Nonequispaced Fast Fourier Transform Research Articles

Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform

The well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various applications such as MRI and solution of PDEs are interested in the inverse problem, i.e., computing Fourier coefficients from given nonequispaced data. In this article, we survey different kinds of approaches to tackle this problem. In contrast to iterative procedures, where multiple iteration steps are needed for computing a solution, we focus especially on so-called direct inversion methods. We review density compensation techniques and introduce a new scheme that leads to an exact reconstruction for trigonometric polynomials. In addition, we consider a matrix optimization approach using Frobenius norm minimization to obtain an inverse NFFT.

Nonequispaced Fast Fourier Transform Boost for the Sinkhorn Algorithm

This contribution features an accelerated computation of Sinkhorn\'s algorithm, which approximates the Wasserstein transportation distance, by employing nonequispaced fast Fourier transforms (NFFT).The proposed algorithm allows approximations of the Wasserstein distance by involving not more than $\mathcal O(n\log n)$ operations for probability measures supported by $n$ points.Furthermore, the proposed method avoids expensive allocations of the characterizing matrices.With this numerical acceleration, the transportation distance is accessible to probability measures out of reach so far.Numerical experiments using synthetic and real data affirm the computational advantage and superiority.

Learning in high-dimensional feature spaces using ANOVA-based fast matrix-vector multiplication

<p style='text-indent:20px;'>Kernel matrices are crucial in many learning tasks such as support vector machines or kernel ridge regression. The kernel matrix is typically dense and large-scale. Depending on the dimension of the feature space even the computation of all of its entries in reasonable time becomes a challenging task. For such dense matrices the cost of a matrix-vector product scales quadratically with the dimensionality <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula>, if no customized methods are applied. We propose the use of an ANOVA kernel, where we construct several kernels based on lower-dimensional feature spaces for which we provide fast algorithms realizing the matrix-vector products. We employ the non-equispaced fast Fourier transform (NFFT), which is of linear complexity for fixed accuracy. Based on a feature grouping approach, we then show how the fast matrix-vector products can be embedded into a learning method choosing kernel ridge regression and the conjugate gradient solver. We illustrate the performance of our approach on several data sets.</p>

Uniform error estimates for nonequispaced fast Fourier transforms

In this paper, we study the error behavior of the nonequispaced fast Fourier transform (NFFT). This approximate algorithm is mainly based on the convenient choice of a compactly supported window function. So far, various window functions have been used and new window functions have recently been proposed. We present novel error estimates for NFFT with compactly supported, continuous window functions and derive rules for convenient choice from the parameters involved in NFFT. The error constant of a window function depends mainly on the oversampling factor and the truncation parameter.

Continuous window functions for NFFT

In this paper, we study the error behavior of the nonequispaced fast Fourier transform (NFFT). This approximate algorithm is mainly based on the convenient choice of a compactly supported window function. Here, we consider the continuous Kaiser–Bessel, continuous exp-type, sinh-type, and continuous cosh-type window functions with the same support and same shape parameter. We present novel explicit error estimates for NFFT with such a window function and derive rules for the optimal choice of the parameters involved in NFFT. The error constant of a window function depends mainly on the oversampling factor and the truncation parameter. For the considered continuous window functions, the error constants have an exponential decay with respect to the truncation parameter.

Semi-supervised Learning for Aggregated Multilayer Graphs Using Diffuse Interface Methods and Fast Matrix-Vector Products

We generalize a graph-based multiclass semi-supervised classification technique based on diffuse interface methods to multilayer graphs. Besides the treatment of various applications with an inherent multilayer structure, we present a very flexible approach that interprets high-dimensional data in a low-dimensional multilayer graph representation. Highly efficient numerical methods involving the spectral decomposition of the corresponding differential graph operators as well as fast matrix-vector products based on the nonequispaced fast Fourier transform (NFFT) enable the rapid treatment of large and high-dimensional data sets. We perform various numerical tests putting a special focus on image segmentation. In particular, we test the performance of our method on data sets with up to 10 million nodes per layer as well as up to 104 dimensions resulting in graphs with up to 52 layers. While all presented numerical experiments can be run on an average laptop computer, the linear dependence per iteration step of the runtime on the network size in all stages of our algorithm makes it scalable to even larger and higher-dimensional problems.

Direct inversion of the nonequispaced fast Fourier transform

Various applications such as MRI, solution of PDEs, etc. need to perform an inverse nonequispaced fast Fourier transform (NFFT), i.e., compute M Fourier coefficients from given N nonequispaced data. In the present paper we consider direct methods for the inversion of the NFFT. We introduce algorithms for the setting M=N as well as for the underdetermined and overdetermined cases. For the setting M=N a direct method of complexity O(Nlog⁡N) is presented, which utilizes Lagrange interpolation and the fast summation. For the remaining cases, we use the matrix representation of the NFFT to deduce our algorithms. Thereby, we are able to compute an inverse NFFT up to a certain accuracy by means of a modified adjoint NFFT in O(Mlog⁡M+N) arithmetic operations. Finally, we show that these approaches can also be explained by means of frame approximation.

NFFT Meets Krylov Methods: Fast Matrix-Vector Products for the Graph Laplacian of Fully Connected Networks

The graph Laplacian is a standard tool in data science, machine learning, and image processing. The corresponding matrix inherits the complex structure of the underlying network and is in certain applications densely populated. This makes computations, in particular matrix-vector products, with the graph Laplacian a hard task. A typical application is the computation of a number of its eigenvalues and eigenvectors. Standard methods become infeasible as the number of nodes in the graph is too large. We propose the use of the fast summation based on the nonequispaced fast Fourier transform (NFFT) to perform the dense matrix-vector product with the graph Laplacian fast without ever forming the whole matrix. The enormous flexibility of the NFFT algorithm allows us to embed the accelerated multiplication into Lanczos-based eigenvalues routines or iterative linear system solvers and even consider other than the standard Gaussian kernels. We illustrate the feasibility of our approach on a number of test problems from image segmentation to semi-supervised learning based on graph-based PDEs. In particular, we compare our approach with the Nystr\"om method. Moreover, we present and test an enhanced, hybrid version of the Nystr\"om method, which internally uses the NFFT.

Random noise attenuation of non-uniformly sampled 3D seismic data along two spatial coordinates using non-equispaced curvelet transform

The attenuation of random noise is important for improving the signal to noise ratio (SNR). However, the precondition for most conventional denoising methods is that the noisy data must be sampled on a uniform grid, making the conventional methods unsuitable for non-uniformly sampled data. In this paper, a denoising method capable of regularizing the noisy data from a non-uniform grid to a specified uniform grid is proposed. Firstly, the denoising method is performed for every time slice extracted from the 3D noisy data along the source and receiver directions, then the 2D non-equispaced fast Fourier transform (NFFT) is introduced in the conventional fast discrete curvelet transform (FDCT). The non-equispaced fast discrete curvelet transform (NFDCT) can be achieved based on the regularized inversion of an operator that links the uniformly sampled curvelet coefficients to the non-uniformly sampled noisy data. The uniform curvelet coefficients can be calculated by using the inversion algorithm of the spectral projected-gradient for ℓ1-norm problems. Then local threshold factors are chosen for the uniform curvelet coefficients for each decomposition scale, and effective curvelet coefficients are obtained respectively for each scale. Finally, the conventional inverse FDCT is applied to the effective curvelet coefficients. This completes the proposed 3D denoising method using the non-equispaced curvelet transform in the source-receiver domain. The examples for synthetic data and real data reveal the effectiveness of the proposed approach in applications to noise attenuation for non-uniformly sampled data compared with the conventional FDCT method and wavelet transformation.

Spherical Field Transformation Above Perfectly Electrically Conducting Ground Planes

A new spherical field transformation approach for near-field measurements above perfectly electrically conducting (PEC) ground planes at arbitrary heights is presented. It provides higher order probe correction and treatment of irregular sample point locations on the measurement surface but has the same low computational complexity as the known fast Fourier transform (FFT) accelerated spherical field transformation with first-order probe correction. The infinite PEC boundary is considered by applying the image principle to the spherical modes of the device under test (DUT). Only the measurements above the ground are required to iteratively solve the inverse problem for the unknown spherical mode coefficients. Due to additional modal translation operators, the scan center of the spherical measurement system does not need to lie in the ground plane. Moreover, if the DUT is positioned at an appropriate height above the ground plane, the free-space sources of the DUT are extractable with good accuracy. By employing the nonequispaced FFT, the near field can be irregularly sampled on the spherical measurement surface. Numerical investigations show the capabilities of the proposed field transformation approach. Its practical relevance and efficiency are demonstrated for simulated automotive antenna measurements and real near-field measurements above a metallic ground.

Automated Software Acceleration in Programmable Logic for an Efficient NFFT Algorithm Implementation: A Case Study.

Non-equispaced Fast Fourier transform (NFFT) is a very important algorithm in several technological and scientific areas such as synthetic aperture radar, computational photography, medical imaging, telecommunications, seismic analysis and so on. However, its computation complexity is high. In this paper, we describe an efficient NFFT implementation with a hardware coprocessor using an All-Programmable System-on-Chip (APSoC). This is a hybrid device that employs an Advanced RISC Machine (ARM) as Processing System with Programmable Logic for high-performance digital signal processing through parallelism and pipeline techniques. The algorithm has been coded in C language with pragma directives to optimize the architecture of the system. We have used the very novel Software Develop System-on-Chip (SDSoC) evelopment tool that simplifies the interface and partitioning between hardware and software. This provides shorter development cycles and iterative improvements by exploring several architectures of the global system. The computational results shows that hardware acceleration significantly outperformed the software based implementation.

INFFTM: Fast evaluation of 3d Fourier series in MATLAB with an application to quantum vortex reconnections

Although Fourier series approximation is ubiquitous in computational physics owing to the Fast Fourier Transform (FFT) algorithm, efficient techniques for the fast evaluation of a three-dimensional truncated Fourier series at a set of arbitrary points are quite rare, especially in MATLAB language. Here we employ the Nonequispaced Fast Fourier Transform (NFFT, by J. Keiner, S. Kunis, and D. Potts), a C library designed for this purpose, and provide a Matlab® and GNU Octave interface that makes NFFT easily available to the Numerical Analysis community. We test the effectiveness of our package in the framework of quantum vortex reconnections, where pseudospectral Fourier methods are commonly used and local high resolution is required in the post-processing stage. We show that the efficient evaluation of a truncated Fourier series at arbitrary points provides excellent results at a computational cost much smaller than carrying out a numerical simulation of the problem on a sufficiently fine regular grid that can reproduce comparable details of the reconnecting vortices. Program summaryProgram Title: INFFTMProgram Files doi:http://dx.doi.org/10.17632/nx5zzp5xxj.1Licensing provisions: GNU GPLv3Programming language: MATLAB/GNU OctaveNature of problem: Evaluation of 3d Fourier series at arbitrary rectilinear grids or sets of arbitrary points, with application to quantum vortex reconnection.Solution method: Fast n-dimensional linear transform of an n-d tensor (rectilinear grids), NFFT (Nonuniform Fast Fourier Transform, arbitrary points), time splitting spectral Fourier method (Gross–Pitaevskii equation for superfluids).External routines/libraries: NFFT (optional but recommended).

A splitting approach for the magnetic Schrödinger equation

The Schrödinger equation in the presence of an external electromagnetic field is an important problem in computational quantum mechanics. It also provides a nice example of a differential equation whose flow can be split with benefit into three parts. After presenting a splitting approach for three operators with two of them being unbounded, we exemplarily prove first-order convergence of Lie splitting in this framework. The result is then applied to the magnetic Schrödinger equation, which is split into its potential, kinetic and advective parts. The latter requires special treatment in order not to lose the conservation properties of the scheme. We discuss several options. Numerical examples in one, two and three space dimensions show that the method of characteristics coupled with a nonequispaced fast Fourier transform (NFFT) provides a fast and reliable technique for achieving mass conservation at the discrete level.

Analytic methods for geometric modeling via spherical decomposition

Analytic methods are emerging in solid and configuration modeling, while providing new insights into a variety of shape and motion related problems by exploiting tools from group morphology, convolution algebras, and harmonic analysis. However, most convolution-based methods have used uniform grid-based sampling to take advantage of the fast Fourier transform (FFT) algorithm. We propose a new paradigm for more efficient computation of analytic correlations that relies on a grid-free discretization of arbitrary shapes as countable unions of balls, in turn described as sublevel sets of summations of smooth radial kernels at adaptively sampled ‘knots’. Using a simple geometric lifting trick, we interpret this combination as a convolution of an impulsive skeletal density and primitive kernels with conical support, which faithfully embeds into the convolution formulation of interactions across different objects. Our approach enables fusion of search-efficient combinatorial data structures prevalent in time-critical collision and proximity queries with analytic methods popular in path planning and protein docking, and outperforms uniform grid-based FFT methods by leveraging nonequispaced FFTs. We provide example applications in formulating holonomic collision constraints, shape complementarity metrics, and morphological operations, unified within a single analytic framework.

Fast Ewald summation based on NFFT with mixed periodicity

In this paper we develop new fast Fourier-based methods for the Coulomb problem. We combine the Ewald summation formulas and the fast summation approach based on the nonequispaced fast Fourier transform (NFFT) in order to develop efficient methods for calculating the Coulomb energies as well as the acting forces in charged particle systems subject to mixed periodic boundary conditions. Therewith, we extend the applicability of NFFT based methods, which already exist for open as well as for 3d-periodic boundary conditions, to arbitrary combinations of periodic and open boundary conditions. We reconsider the derivation of the Ewald formulas for 2d- and 1d-periodic systems, introduce the new algorithms and present high precision numerical results.

Fast simulation of atmospheric turbulence phase screen based on non-uniform sampling

The generation of atmosphere turbulence wave-front is important for studying the light propagation and imaging through the atmosphere, and correcting the atmosphere turbulence, such as the adaptive optics system. The power spectral density method generates phase screens quickly for using the fast Fourier transform (FFT). The main drawback to this approach is that lower order aberrations such as tilt are often under represented. The reason is that the low frequency is sampled inadequately. Since the low order aberrations include a major percentage of the atmospheric energy spectrum, the error of simulated phase screens makes this method less desirable to use. To overcome this shortcoming, a non-uniform sampling method is proposed to generate phase screens accurately. Unfortunately, when the sampling is nonuniform, the FFT does not apply directly. Generating such a phase screen is computation intensive which greatly reduces simulation speed. In this paper, we develop a fast, more accurate method to generate atmospheric turbulence phase screens, according to non-uniforming sampling. The nonequispaced fast Fourier transform (NUFFT) arises in a variety of application areas, ranging from medical imaging to radio astronomy to the numerical solution of partial differential equations. Speeding up the simulation of atmospheric turbulence phase screens is possible by using the non-uniform fast Fourier transform. In this paper, the atmospheric turbulence phase screen is decomposed into a series of harmonics. Then the non-uniform distributed harmonics are projected onto over-sampled uniform grid by using the Gaussian kernel function. Atmospheric turbulence phase screen will be generated using the standard fast Fourier transform on the over-sampled uniform grid. The atmospheric turbulence phase screens can be generated quickly. Using Kolmogorov spectrum model in this paper, the phase screens can be generated quickly. The performances of generated phase screens are analyzed through their phase structure functions. The statistical results are in very good agreement with the theoretical values. The relative error curve of simulation phase screens is calculated and analyzed. The more the oversampling grid, the more the relative error is. Compared with the result from the direct harmonics summation method, the error here mainly concentrates in high-frequency region where the sampling frequency points are sparse. However, the atmosphere turbulence phase screen is simulated in high accuracy on the whole. Compared with the time cost of the harmonics summation, the time using NUFFT is decreased to about 800 times. The simulated phase screens indicate that non-uniform fast Fourier transform is able to generate atmospheric turbulence phase screen with high accuracy and fast speed.

The Fourier approximation of smooth but non-periodic functions from unevenly spaced data

We develop an algorithm to extend, to the nonequispaced case, a recently-introduced fast algorithm for constructing spectrally-accurate Fourier approximations of smooth, but nonperiodic, data. Fast Fourier continuation algorithms, which allow for the Fourier approximation to be periodic in an extended domain, are combined with the underlying ideas behind nonequispaced fast Fourier transform (NFFT) algorithms. The result is a method which allows for the fast and accurate approximation of unevenly sampled nonperiodic multivariate data by Fourier series. A particular contribution of the proposed method is that its formulation avoids the difficulties related to the conditioning of the linear systems that must be solved in order to construct a Fourier continuation. The efficiency, essentially equivalent to that of an NFFT, and accuracy of the algorithm is shown through a number of numerical examples. Numerical results demonstrate the spectral rate of convergence of this method for sufficiently smooth functions. The accuracy, for sufficiently large data sets, is shown to be improved by several orders of magnitudes over previously published techniques for scattered data interpolation.

The nonequispaced FFT on graphics processing units

AbstractWithout doubt, the fast Fourier transform (FFT) belongs to the algorithms with large impact on science and engineering. By appropriate approximations, this scheme has been generalized for arbitrary spatial sampling points. This so called nonequispaced FFT is the core of the sequential NFFT3 library and we discuss its computational costs in detail. On the other hand, programmable graphics processing units have evolved into highly parallel, multithreaded, manycore processors with enormous computational capacity and very high memory bandwidth. By means of the so called Compute Unified Device Architecture (CUDA), we parallelized the nonequispaced FFT using the CUDA FFT library and a dedicated parallelization of the approximation scheme. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A New Algorithm for the Nonequispaced Fast Fourier Transform on the Rotation Group

We develop an approximate algorithm to efficiently calculate the discrete Fourier transform on the rotation group $\text{SO}(3)$. Our method needs $\mathcal{O}(L^3 \log L \log (1/\varepsilon) + \log^3 (1/\varepsilon)\,Q)$ arithmetic operations for a degree-$L$ transform at $Q$ nodes free of choice and with desired accuracy $\varepsilon$. Our main contribution is a method that allows us to replace finite expansions in Wigner-$d$ functions of arbitrary orders with those of low orders. It is based on new insights into the structure of related semiseparable matrices. This enables us to employ an established divide-and-conquer algorithm for symmetric semiseparable eigenproblems together with the fast multipole method to achieve an efficient algorithm.

Fast electrostatic halftoning

Electrostatic halftoning is a high-quality method for stippling, dithering, and sampling, but it suffers from a high runtime. This made the technique difficult to use for most real-world applications. A recently proposed minimisation scheme based on the non-equispaced fast Fourier transform (NFFT) lowers the complexity in the particle number M from $$\mathcal{O}(M^2)$$ to $$\mathcal{O}(M \log M).$$ However, the NFFT is hard to parallelise, and the runtime on modern CPUs lies still in the orders of an hour for about 50,000 particles, to a day for 1 million particles. Our contributions to remedy this problem are threefold: we design the first GPU-based NFFT algorithm without special structural assumptions on the positions of nodes, we introduce a novel nearest-neighbour identification scheme for continuous point distributions, and we optimise the whole algorithm for n-body problems such as electrostatic halftoning. For 1 million particles, this new algorithm runs 50 times faster than the most efficient technique on the CPU, and even yields a speedup of 7,000 over the original algorithm.