing iterative reconstruction techniques for divergent-beam geometries can be easily adapted to the proposed framework. The significant advantage of this formulation is the possibility of exploiting efficient Fourier-based recovery methods without rebinning of the projections. In case of highly sparse measurements (few-view data), rebinning methods are not suitable due to error-prone angular interpolation involved. In this work, three new methods based on the novel analytical framework for fan-beam geometry are presented: the Gerchberg-Papoulis algorithm, the Neumann decomposition method and its total variation regularized version. Presented numerical experiments demonstrate that the methods can be competitive in reconstructing from few-view noisy tomographic measurements.

Improved Fourier Analysis of Periodically Patterned Graphene Sheets Embedded in Multilayered Structures and Its Application to the Design of a Broadband Tunable Wide-Angle Polarizer

Numerical modeling of periodically patterned graphene sheets (PPGS) embedded in planar multilayered media using Fourier-based methods suffers from very slow convergence because of the fact that the conductivity is zero in unfilled areas of the patterned surface and, thus, the so-called Li’s inverse rule is not applicable. In this paper, a simple and efficient approach is proposed to overcome this problem such that the exact boundary condition can be applied and the surface current density on PPGS can be obtained accurately. Here, the PPGS is modeled as a conductive surface and only its conductivity representation by the Fourier series is modified. The proposed method can be used easily for 1-D and 2-D periodic structures without the need to change the basic formulations of Fourier-based methods. Fast convergence and accuracy of the method will be demonstrated by computing the absorption of 1-D and 2-D PPGS. Moreover, the proposed method is utilized to design a wideband tunable wide-angle polarizer consisting of two-parallel PPGS separated by a 0.5- $\mu \text{m}$ layer of SiO2. The transmittance of the structure exceeds 95% from microwaves up to 2 THz.

Nonsinusoidal Beta Oscillations Reflect Cortical Pathophysiology in Parkinson's Disease.

Oscillations in neural activity play a critical role in neural computation and communication. There is intriguing new evidence that the nonsinusoidal features of the oscillatory waveforms may inform underlying physiological and pathophysiological characteristics. Time-domain waveform analysis approaches stand in contrast to traditional Fourier-based methods, which alter or destroy subtle waveform features. Recently, it has been shown that the waveform features of oscillatory beta (13-30 Hz) events, a prominent motor cortical oscillation, may reflect near-synchronous excitatory synaptic inputs onto cortical pyramidal neurons. Here we analyze data from invasive human primary motor cortex (M1) recordings from patients with Parkinson's disease (PD) implanted with a deep brain stimulator (DBS) to test the hypothesis that the beta waveform becomes less sharp with DBS, suggesting that M1 input synchrony may be decreased. We find that, in PD, M1 beta oscillations have sharp, asymmetric, nonsinusoidal features, specifically asymmetries in the ratio between the sharpness of the beta peaks compared with the troughs. This waveform feature is nearly perfectly correlated with beta-high gamma phase-amplitude coupling (r = 0.94), a neural index previously shown to track PD-related motor deficit. Our results suggest that the pathophysiological beta generator is altered by DBS, smoothing out the beta waveform. This has implications not only for the interpretation of the physiological mechanism by which DBS reduces PD-related motor symptoms, but more broadly for our analytic toolkit in general. That is, the often-overlooked time-domain features of oscillatory waveforms may carry critical physiological information about neural processes and dynamics.SIGNIFICANCE STATEMENT To better understand the neural basis of cognition and disease, we need to understand how groups of neurons interact to communicate with one another. For example, there is evidence that parkinsonian bradykinesia and rigidity may arise from an oversynchronization of afferents to the motor cortex, and that these symptoms are treatable using deep brain stimulation. Here we show that the waveform shape of beta (13-30 Hz) oscillations, which may reflect input synchrony onto the cortex, is altered by deep brain stimulation. This suggests that mechanistic inferences regarding physiological and pathophysiological neural communication may be made from the temporal dynamics of oscillatory waveform shape.

Edge Detection of Piecewise Smooth Functions from UnderSampled Fourier Data Using Variance Signatures

Detecting edges in piecewise smooth images is important in a variety of applications. In some cases, such as magnetic resonance imaging, data are collected as Fourier samples, and are often oversampled at low frequencies but more sparsely acquired in the high frequency range. This undersampling causes Fourier-based edge detection algorithms to yield false positives in smooth regions. The problem is compounded when the data are noisy. In this paper, we demonstrate that even when the given data are noisy and undersampled, the edge response behavior obtained using certain Fourier-based edge detection methods can be accurately characterized. In particular, the responses of different high order methods vary little, both in regions away from the discontinuities and at true edges. By contrast, the variance is large in cells neighboring each true edge. By locating these “double peaks” in variance, it is possible to determine edges from noisy and undersampled Fourier data. Alternative methods of generating and ref...

High-resolution seismic complex trace analysis by adaptive fast sparse S-transform

Complex trace analysis provides seismic interpreters with a view to identify the nature of challenging subsurface geologic features. However, the conventional procedure based on the Hilbert transform (HT) is highly sensitive to random noise and sudden frequency variations in seismic data. Generally, conventional filtering methods reduce the spectral bandwidth while stabilizing complex trace analysis, whereas obtaining high-resolution images of multiple thin-bed layers requires wideband data. It is thus a challenging problem to reconcile the conflict between the two purposes, and a powerful signal processing device is required. To overcome the issue, I first introduced the fast sparse S-transform (ST) as a powerful time-frequency decomposition method to improve the windowed Hilbert transform (WHT). Then, in addition to the mixed-norm higher resolution provided by the fast sparse ST, I have developed a novel sparsity-based optimization for window parameters. The process adaptively regularizes sudden changes in frequency content of nonstationary signals with the same computational complexity of the nonoptimized algorithm. The performance of the proposed windowing optimization is compared with those of available methods that have so far been used for adaptivity enhancement of Fourier-based spectral decomposition methods. The final adaptive and sparse version of WHT is used to achieve high-resolution complex trace analysis and address the above-mentioned conflict. The instantaneous complex attributes obtained by the proposed method for several synthetic and real data sets of which multiple thin-bed layers contain wedges, trapped gas reservoirs, and faults are superior to those obtained by WHT via adaptive sparse STFT, robust adaptive WHT, and conventional HT. Potential applications of the adaptive double-sparse ST as a new spectral decomposition method were also evaluated.

Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise

In nonlinear dynamical systems the determination of stable and unstable periodic orbits as part of phase space prediction is problematic in particular if perturbed by noise. Fourier spectra of the time series or its autocorrelation function have shown to be of little use if the dynamic process is not strictly wide-sense stationary or if it is nonlinear. To locate unstable periodic orbits of a chaotic attractor in phase space the least stable eigenvalue can be determined by approximating locally the trajectory via linearisation. This approximation can be achieved by employing a Gaussian kernel estimator and minimising the summed up distances of the measured time series i.e. its estimated trajectory (e.g. via Levenberg-Marquardt). Noise poses a significant problem here. The application of the Wiener-Khinchin theorem to the time series in combination with recurrence plots, i.e. the Fourier transform of the recurrence times or rates, has been shown capable of detecting higher order dynamics (period-2 or period-3 orbits), which can fail using classical FouRiER-based methods. However little is known about its parameter sensitivity, e.g. with respect to the time delay, the embedding dimension or perturbations.Here we provide preliminary results on the application of the recurrence time spectrum by analysing the Hénon and the Rössler attractor. Results indicate that the combination of recurrence time spectra with a nonlinearly filtered plot of return times is able to estimate the unstable periodic orbits. Owing to the use of recurrence plot based measures the analysis is more robust against noise than the conventional Fourier transform.

Real-time multi-signal frequency tracking with a bank of notch filters to estimate the respiratory rate from the ECG

Measuring the instantaneous frequency of a signal rapidly and accurately is essential in many applications. However, the instantaneous frequency by definition is a parameter difficult to determine. Fourier-based methods introduce estimation delays as computations are performed in a time-window. Instantaneous methods based on the Hilbert transform lack robustness. State-of-the-art adaptive filters yield accurate estimates, however, with an adaptation delay. In this study we propose an algorithm based on short length-3 FIR notch filters to estimate the instantaneous frequency of a signal at each sample, in a real-time manner and with very low delay. The output powers of a bank of the above-mentioned filters are used in a recursive weighting scheme to estimate the dominant frequency of the input. This scheme has been extended to process multiple inputs containing a common frequency by introducing an additional weighting scheme upon the inputs. The algorithm was tested on synthetic data and then evaluated on real biomedical data, i.e. the estimation of the respiratory rate from the electrocardiogram. It was shown that the proposed method provided more accurate estimates with less delay than those of state-of-the-art methods. By virtue of its simplicity and good performance, the proposed method is a worthy candidate to be used in biomedical applications, for example in health monitoring developments based on portable and automatic devices.

Computation of Greeks in LIBOR models driven by time–inhomogeneous Lévy processes

ABSTRACTThe aim of this article is to compute Greeks, i.e. price sensitivities in the framework of the Lévy LIBOR model. Two approaches are discussed. The first approach is based on the integration-by-parts formula, which lies at the core of the application of the Malliavin calculus to finance. The second approach consists of using Fourier-based methods for pricing derivatives. We illustrate the result by applying the formula to a caplet price where the jump part of the driving process of the underlying model is given by a time–inhomogeneous Gamma process and alternatively by a Variance Gamma process.

Alternating wavelet-time finite element method: Modeling and analysis of nonlinear wave propagation in one and two-dimensional waveguides

In this paper, a novel method named the alternating wavelet-time finite element method (AWT-FEM) is proposed for studying elastic wave propagation in nonlinear structures. An alternating iterative procedure between the time-domain and a wavelet-domain combined with the spectral finite element method (SFEM) is employed to solve wave equations with general nonlinearities. The advantages of the proposed method are (1) the potential to provide high fidelity results for impacts with high frequency content through the use of the spectral finite element method; (2) nonlinear structures with physically realistic boundary conditions can easily be studied by circumventing wrap-around issues associated with Fourier-based methods; (3) the parallel computing compatible framework and the semi-analytical nature of SFEM make it more computationally efficient for nonlinear systems modeled with structural components. Simulations using the proposed method are conducted to demonstrate its applicability to study nonlinear wave propagation in one-dimensional and two-dimensional systems.

Fast Algorithms and Efficient GPU Implementations for the Radon Transform and the Back-Projection Operator Represented as Convolution Operators

The Radon transform and its adjoint, the back-projection operator, can both be expressed as convolutions in log-polar coordinates. Hence, fast algorithms for the application of these operators can be constructed by using the FFT, if data is resampled at log-polar coordinates. Radon data is typically measured on an equally spaced grid in polar coordinates, and reconstructions are represented (as images) in Cartesian coordinates. Therefore, in addition to FFT, several steps of interpolation have to be conducted in order to apply the Radon transform and the back-projection operator by means of convolutions. However, in comparison to the interpolation conducted in Fourier-based gridding methods, the interpolation performed in the Radon and image domains will typically deal with functions that are substantially less oscillatory. Reasonable reconstruction results can thus be expected using interpolation schemes of moderate order. The approach also provides better control over the artifacts that can appear due t...

Fourier-Based 2-D Imaging Algorithm for Circular Synthetic Aperture Radar: Analysis and Application

Circular synthetic aperture radar (CSAR) has become a research hotspot in the SAR community. Since the Fourier-based imaging methods of CSAR are underdeveloped, further developments have to be investigated. In this paper, the exact expression and property analysis of the spectrum of CSAR are presented first. It turns out that the challenges of the Fourier-based 2-D imaging processing of CSAR lie in the projection of spectrum from the slant plane into the ground plane. Correspondingly, a spectrum-projection method by matched filtering as well as the practical subaperture-based imaging flow of CSAR are proposed, during which the trajectory deviation has been taken into account. Simulation and experimental tests are carried out to measure the efficiency and accuracy of the proposed method. The corresponding results verify that the proposed signal model is correct and the presented imaging flow is a good candidate to produce well-focused CSAR imagery with much less time consumption.

Computationally efficient autoregressive method for generating phase screens with frozen flow and turbulence in optical simulations.

We present a sample-based, autoregressive (AR) method for the generation and time evolution of atmospheric phase screens that is computationally efficient and uses a single parameter per Fourier mode to vary the power contained in the frozen flow and stochastic components. We address limitations of Fourier-based methods such as screen periodicity and low spatial frequency power content. Comparisons of adaptive optics (AO) simulator performance when fed AR phase screens and translating phase screens reveal significantly elevated residual closed-loop temporal power for small increases in added stochastic content at each time step, thus displaying the importance of properly modeling atmospheric "boiling". We present preliminary evidence that our model fits to AO telemetry are better reflections of real conditions than the pure frozen flow assumption.

Low-Cost Diagnosis of Rotor Asymmetries in Induction Machines Working at a Very Low Slip Using the Reduced Envelope of the Stator Current

Fault diagnosis of rotor asymmetries in induction machines working at a very low slip, through Fourier-based methods, usually requires a long acquisition time to achieve a high spectral resolution and a high sampling frequency to reduce aliasing effects. However, this approach generates a huge amount of data, which makes its implementation difficult using embedded devices with small internal memory, such as digital signal processors and field programmable gate arrays or devices with low computing power. In this paper, a new simplified diagnostic signal designated as the reduced envelope of the stator current is introduced to address this problem. The reduced envelope signal is built using only one sample of the current per cycle without any further processing, and it is demonstrated that it carries the same spectral information about the fault as the full-length current signal. Based on this approach, an embedded device has only to store and process a minimal set of samples compared with the raw current signal for a desired resolution. In this paper, the theoretical basis of the proposed method is presented, as well as its experimental validation using two different motors with broken bars: 1) a high-power induction motor working in a factory; and 2) a low-power induction motor mounted in a laboratory test bed.

Optical Response of a Hematite Coating: Ellipsometry Data versus Fourier-Based Computations

The optical properties of a hematite-epoxy coating are predicted numerically and compared with el-lipsometry measurements. The highly-heterogeneous dispersion of nanocubic hematite particles in the epoxy resin is simulated using a previously developed two-scales random model. The local anisotropic permittivity tensor of hematite particles, and that of the epoxy, are estimated by ellipsometry measurements carried out on a macroscopic hematite and epoxy samples. Fourier-based methods using a discrete Green operator are used to treat complex permittivities. They predict the effective and local electric displacement field in the quasi-static approximation. The former is close to two estimates based on the Hashin-Shtrikman bounds and a self-consistent approximation. Good agreement is found between experimental data and FFT computations in the whole range of the visible spectrum.

Dynamic signal analysis for speed-varying machinery: A review

For periodic signals from machinery, those sampled by A/D converter are no longer periodic under speed-varying condition. For this reason, Fourier-based methods are not applicable any more. Many methods are proposed to tackle this problem especially in recent years. Different kinds of machinery are with different speed variations, also with different measuring systems. Up to now, however, there is no deep investigation on how current methods are applicable to different working conditions and what direction shoud follow in the future. In this paper, current methods for speed-varying machinery dynamic signal analysis are reviewed, including order tracking, cyclostationary analysis, time-frequency analysis, EMD, etc. Finally, further work along this direction is discussed.

Enhancing the Detectability of Clouds and Their Shadows in Multitemporal Dryland Landsat Imagery: Extending Fmask

We developed a new two-step approach for automated masking of clouds and their shadows in Landsat imagery. The first step consists of detecting clouds and cloud shadows in every Landsat image independently by using the Fmask algorithm. We modified two features of the original Fmask: we dropped the termination criterion for shadow matching, and we appended a darkness filter to counteract false positives in bifidly structured dryland areas. The second step utilizes the scene-by-scene detections of the first step and additional time series of cloud and cloud shadow probabilities. All clear-sky observations of a pixel are used to estimate the probabilities' median and standard deviation. Any observation that deviates more than a multiple of the standard deviation from the median is considered an outlier and thus a remaining cloud or cloud shadow. The method was specifically designed for use in water-limited dryland areas, where event-based precipitation is predominant. As an effect, green vegetation peaks are highly variable, in timing, magnitude, and frequency, with adverse effects on commonly used Fourier-based outlier detection methods. The method is designed to be robust even if temporally dense data coverage is not available.