Fourier Basis Research Articles

Two-dimensional grid-free compressive beamforming with spherical microphone arrays

Compressive beamforming based on microphone array measurements and compressive sensing theory is an emerging and practical acoustic source identification technology. The grid-free compressive beamforming that treats the target source region as a continuum has attracted tremendous attention due to the advantage of circumventing the basis mismatch defect caused by the gridding. The grid-free methods have been developed for the linear and the planar microphone array measurements to identify sources in a local region. In contrast, for the spherical microphone array measurements suitable for 360° panoramic identification, the current research mainly focuses on grid-based methods. In this paper, we propose a two-dimensional grid-free compressive beamforming method with spherical microphone arrays. First of all, the relationship between the spherical harmonic function vector and the two-dimensional Fourier basis function vector is determined. Then, the mathematical model relating the measured sound pressures to the source distribution is built under a continuous setting. Next, a source sparsity constraint is imposed based on the total variation norm to solve the mathematical model, and its dual problem is deduced. Subsequently, the dual problem is solved by positive semidefinite programming. Finally, the DOAs of sources are estimated through polynomial rooting and utilizing the dual optimal variables, and the source strengths are quantified via least-square fitting. An experimental case is conducted to examine the practicability and superiority of the proposed method to the existing grid-based compressive beamforming method. Plenty of Monte Carlo simulations are carried out to disclose the influence of the source coherence, the angular distance between sources, the noise, the number of snapshots and the number of microphones. This study offers a new approach for the accurate and 360° panoramic identification of acoustic sources.

Multi-Target Tracking Using Windowed Fourier Single-Pixel Imaging.

The single-pixel imaging (SPI) technique enables the tracking of moving targets at a high frame rate. However, when extended to the problem of multi-target tracking, there is no effective solution using SPI yet. Thus, a multi-target tracking method using windowed Fourier single-pixel imaging (WFSI) is proposed in this paper. The WFSI technique uses a series of windowed Fourier basis patterns to illuminate the target. This method can estimate the displacements of K independently moving targets by implementing measurements and calculating windowed Fourier coefficients, which is a measurement method with low redundancy. To enhance the capability of the proposed method, we propose a joint estimation approach for multi-target displacement, which solves the problem where different targets in close proximity cannot be distinguished. Using the independent and joint estimation approaches, multi-target tracking can be implemented with WFSI. The accuracy of the proposed multi-target tracking method is verified by numerical simulation to be less than 2 pixels. The tracking effectiveness is analyzed by a video experiment. This method provides, for the first time, an effective idea of multi-target tracking using SPI.

Sparse representation and reproduction of speech signals in complex Fourier basis

Sparse representation and reproduction of speech signals in complex Fourier basis

Diagrammatic strong coupling expansion of a U(1) lattice model in the Fourier basis

The transfer-matrix of the U(1) lattice gauge theory is investigated in the field Fourier space, the basis of which consists of the quantized currents on lattice links. Based on a lattice version of the current conservation, the transfer-matrix elements are shown to be nonzero only between current states that differ in circulating currents inside plaquettes. In the strong coupling limit, a series expansion is developed for the elements of the transfer matrix, to which a diagrammatic representation based on the occurrence of virtual link and loop currents can be associated. With $g$ as the coupling, the weight of each virtual current in the expansion is $1/{g}^{2}$, by which at any given order the relevant diagrams are determined. Either by interpretation or through their role in fixing the relevant terms, the diagrams are reminiscent of the Feynman ones of the perturbative small coupling expansions.

Reduced‐rank space‐time adaptive processing algorithm based on multistage selections of angle‐Doppler filters

A new reduced-rank (RR) space-time adaptive processing (STAP) algorithm based on multistage selections of angle-Doppler filters is proposed in the form of a generalised sidelobe canceller. First, two types of the RR auxiliary angle-Doppler filters are designed based on the discrete Fourier basis functions. Then, a novel multistage method is implemented to select the angle-Doppler filters by using the cross-correlation observed at the current stage and the residual output observed at the previous stage. Finally, the asymptotic statistic performance of target detection is evaluated in terms of the conventional framework of STAP for the cases of both known and unknown covariance, which shows that the proposed algorithm is capable of providing the constant false alarm rate. Numerical examples are provided, and they demonstrate that the proposed algorithm is able to offer better performance than the existing sparsity-based and the conventional reduced-dimension algorithms under the intrinsic clutter motion and array gain and phase errors.

Optimal rates for independence testing via U-statistic permutation tests

We study the problem of independence testing given independent and identically distributed pairs taking values in a σ-finite, separable measure space. Defining a natural measure of dependence D(f) as the squared L2-distance between a joint density f and the product of its marginals, we first show that there is no valid test of independence that is uniformly consistent against alternatives of the form {f:D(f)≥ρ2}. We therefore restrict attention to alternatives that impose additional Sobolev-type smoothness constraints, and define a permutation test based on a basis expansion and a U-statistic estimator of D(f) that we prove is minimax optimal in terms of its separation rates in many instances. Finally, for the case of a Fourier basis on [0,1]2, we provide an approximation to the power function that offers several additional insights. Our methodology is implemented in the R package USP.

Fast solver for quasi-periodic 2D-Helmholtz scattering in layered media

We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding cut-off frequencies (also known as Rayleigh-Wood frequencies), we rigorously establish the well-posedness of both continuous and discrete problems, and prove super-algebraic error convergence rates for the proposed scheme. Through several numerical examples, we confirm our findings and show performances competitive to those attainedviaNyström methods.

Statistical learning for fluid flows: Sparse Fourier divergence-free approximations

We reconstruct the velocity field of incompressible flows given a finite set of measurements. For the spatial approximation, we introduce the Sparse Fourier divergence-free approximation based on a discrete L2 projection. Within this physics-informed type of statistical learning framework, we adaptively build a sparse set of Fourier basis functions with corresponding coefficients by solving a sequence of minimization problems where the set of basis functions is augmented greedily at each optimization problem. We regularize our minimization problems with the seminorm of the fractional Sobolev space in a Tikhonov fashion. In the Fourier setting, the incompressibility (divergence-free) constraint becomes a finite set of linear algebraic equations. We couple our spatial approximation with the truncated singular-value decomposition of the flow measurements for temporal compression. Our computational framework thus combines supervised and unsupervised learning techniques. We assess the capabilities of our method in various numerical examples arising in fluid mechanics.

A finite-element model for the Hasegawa–Mima wave equation

In a recent work Karakazian and Nassif (2021)[1], two of the authors have formulated the non-linear space-time Hasegawa–Mima plasma equation as a coupled system of two linear PDEs, a solution of which is a pair (u,w), with w=(I−Δ)u. The first equation is of hyperbolic type and the second of elliptic type. Variational frames for obtaining weak solutions to the initial value Hasegawa–Mima problem with periodic boundary conditions were also derived. Using the Fourier basis in the space variables, existence of solutions were obtained. Implementation of algorithms based on Fourier series leads to systems of dense matrices. In this paper, we use a finite element space-domain approach to semi-discretize the coupled variational Hasegawa–Mima model, obtaining global existence of solutions in H2 on any time interval [0,T],∀T. In the sequel, full-discretization using an implicit time scheme on the semi-discretized system leads to a nonlinear full space-time discrete system. For the existence proof, a condition on the time step is imposed that is independent from the space mesh size h. For the uniqueness proof, a CFL-type of condition on the time step is imposed. Tests on a semi-linear version of the implicit nonlinear full-discrete system are conducted for several initial data, assessing the efficiency of our approach.

Efficient Fourier Single-Pixel Imaging with Gaussian Random Sampling

Fourier single-pixel imaging (FSI) is a branch of single-pixel imaging techniques. It allows any image to be reconstructed by acquiring its Fourier spectrum by using a single-pixel detector. FSI uses Fourier basis patterns for structured illumination or structured detection to acquire the Fourier spectrum of image. However, the spatial resolution of the reconstructed image mainly depends on the number of Fourier coefficients sampled. The reconstruction of a high-resolution image typically requires a number of Fourier coefficients to be sampled. Consequently, a large number of single-pixel measurements lead to a long data acquisition time, resulting in imaging of a dynamic scene challenging. Here we propose a new sampling strategy for FSI. It allows FSI to reconstruct a clear and sharp image with a reduced number of measurements. The key to the proposed sampling strategy is to perform a density-varying sampling in the Fourier space and, more importantly, the density with respect to the importance of Fourier coefficients is subject to a one-dimensional Gaussian function. The final image is reconstructed from the undersampled Fourier spectrum through compressive sensing. We experimentally demonstrate the proposed method is able to reconstruct a sharp and clear image of 256 × 256 pixels with a sampling ratio of 10%. The proposed method enables fast single-pixel imaging and provides a new approach for efficient spatial information acquisition.

Exact Frequency Estimation in the i.i.d. Noise via KL Divergence of Accumulated Power

To solve the frequency estimation deterioration caused by independent and identically distributed (i.i.d.) noise, this letter proposes an exact and robust frequency estimation algorithm based on the accumulated phase-difference power on Kullback-Leibler divergence (APP-KLD). First, the frequency estimation problem is modeled by the accumulated power of phase-difference between the receiving signal and Fourier basis function. Furthermore, the true frequency is estimated iteratively by maximizing the Kullback–Leibler divergence, which measures the distance between the accumulated power and a constant reference vector. Simulations demonstrate the robust performance of the APP-KLD algorithm at various levels of heavy-tailed noise and sample numbers.

Active control of separated flow on a symmetric airfoil by pitching oscillation

Experiments are conducted on the control of massively separated flow over a symmetric airfoil by imposing pitching oscillation at smaller amplitudes. Two symmetric airfoils of different thickness to chord ratios are considered to account for the effect of thickness and the stall-type on flow control. Instantaneous aerodynamic forces and moments are calculated by integrating surface and wake pressure distributions. Time-resolved particle image velocimetry and unsteady pressure measurements are carried out simultaneously to characterize the flow field over the pitching airfoil. The results presented here provide an insight into flow dynamics and the dynamic response of a stalled airfoil to pitching oscillation at different reduced frequencies. The present research adopts a dynamical system approach and attempts to estimate an optimum pitching oscillation frequency for better flow control on a stalled airfoil. The adopted methodology can be used to determine the optimal value of input parameters pertaining to flow control on a flat plate/wing by any unsteady excitation. Besides, a novel data-driven aerodynamic model has been developed on a Fourier basis for a symmetrical airfoil undergoing pitching oscillation at the post-stall regime.

A Conditional Simulation Method for Predicting Wind Pressure Fields of Large-Span Spatial Structures

Wind load is among the control loads for large-span spatial structures. Wind tunnel test is one of the commonly used methods for measuring wind pressure fields of different kinds of structures. However, due to the limited wind pressure data obtained from wind tunnel testing, it is quite meaningful to employ the limited measured data to predict the unknown wind pressure at target points. Considering the complexity of wind pressure fields of large-span spatial structures, a simplified nonparametric method based on conditional simulation is proposed to predict the unknown pressures using the existing data. The Karhunen–Loève (KL for short) expansion is employed to represent wind pressure random variants as eigenfunctions of the covariance operator. To reduce the variant dimensionality, the nearest neighboring estimator is given for the transition distribution of the KL expansion. The targeted wind pressure fields are obtained by expanding the Fourier basis of the eigenfunction and estimating its expansion coefficients. The proposed method is applied to estimate wind pressures on a gable roof building. The relevant parameters of the wind pressure field are obtained, and the results compare well with those from wind tunnel testing, with higher efficiency. The proposed method effectively reduces the dimensionality of the predicted wind pressures, with reduced errors, higher accuracy, and increased efficiency.

Convexification numerical algorithm for a 2D inverse scattering problem with backscatter data

This paper is concerned with the inverse scattering problem which aims to determine the spatially distributed dielectric constant coefficient of the 2D Helmholtz equation from multifrequency backscatter data associated with a single direction of the incident plane wave. We propose a globally convergent convexification numerical algorithm to solve this nonlinear and ill-posed inverse problem. The key advantage of our method over conventional optimization approaches is that it does not require a good first guess about the solution. First, we eliminate the coefficient from the Helmholtz equation using a change of variables. Next, using a truncated expansion with respect to a special Fourier basis, we approximately reformulate the inverse problem as a system of quasilinear elliptic PDEs, which can be numerically solved by a weighted quasi-reversibility approach. The cost functional for the weighted quasi-reversibility method is constructed as a Tikhonov-like functional that involves a Carleman Weight Function. Our numerical study shows that, using a version of the gradient descent method, one can find the minimizer of this Tikhonov-like functional without any advanced a priori knowledge about it.

CVNet: confidence voting convolutional neural network for camera spectral sensitivity estimation.

Spectral sensitivity, as one of the most important parameters of a digital camera, is playing a key role in many computer vision applications. In this paper, a confidence voting convolutional neural network (CVNet) is proposed to rebuild the spectral sensitivity function, modeled as the sum of weighted basis functions. By evaluating useful information supplied by different image segments, disparate confidence is calculated to automatically learn basis functions' weights, only using one image captured by the object camera. Three types of basis functions are made up and employed in the network, including Fourier basis function (FBF), singular value decomposition basis function (SVDBF), and radial basis function (RBF). Results show that the accuracy of the proposed method with FBF, SVDBF, and RBF is 97.92%, 98.69%, and 99.01%, respectively. We provide theory for network design, build a dataset, demonstrate training process, and present experimental results with high precision. Without bulky benchtop setups and strict experimental limitations, this proposed simple and effective method could be an alternative in the future for spectral sensitivity function estimation.

Learning Set Functions that are Sparse in Non-Orthogonal Fourier Bases

Many applications of machine learning on discrete domains, such as learning preference functions in recommender systems or auctions, can be reduced to estimating a set function that is sparse in the Fourier domain. In this work, we present a new family of algorithms for learning Fourier-sparse set functions. They require at most nk − k log k + k queries (set function evaluations), under mild conditions on the Fourier coefficients, where n is the size of the ground set and k the number of non-zero Fourier coefficients. In contrast to other work that focused on the orthogonal Walsh-Hadamard transform (WHT), our novel algorithms operate with recently introduced non-orthogonal Fourier transforms that offer different notions of Fourier-sparsity. These naturally arise when modeling, e.g., sets of items forming substitutes and complements. We demonstrate effectiveness on several real-world applications.

Gradients of connectivity as graph Fourier bases of brain activity.

The application of graph theory to model the complex structure and function of the brain has shed new light on its organization, prompting the emergence of network neuroscience. Despite the tremendous progress that has been achieved in this field, still relatively few methods exploit the topology of brain networks to analyze brain activity. Recent attempts in this direction have leveraged on the one hand graph spectral analysis (to decompose brain connectivity into eigenmodes or gradients) and the other graph signal processing (to decompose brain activity “coupled to” an underlying network in graph Fourier modes). These studies have used a variety of imaging techniques (e.g., fMRI, electroencephalography, diffusion-weighted and myelin-sensitive imaging) and connectivity estimators to model brain networks. Results are promising in terms of interpretability and functional relevance, but methodologies and terminology are variable. The goals of this paper are twofold. First, we summarize recent contributions related to connectivity gradients and graph signal processing, and attempt a clarification of the terminology and methods used in the field, while pointing out current methodological limitations. Second, we discuss the perspective that the functional relevance of connectivity gradients could be fruitfully exploited by considering them as graph Fourier bases of brain activity.

Compressive Covariance Sensing-Based Power Spectrum Estimation of Real-Valued Signals Subject to Sub-Nyquist Sampling

In this work, an estimate of the power spectrum of a real-valued wide-sense stationary autoregressive signal is computed from sub-Nyquist or compressed measurements in additive white Gaussian noise. The problem is formulated using the concepts of compressive covariance sensing and Blackman-Tukey nonparametric spectrum estimation. Only the second-order statistics of the original signal, rather than the signal itself, need to be recovered from the compressed signal. This is achieved by solving the resulting overdetermined system of equations by application of least squares, thereby circumventing the need for applying the complicated ℓ 1 -minimization otherwise required for the reconstruction of the original signal. Moreover, the signal need not be spectrally sparse. A study of the performance of the power spectral estimator is conducted taking into account the properties of the different bases of the covariance subspace needed for compressive covariance sensing, as well as different linear sparse rulers by which compression is achieved. A method is proposed to benefit from the possible computational efficiency resulting from the use of the Fourier basis of the covariance subspace without considerably affecting the spectrum estimation performance.

Nonstationary local time-frequency transform

Time-frequency analysis is a fundamental approach to many seismic problems. Time-frequency decomposition transforms input seismic data from the time domain to the time-frequency domain, offering a new dimension to probe the hidden information inside the data. Considering the nonstationary nature of seismic data, time-frequency spectra can be obtained by applying a local time-frequency transform (LTFT) method that matches the input data by fitting the Fourier basis with nonstationary Fourier coefficients in the shaping regularization framework. The key part of LTFT is the temporal smoother with a fixed smoothing radius that guarantees the stability of the nonstationary least-squares fitting. We have developed a new LTFT method to handle the nonstationarity in all time, frequency, and space ( x and y) directions of the input seismic data by extending fixed-radius temporal smoothing to nonstationary smoothing with a variable radius in all physical dimensions. The resulting time-frequency transform is referred to as the nonstationary LTFT method, which could significantly increase the resolution and antinoise ability of time-frequency transformation. There are two meanings of nonstationarity, i.e., coping with the nonstationarity in the data by LTFT and dealing with the nonstationarity in the model by nonstationary smoothing. We evaluate the performance of our nonstationary LTFT method in several standard seismic applications via synthetic and field data sets, e.g., arrival picking, quality factor estimation, low-frequency shadow detection, channel detection, and multicomponent data registration, and we benchmark the results with the traditional stationary LTFT method.

О зависимости качества машинного перевода текста от используемого преобразования Фурье

Машинный перевод широко используется при переводе различной коммерческой, технической, научной информации, что связано с процессом глобализации и соответственно расширением сети деловых отношений. Математические методы, касающиеся машинных переводов текстов, в последнее время получили новый толчок в связи с интенсивным развитием теории преобразования Фурье. Так, при обработке контрастных сигналов и изображений возросли требования к точности фильтрации, которые впоследствии позволят создавать более эффективные алгоритмы фильтрации. Среди существующих алгоритмов фильтрации наиболее эффективны частотные алгоритмы, т.е. такие, в которых обработке подлежат коэффициенты разложения зашумленного сигнала по Фурье-базису. При использовании алгоритмов Фурье фильтрации важную роль играют свойства преобразования Фурье, которые, в свою очередь, зависят от принадлежности к тому или иному классу дифференцируемых функций. Необходимым условием для существования непрерывного преобразования Фурье является абсолютное сходимость некоторых функций, с помощью которого описывается исследуемый реальный процесс. В роли таких моделируемых функций на практике очень часто используют так называемые «суммирующие функции», которые очень удобно строить с помощью линейного матричного суммирования рядов Фурье. Что касается последних, то здесь принято различать как треугольные, так и прямоугольные линейные матричные методы. Работа посвящена исследованию условий сходимости преобразований Фурье как треугольных, так и прямоугольных линейных матричных методов суммирования рядов Фурье. Кроме того, в данной статье показано, что скорость сходимости преобразования Фурье прямоугольного линейного метода Абеля-Пуассона в раз быстрее скорости сходимости аналогичного треугольного линейного метода Абеля-Пуассона. Этот результат в дальнейшем может существенно влиять на выбор более эффективного преобразования Фурье, которое можно будет использовать в процессе машинного перевода текста.