Fourier Basis Research Articles

Graph Signal Processing Over a Probability Space of Shift Operators

Graph signal processing (GSP) uses a shift operator to define a Fourier basis for the set of graph signals. The shift operator is often chosen to capture the graph topology. However, in many applications, the graph topology may be unknown <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> , its structure uncertain, or generated randomly from a predefined set for each observation. Each graph topology gives rise to a different shift operator. In this paper, we develop a GSP framework over a probability space of shift operators. We develop the corresponding notions of Fourier transform, MFC filters, and band-pass filters, which subsumes classical GSP theory as the special case where the probability space consists of a single shift operator. We show that an MFC filter under this framework is the expectation of random convolution filters in classical GSP, while the notion of bandlimitedness requires additional wiggle room from being simply a fixed point of a band-pass filter. We develop a mechanism that facilitates mapping from one space of shift operators to another, which allows our framework to be applied to a rich set of scenarios. We demonstrate how the theory can be applied by using both synthetic and real datasets.

A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems

&lt;p style='text-indent:20px;'&gt;Based on high order polynomial approximation and dimension reduction technique, we propose a novel numerical method for the fourth order Steklov problems in the circular domain. We first decompose the primal problem into a set of 1D problems via polar coordinate transformation and Fourier basis functions expansion. Then, by introducing a non-uniformly weighed Sobolev space, the variational form and corresponding discrete scheme are derived. Employing the Lax-Milgram lemma and approximation properties of the projection operators, we further prove existence and uniqueness of weak solutions and approximation solutions for each one-dimensional problems, and the error estimation between them, respectively. We also carry out ample numerical experiments which illustrate that the numerical algorithm is efficient and highly accurate.&lt;/p&gt;

An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients

&lt;abstract&gt;&lt;p&gt;We propose in this paper an efficient algorithm based on the Fourier spectral-Galerkin approximation for the fourth-order elliptic equation with periodic boundary conditions and variable coefficients. First, by using the Lax-Milgram theorem, we prove the existence and uniqueness of weak solution and its approximate solution. Then we define a high-dimensional $ L^2 $ projection operator and prove its approximation properties. Combined with Céa lemma, we further prove the error estimate of the approximate solution. In addition, from the Fourier basis function expansion and the properties of the tensor, we establish the equivalent matrix form based on tensor product for the discrete scheme. Finally, some numerical experiments are carried out to demonstrate the efficiency of the algorithm and correctness of the theoretical analysis.&lt;/p&gt;&lt;/abstract&gt;

Imaging Algorithm for Inverse Synthetic Aperture Radar in Condition of Non-Uniform Data Rate

Inverse synthetic aperture radar imaging is an important technique for radar detection and recognition of non-cooperative targets. However, for most of the existing imaging methods, pulse signals are transmitted and received at equal intervals, without considering the possible problems under condition of non-uniform data rate, such as accuracy decline of echo's range alignment and high side lobes in Doppler spectrum. The resulting vertical and horizontal defocusing can reduce the imaging quality and affect the feature extraction of the target. Therefore, an inverse synthetic aperture radar imaging method for non-uniform data rate is proposed. First, by designing the weight to simulate the alignment situation of uniform data rate, the correlation method with weighted sliding window is used to solve the problems of declined range alignment accuracy and vertical defocusing. Then, based on the linear form of non-uniform Fourier transform, the Fourier base matrix of non-uniform data rate is built, and CLEAN technology is used to complete the projection of echo signal on the bases iteratively. Finally, the Fourier basis matrix with uniform data rate is used to reconstruct the echo signal, which solves the problem of horizontal defocusing and makes the data rate of echo signal uniform. The effectiveness of the proposed method is verified by experiments using measured data.

Fourier coded aperture transform hyperspectral imaging system

We consider using Fourier coded aperture transform to solve the acquisition and reconstruction problem of three-dimensional (3-D) hyperspectral imaging. In recent years, Coded aperture snapshot spectral imaging (CASSI) system has received increasing attention. However, reconstruction errors are unavoidable due to its ill-posed problems and the number of dispersion channels is limited. The Hadamard transform spectral (HTS) imaging method has no limitation on the number of dispersion channels, but it usually requires a long data acquisition time. By exploring the process of coded spectral imaging, in this paper, we propose Fourier coded aperture transform spectral imaging (FCTS). We compared the performance of HTS and FCTS by theoretical analysis and experiments. The results show that FCTS has better concentrated energy than HTS on natural hyperspectral data. Compared with the Hadamard transform method, the Fourier basis mode has better physical correlation with the natural image, and the reconstruction quality of FCTS is better than that of HTS in the case of under-sampling.

Full-resolution, full-field-of-view, and high-quality fast Fourier single-pixel imaging.

Fourier single-pixel imaging (FSI) uses Fourier basis patterns for spatial light modulation to acquire the Fourier spectrum of the object image. The object image can be reconstructed via an inverse Fourier transform. However, the Fourier basis patterns are inherently gray scale, which results in the difficulty that the patterns can hardly be generated at a high speed by using a commonly used spatial light modulator-digital micromirrors device. To tackle this problem, fast FSI, which uses upsampled and dithered Fourier basis patterns to approximate the gray scale patterns, has been reported, but the achievable spatial resolution has to be sacrificed in the pattern upsampling process. Here we propose a method that can achieve not only full-resolution but also full-field-of-view and high-quality FSI. The key to the proposed method is to use a new, to the best of our knowledge, error diffusion dithering algorithm combined with two different scanning strategies to generate two sets of binarized Fourier basis patterns for spatial light modulation. As a result, two images with a sub-pixel shift from each other are reconstructed. It results in the final high-quality reconstruction by synthesizing the two images. We experimentally demonstrate the method can produce a high-quality 1024 × 768-pixel and full resolution image with a digital micromirror device with 1024 × 768 micromirrors.

Band-pass filtering based on the Fourier basis pursuit spectrum

Band-pass filtering based on the Fourier basis pursuit spectrum

Fast solution of three‐dimensional elliptic equations with randomly generated jumping coefficients by using tensor‐structured preconditioners

AbstractIn this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in , . Both the two‐dimensional (2D) and three‐dimensional (3D) elliptic problems are considered for the jumping equation coefficients built as a checkerboard type configuration of bumps randomly distributed on a large , or lattice, respectively. The finite element method discretization procedure on a 3D uniform tensor grid is described in detail, and the Kronecker tensor product approach is proposed for fast generation of the stiffness matrix. We introduce tensor techniques for the construction of the low Kronecker rank spectrally equivalent preconditioner in a periodic setting to be used in the framework of the preconditioned conjugate gradient iteration. The discrete 3D periodic Laplacian pseudo‐inverse is first diagonalized in the Fourier basis, and then the diagonal matrix is reshaped into a fully populated third‐order tensor of size . The latter is approximated by a low‐rank canonical tensor by using the multigrid Tucker‐to‐canonical tensor transform. As an example, we apply the presented solver in numerical analysis of stochastic homogenization method where the 3D elliptic equation should be solved many hundred times, and where for every random sampling of the equation coefficient one has to construct the new stiffness matrix and the right‐hand side. The computational characteristics of the presented solver in terms of a lattice parameter and the grid‐size, , in both 2D and 3D cases are illustrated in numerical tests. Our solver can be used in various applications where the elliptic problem should be solved for a number of different coefficients for example, in many‐particle dynamics, protein docking problems or stochastic modeling.

Comparison of LED-based and Reflective Colour Targets for Camera Spectral Sensitivities Estimation

Spectral sensitivities represent the spectral property of a digital camera. Most of the prior art spectral sensitivities estimation algorithms were applied to reflective colour charts, while some algorithms used a LED-based target. In this study, the spectral sensitivities of camera were estimated from both the LED-based and reflective colour targets. Four algorithms including Tikhonov Regularization based on Derivatives, Fourier basis function, Principal Component Analysis (PCA) and Singular Value Decomposition (SVD), were implemented. The estimated accuracy was compared between different types of colour targets, and between different algorithms. It was found that the optimal algorithm was different when using LED-based and reflective colour targets.

Tensor dictionary learning for representing three-dimensional sound speed fields.

Ocean sound speed field (SSF) representation is often plagued with low resolution (i.e., the capability of explaining fine-scale fluctuations). This drawback, however, is inherent in a number of classical SSF basis functions, e.g., empirical orthogonal functions, Fourier basis functions, and more recent tensor-based basis functions learned via the higher-order orthogonal iterative algorithm. For two-dimensional depth-time SSF representation, recent attempts relying on dictionary learning have shown that fine-scale sound speed information can be well preserved by a large number of basis functions. They are learned from the historical data without imposing rigid constraints on their shapes, e.g., the orthogonal constraints. However, generalizing the dictionary learning idea to represent three-dimensional (3D) spatial ocean SSF is non-trivial, in terms of both problem formulation and algorithm development. It calls for integrating the dictionary learning framework and the tensor-based basis function learning framework, a recently proposed one that captures the 3D sound speed correlations well. To achieve this goal, we develop a 3D SSF-tailored tensor dictionary learning algorithm that learns a large number of tensor-based basis functions with flexible shapes in a data-driven fashion. Numerical results based on the South China Sea 3D SSF data have showcased the superiority of the proposed approach over the prior method.

Time-frequency representation with reweighted regularization for fault diagnosis in a sparse way

In the field of fault diagnosis, time–frequency representation (TFR) is a very popular tool due to its good performance under time-varying condition. TFR provides both time and frequency information of fault signal, and the component related to fault characteristic frequency (FCF) can be found in the time–frequency image. To improve the time–frequency resolution of TFR, a special kind of TFR method named sparse time frequency representation (STFR) combines TFR and sparse theory, where the TFR is processed in a sparse way and the energy is more concentrated on FCF. However, the underestimation problem from the optimization process limits the efficiency of STFR. Concretely, the coefficients of the components of FCF are easy to be underestimated in the iterative process of solving coefficient, which will enhance other interference components. Especially for the weak faults from the early stage or the serious noise pollution, STFR is easy to be contaminated and is hard to reveal the components of FCF because of the underestimation problem. To solve the underestimation problem, we propose a novel STFR with reweighted regularization (RSTFR) for fault diagnosis. In the proposed method, the envelope of signal is divided into small segments, and each segment is decomposed into the Fourier basis. Naturally, the sparse coefficient represents the envelope spectrum of each segment. Meanwhile, the reweighted scheme is used to solve the underestimation problem. The components related to FCF are encouraged with low thresholds while other interference components are punished with high thresholds. At last, the envelope spectrums of all segments at different time form the TFR. The proposed method can provide a higher time–frequency resolution and diagnose the fault precisely. RSTFR is applied in the both simulation cases and experimental cases of bearing fault diagnosis, and it shows the overwhelming superiority compared to other four state-of-art TFR methods.

Weak coupling limit of U(1) lattice model in Fourier basis

The transfer-matrix of the U(1) lattice model is considered in the Fourier basis and in the weak coupling limit. The issues of Gauss law constraint and gauge invariant states are addressed in the Fourier basis. In particular, it is shown that in the strong coupling limit the gauge invariant Fourier states are effectively the finite size closed loop currents. In the weak coupling limit, however, the link-currents along periodic or infinite spatial directions find comparable roles as gauge invariant states. The subtleties related to the extreme weak coupling of the transfer-matrix in the Fourier basis are discussed. A careful analysis of the zero eigenvalues of the matrix in the quadratic action leads to a safe extraction of the diverging group volume in the limit $g\to 0$. By means of the very basic notions and tools of the lattice model, the spectrum at the weak coupling limit for any dimension and size of lattice is obtained analytically. The spectrum at the weak coupling limit corresponds to the expected one by the continuum model in the large lattice limit.

The uniform sparse FFT with application to PDEs with random coefficients

We develop the uniform sparse Fast Fourier Transform (usFFT), an efficient, non-intrusive, adaptive algorithm for the solution of elliptic partial differential equations with random coefficients. The algorithm is an adaption of the sparse Fast Fourier Transform (sFFT), a dimension-incremental algorithm, which tries to detect the most important frequencies in a given search domain and therefore adaptively generates a suitable Fourier basis corresponding to the approximately largest Fourier coefficients of the function. The usFFT does this w.r.t. the stochastic domain of the PDE simultaneously for multiple fixed spatial nodes, e.g., nodes of a finite element mesh. The key idea of joining the detected frequency sets in each dimension increment results in a Fourier approximation space, which fits uniformly for all these spatial nodes. This strategy allows for a faster and more efficient computation due to a significantly smaller amount of samples needed, than just using other algorithms, e.g., the sFFT for each spatial node separately. We test the usFFT for different examples using periodic, affine and lognormal random coefficients in the PDE problems.

SIS-BAM Algorithm for Angular Parameter Estimation of 2-D Incoherently Distributed Sources

For two-dimensional (2-D) incoherently distributed sources, this paper presents an effective angular parameter estimation method based on shift invariant structure (SIS) of the beamspace array manifold (BAM), named as SIS-BAM algorithm. In the proposed method, a shift invariance structure of the observed vectors is established utilizing a generalized array manifold of a uniform linear orthogonal array. Then, based on Fourier basis vectors and the SIS, a beamspace transformation matrix can be obtained. It projects received signals into the corresponding beamspace, so as to carry out dimension reduction of observed signals in beamspace domain. Finally, according to the SIS of beamspace observed vectors, the closed form solutions of the nominal azimuth and elevation are derived. Compared with the previous works, the presented SIS-BAM method provides better estimation performance, for example: 1) the computational complexity is reduced due to dealing with low-dimension beamspace signals and avoiding spectral search; 2) it can not only improve the angular parameter estimation accuracy but also have excellent robustness to the change of signal-to-noise ratio (SNR) and snapshot number. The theoretical analysis and simulation results confirm the effectiveness of the proposed method.

Bayesian Dependent Functional Mixture Estimation for Area and Time-Indexed Data: An Application for the Prediction of Monthly County Employment

The U.S. Bureau of Labor Statistics (BLS) publishes employment totals for all U.S. counties on a monthly basis. BLS use the Quarterly Census of Employment and Wages, where responses are received on a 6–7 month lagged basis and aggregated to county, and apply a time series forecast model to each county and project forward to the current month, which ignores the dependence among counties. Our approach treats these by-county employment time series as a collection of area indexed noisy functions that we co-model. Our model includes predictor, trend and seasonality terms indexed by county. This application is among the first in the U.S. Federal Statistical System to address the joint modeling of a collection of time series expressing heterogenous seasonality patterns between them. We demonstrate that use of a Fourier basis to model seasonality outperforms a locally-adaptive, intrinsic conditional autoregressive construction on our collection of time series where the degree of expressed seasonality varies. County-indexed parameters of the 3 terms are drawn from a dependent Dirichlet process (DDP) prior to allow the borrowing of information. We show that employment of both spatial and industry concentration predictors into the prior probabilities for co-clustering among the counties produces better prediction accuracy. Our DDP prior accounts for the possibility that nearby counties may express distinct underlying economic structures. A feature of our joint modeling framework is that it computes efficiently to support the monthly BLS production cycle. We compare the performances of alternative formulations for the dependent Dirichlet process prior on monthly, county employment data from 2002–2016.

A time-varying autoregressive model for groundwater depth prediction

The nonstationarity of hydrological variables makes the application of autoregressive (AR) models challenging. Therefore, this study introduces a new time-varying AR (TVAR) model in the field of hydrology. Specifically, in this study, we focus the parameter estimation of the TVAR model and exploring the model’s performance for predicting groundwater depth. We demonstrate the application of the model to the monthly groundwater depth series obtained on the Guanzhong Plain, China. We summarize the process of parameter estimation of the TVAR model. First, the TVAR model is transformed into the time-invariance regression problem by expanding the time-varying coefficients into a set of Fourier or Legendre basis functions. Then, a fading memory recursive least squares (FMRLS) algorithm is used to estimate the parameters of the regression problem. In this process, the model order and dimension of the basis function are determined by minimizing our proposed improved Bayesian information criterion (IBIC) with a range of dimensions greater than 0. To further demonstrate the effectiveness of the parameter estimation method and the generalizable performance of the model, the method is applied to nonstationary series simulated in statistical experiments. The study results indicate that the TVAR model based on such a parameter estimation process exhibits better prediction performance, lower model complexity and more straight-forward application compared with the autoregressive integrated (ARI) and seasonal ARI (SARI) models. In conclusion, using the TVAR model as an alternative to the time-invariance ARI and SARI models results in a model that is more flexible and suitable for nonstationary groundwater depth prediction.

An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem

AbstractIn this article, an effective finite element method based on dimension reduction scheme is proposed for a fourth-order Steklov eigenvalue problem in a circular domain. By using the Fourier basis function expansion and variable separation technique, the original problem is transformed into a series of radial one-dimensional eigenvalue problems with boundary eigenvalue. Then we introduce essential polar conditions and establish the discrete variational form for each radial one-dimensional eigenvalue problem. Based on the minimax principle and the approximation property of the interpolation operator, we prove the error estimates of approximation eigenvalues. Finally, some numerical experiments are provided, and the numerical results show the efficiency of the proposed algorithm.

FIESTA II. Disentangling Stellar and Instrumental Variability from Exoplanetary Doppler Shifts in the Fourier Domain

The radial velocity (RV) detection of exoplanets is challenged by stellar spectroscopic variability that can mimic the presence of planets and by instrumental instability that can further obscure the detection. Both stellar and instrumental changes can distort the spectral line profiles and be misinterpreted as apparent RV shifts. We present an improved FourIEr phase SpecTrum Analysis (FIESTA, aka ϕESTA) to disentangle apparent velocity shifts due to a line deformation from a true Doppler shift. ϕESTA projects a stellar spectrum’s cross-correlation function (CCF) onto a truncated set of Fourier basis functions. Using the amplitude and phase information from each Fourier mode, we can trace the line variability at different CCF width scales to robustly identify and mitigate multiple sources of RV contamination. For example, in our study of the 3 yr of HARPS-N solar data, ϕESTA reveals the solar rotational effect, the long-term trend due to solar magnetic cycle, instrumental instability, and apparent solar rotation rate changes. Applying a multiple linear regression model on ϕESTA metrics, we reduce the weighted rms noise from 1.89 to 0.98 m s−1. In addition, we observe a ∼3-day lag in the ϕESTA metrics, similar to the findings from previous studies on the bisector inverse slope and FWHM.

A novel approach to radially global gyrokinetic simulation using the flux-tube code stella

A novel approach to global gyrokinetic simulation is implemented in the flux-tube code stella. This is done by using a subsidiary expansion of the gyrokinetic equation in the perpendicular scale length of the turbulence, originally derived by Parra and Barnes (2015) [23], which allows the use of Fourier basis functions while enabling the effect of radial profile variation to be included in a perturbative way. Radial variation of the magnetic geometry is included by utilizing a global extension of the Grad-Shafranov equation and the Miller equilibrium equations which is obtained through Taylor expansion. Radial boundary conditions that employ multiple flux-tube simulations are also developed, serving as a more physically motivated replacement to the conventional Dirichlet radial boundary conditions that are used in global simulation. It is shown that these new boundary conditions eliminate much of the numerical artefacts generated near the radial boundary when expressing a non-periodic function using a spectral basis. We then benchmark the new approach both linearly and non-linearly using a number of standard test cases.

Using Reinforcement Learning to Control Traffic Signals in a Real-World Scenario: An Approach Based on Linear Function Approximation

Reinforcement learning is an efficient, widely used machine learning technique that performs well in problems with a reasonable number of states and actions. This is rarely the case regarding control-related problems, as for instance controlling traffic signals, where the state space can be very large. One way to deal with the curse of dimensionality is to use generalization techniques such as function approximation. In this paper, a linear function approximation is used by traffic signal agents in a network of signalized intersections. Specifically, a true online SARSA <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\lambda)$ </tex-math></inline-formula> algorithm with Fourier basis functions (TOS( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> )-FB) is employed. This method has the advantage of having convergence guarantees and error bounds, a drawback of non-linear function approximation. In order to evaluate TOS( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> )-FB, we perform experiments in variations of an isolated intersection scenario and a scenario of the city of Cottbus, Germany, with 22 signalized intersections, implemented in MATSim. We compare our results not only to fixed-time controllers, but also to a state-of-the-art rule-based adaptive method, showing that TOS( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> )-FB shows a performance that is highly superior to the fixed-time, while also being at least as efficient as the rule-based approach. For more than half of the intersections, our approach leads to less congestion and delay, without the need for the knowledge that underlies the rule-based approach.