Nonuniform Fast Fourier Transform Research Articles

Reduction of ringing artifacts induced by diaphragm drifting in free-breathing dynamic pulmonary MRI using 3D koosh-ball acquisition.

To reduce the ringing artifacts of the motion-resolved images in free-breathing dynamic pulmonary MRI. A golden-step based interleaving (GSI) technique was proposed to reduce ringing artifacts induced by diaphragm drifting. The pulmonary MRI data were acquired using a superior-inferior navigated 3D radial UTE sequence in an interleaved manner during free breathing. Successive interleaves were acquired in an incoherent fashion along the polar direction. Four-dimensional images were reconstructed from the motion-resolved k-space data obtained by retrospectively binning. The reconstruction algorithms included standard nonuniform fast Fourier transform (NUFFT), Voronoi-density-compensated NUFFT, extra-dimensional UTE, and motion-state weighted motion-compensation reconstruction. The proposed interleaving technique was compared with a conventional sequential interleaving (SeqI) technique on a phantom and eight subjects. The quantified ringing artifacts level in the motion-resolved image is positively correlated with the quantified nonuniformity level of the corresponding k-space. The nonuniformity levels of the end-expiratory and end-inspiratory k-space binned from GSI data (0.34 ± 0.07, 0.33 ± 0.05) are significantly lower with statistical significance (p < 0.05) than that binned from SeqI data (0.44 ± 0.11, 0.42 ± 0.12). Ringing artifacts are substantially reduced in the dynamic images of eight subjects acquired using the proposed technique in comparison with that acquired using the conventional SeqI technique. Ringing artifacts in the motion-resolved images induced by diaphragm drifting can be reduced using the proposed GSI technique for free-breathing dynamic pulmonary MRI. This technique has the potential to reduce ringing artifacts in free-breathing liver and kidney MRI based on full-echo interleaved 3D radial acquisition.

Nifty-ls: Fast and Accurate Lomb–Scargle Periodograms Using a Non-uniform FFT

We present nifty-ls, a software package for fast and accurate evaluation of the Lomb–Scargle periodogram. nifty-ls leverages the fact that Lomb–Scargle can be computed using a non-uniform fast Fourier transform (NUFFT), which we evaluate with the Flatiron Institute NUFFT package (finufft). This approach achieves a many-fold speedup over the Press & Rybicki method as implemented in Astropy and is simultaneously many orders of magnitude more accurate. nifty-ls also supports fast evaluation on GPUs via CUDA and integrates with the Astropy Lomb–Scargle interface. nifty-ls is publicly available at https://github.com/flatironinstitute/nifty-ls/.

Ray-tracing versus Born approximation in full-sky weak lensing simulations of the MillenniumTNG project

ABSTRACT Weak gravitational lensing is a powerful tool for precision tests of cosmology. As the expected deflection angles are small, predictions based on non-linear N-body simulations are commonly computed with the Born approximation. Here, we examine this assumption using DORIAN, a newly developed full-sky ray-tracing scheme applied to high-resolution mass-shell outputs of the two largest simulations in the MillenniumTNG suite, each with a 3000 Mpc box containing almost 1.1 trillion cold dark matter particles in addition to 16.7 billion particles representing massive neutrinos. We examine simple two-point statistics like the angular power spectrum of the convergence field, as well as statistics sensitive to higher order correlations such as peak and minimum statistics, void statistics, and Minkowski functionals of the convergence maps. Overall, we find only small differences between the Born approximation and a full ray-tracing treatment. While these are negligibly small at power-spectrum level, some higher order statistics show more sizeable effects; ray-tracing is necessary to achieve per cent level precision. At the resolution reached here, full-sky maps with 0.8 billion pixels and an angular resolution of 0.43 arcmin, we find that interpolation accuracy can introduce appreciable errors in ray-tracing results. We therefore implemented an interpolation method based on non-uniform fast Fourier transforms (NUFFT) along with more traditional methods. Bilinear interpolation introduces significant smoothing, while nearest grid point sampling agrees well with NUFFT, at least for our fiducial source redshift, $z_s=1.0$, and for the 1 arcmin smoothing we use for higher order statistics.

AutoSamp: Autoencoding k-space Sampling via Variational Information Maximization for 3D MRI.

Accelerated MRI protocols routinely involve a predefined sampling pattern that undersamples the k-space. Finding an optimal pattern can enhance the reconstruction quality, however this optimization is a challenging task. To address this challenge, we introduce a novel deep learning framework, AutoSamp, based on variational information maximization that enables joint optimization of sampling pattern and reconstruction of MRI scans. We represent the encoder as a non-uniform Fast Fourier Transform that allows continuous optimization of k-space sample locations on a non-Cartesian plane, and the decoder as a deep reconstruction network. Experiments on public 3D acquired MRI datasets show improved reconstruction quality of the proposed AutoSamp method over the prevailing variable density and variable density Poisson disc sampling for both compressed sensing and deep learning reconstructions. We demonstrate that our data-driven sampling optimization method achieves 4.4dB, 2.0dB, 0.75dB, 0.7dB PSNR improvements over reconstruction with Poisson Disc masks for acceleration factors of R = 5, 10, 15, 25, respectively. Prospectively accelerated acquisitions with 3D FSE sequences using our optimized sampling patterns exhibit improved image quality and sharpness. Furthermore, we analyze the characteristics of the learned sampling patterns with respect to changes in acceleration factor, measurement noise, underlying anatomy, and coil sensitivities. We show that all these factors contribute to the optimization result by affecting the sampling density, k-space coverage and point spread functions of the learned sampling patterns.

Bayesian-guided operational modal identification of a highway bridge considering non-uniform sampling

During the structural health monitoring of bridges, it has been observed that the vibration data collected can sometimes be randomly lost or sampled non-uniformly. This leads to a low signal-to-noise ratio in the spectral functions of the measured data, making it difficult to identify weak modes. To address this issue, a framework for operational modal identification is proposed in this study. It utilizes the fast Bayesian fast Fourier transform (FFT) method to estimate the modal parameters of highway bridges considering the non-uniform monitoring data. The initial frequency parameters for the fast Bayesian FFT approach are automatically determined using the proposed autoregressive (AR) power spectral density (PSD)-guided peak picking method. This overcomes the challenge of capturing initial frequencies related to weakly contributed modes. Additionally, the bandwidth parameter for each mode is determined using the modal assurance criterion (MAC) of the first left singular vectors of PSD matrices. Furthermore, when analyzing non-uniform vibration data, it is recommended to use the non-uniform FFT (NUFFT) for calculating PSD functions in order to improve identification accuracy. The proposed method is validated using acceleration data from both a numerical model and a real-world bridge. The results demonstrate that the identification uncertainty of modal parameters increases with higher non-uniform levels.

Simulation of multivariate ergodic stochastic processes using adaptive spectral sampling and non-uniform fast Fourier transform

The simulation of multivariate ergodic stochastic processes is critical for structural dynamic analysis and reliability evaluation. Although the traditional spectral representation method (SRM) has a wide application in many areas, it is highly inefficient in simulating stochastic processes with many simulation points or long durations due to the significant computational cost associated with matrix factorizations concerning frequency. To address the encountered challenge, this paper presents an efficient approach for simulating ergodic stochastic processes with limited frequencies. Central to this approach is a fusion of the adaptive spectral sampling and the non-uniform fast Fourier transform (NUFFT) techniques. The adaptive spectral sampling of the envelope spectrum enables the determination of limited non-equispaced frequencies, which are randomly sampled according to a uniform distribution. Thus, the Cholesky decomposition is only required at limited specific frequencies, which dramatically reduces the computational cost of matrix factorizations. Since the randomly sampled frequencies are not equispaced, utilizing FFT to accelerate the summation of trigonometric functions becomes impractical. Then, the NUFFT that adapts the non-equispaced sampling points is employed instead to expedite this process with the non-uniform increment approximated through reduced interpolation. By taking the wind field simulation of a long-span suspension bridge as an example, a parametric analysis is conducted to investigate the effect of random frequencies on the simulation error of the developed approach and the convergence of spectra. Finally, the developed approach is further validated by focusing on the spectra and probabilistic density functions of the simulated wind samples, and the simulation performance is compared with that of the traditional approach. The analytical results demonstrate the efficiency and accuracy of the developed approach in simulating ergodic stochastic processes.

A Fourier spectral immersed boundary method with exact translation invariance, improved boundary resolution, and a divergence-free velocity field

This paper introduces a new immersed boundary (IB) method for viscous incompressible flow, based on a Fourier spectral method for the fluid solver and on the nonuniform fast Fourier transform (NUFFT) algorithm for coupling the fluid with the immersed boundary. The new Fourier spectral immersed boundary (FSIB) method gives improved boundary resolution in comparison to the standard IB method. The interpolated velocity field, in which the boundary moves, is analytically divergence-free. The FSIB method is gridless and has the meritorious properties of volume conservation, exact translation invariance, conservation of momentum, and conservation of energy. We verify these advantages of the FSIB method numerically both for the Stokes equations and for the Navier-Stokes equations in both two and three space dimensions. The FSIB method converges faster than the IB method. In particular, we observe second-order convergence in various problems for the Navier-Stokes equations in three dimensions. The FSIB method is also computationally efficient with complexity of O(N3log⁡(N)) per time step for N3 Fourier modes in three dimensions.

Band-amplified angle spectrum method for the phase hologram design to achieve high-quality holographic imaging in long distance

The angular spectrum method (ASM) is widely used in the design of computer-generated holograms (CGH) due to its strict adherence to the Helmholtz equations and the potential for accelerated computation using fast Fourier transform (FFT). However, when dealing with long distance propagation conditions, the reduction of effective spectral components and spectral aliasing will introduce significant computation errors, leading to a substantial degradation in holographic imaging quality. To achieve high-quality holographic imaging design in long distance, we introduce a novel angular spectrum computation method, the calculation process is divided into two steps: (1) First propagate a longer distance from the CGH plane to an intermediate plane. (2) Then continue to propagate to the observation plane after incorporating a spread spectrum phase factor to amplify the effective non-aliased bandwidth in the intermediate plane and eliminating the aliasing artifacts. Furthermore, the non-uniform fast Fourier transform (NUFFT) algorithm is adopted which allows for the flexible adjustments of the sampling interval and quantity in the spatial frequency domain, all while maintaining compliance with the Nyquist sampling theorem even in the context of bandwidth amplification. Both simulations and optical experiments are conducted to validate the feasibility and effectiveness of our proposed method.

Rapid 2D 23Na MRI of the calf using a denoising convolutional neural network

Purpose23Na MRI can be used to quantify in-vivo tissue sodium concentration (TSC), but the inherently low 23Na signal leads to long scan times and/or noisy or low-resolution images. Reconstruction algorithms such as compressed sensing (CS) have been proposed to mitigate low signal-to-noise ratio (SNR); although, these can result in unnatural images, suboptimal denoising and long processing times. Recently, machine learning has been increasingly used to denoise 1H MRI acquisitions; however, this approach typically requires large volumes of high-quality training data, which is not readily available for 23Na MRI. Here, we propose using 1H data to train a denoising convolutional neural network (CNN), which we subsequently demonstrate on prospective 23Na images of the calf. Methods1893 1H fat-saturated transverse slices of the knee from the open-source fastMRI dataset were used to train denoising CNNs for different levels of noise. Synthetic low SNR images were generated by adding gaussian noise to the high-quality 1H k-space data before reconstruction to create paired training data. For prospective testing, 23Na images of the calf were acquired in 10 healthy volunteers with a total of 150 averages over ten minutes, which were used as a reference throughout the study. From this data, images with fewer averages were retrospectively reconstructed using a non-uniform fast Fourier transform (NUFFT) as well as CS, with the NUFFT images subsequently denoised using the trained CNN. ResultsCNNs were successfully applied to 23Na images reconstructed with 50, 40 and 30 averages. Muscle and skin apparent TSC quantification from CNN-denoised images were equivalent to those from CS images, with <0.9 mM bias compared to reference values. Estimated SNR was significantly higher in CNN-denoised images compared to NUFFT, CS and reference images. Quantitative edge sharpness was equivalent for all images. For subjective image quality ranking, CNN-denoised images ranked equally best with reference images and significantly better than NUFFT and CS images. ConclusionDenoising CNNs trained on 1H data can be successfully applied to 23Na images of the calf; thus, allowing scan time to be reduced from ten minutes to two minutes with little impact on image quality or apparent TSC quantification accuracy.

Model-based reconstruction for looping-star MRI.

The aim of this study was to develop a reconstruction method that more fully models the signals and reconstructs gradient echo (GRE) images without sacrificing the signal to noise ratio and spatial resolution, compared to conventional gridding and model-based image reconstruction method. By modeling the trajectories for every spoke and simplifying the scenario to only echo-in and echo-out mixture, the approach explicitly models the overlapping echoes. After modeling the overlapping echoes with two system matrices, we use the conjugate gradient algorithm (CG-SENSE) with the nonuniform FFT (NUFFT) to optimize the image reconstruction cost function. The proposed method is demonstrated in phantoms and in-vivo volunteer experiments for three-dimensional, high-resolution T2*-weighted imaging and functional MRI tasks. Compared to the gridding method, the high resolution protocol exhibits improved spatial resolution and reduced signal loss as a result of less intra-voxel dephasing. The fMRI task shows that the proposed model-based method produced images with reduced artifacts and blurring as well as more stable and prominent time courses. The proposed model-based reconstruction results shows improved spatial resolution and reduced artifacts. The fMRI task shows improved time series and activation map due to the reduced overlapping echoes and under-sampling artifacts.

Accelerated multiple-quantum-filtered sodium magnetic resonance imaging using compressed sensing at 7 T

PurposeMultiple-quantum-filtered (MQF) sodium magnetic resonance imaging (MRI), such as enhanced single-quantum and triple-quantum-filtered imaging of 23Na (eSISTINA), enables images to be weighted towards restricted sodium, a promising biomarker in clinical practice, but often suffers from clinically infeasible acquisition times and low image quality. This study aims to mitigate the above limitation by implementing a novel eSISTINA sequence at 7 T with the application of compressed sensing (CS) to accelerate eSISTINA acquisitions without a noticeable loss of information. MethodsA novel eSISTINA sequence with a 3D spiral-based sampling scheme was implemented at 7 T for the application of CS. Fully sampled datasets were obtained from one phantom and ten healthy subjects, and were then retrospectively undersampled by various undersampling factors. CS undersampled reconstructions were compared to fully sampled and undersampled nonuniform fast Fourier transform (NUFFT) reconstructions. Reconstruction performance was evaluated based on structural similarity (SSIM), signal-to-noise ratio (SNR), weightings towards total and compartmental sodium, and in vivo quantitative estimates. ResultsCS-based phantom and in vivo images have less noise and better structural delineation while maintaining the weightings towards total, non-restricted (predominantly extracellular), and restricted (primarily intracellular) sodium. CS generally outperforms NUFFT with a higher SNR and a better SSIM, except for the SSIM in TQ brain images, which is likely due to substantial noise contamination. CS enables in vivo quantitative estimates with <15% errors at an undersampling factor of up to two. ConclusionsSuccessful implementation of an eSISTINA sequence with an incoherent sampling scheme at 7 T was demonstrated. CS can accelerate eSISTINA by up to twofold at 7 T with reduced noise levels compared to NUFFT, while maintaining major structural information, reasonable weightings towards total and compartmental sodium, and relatively reliable in vivo quantification. The associated reduction in acquisition time has the potential to facilitate the clinical applicability of MQF sodium MRI.

Acquisition Acceleration of Ultra-low Field MRI with Parallel Imaging and Compressed Sensing in Microtesla Fields.

In recent years, ultra-low field (ULF) magnetic resonance imaging (MRI) has gained widespread attention due to its advantages, such as low cost, light weight, and portability. However, the low signal-to-noise ratio (SNR) leads to a long scan time. Herein, we study the acceleration performance of parallel imaging (PI) and compressed sensing (CS) in different kspace sampling strategies at 0.12 mT. This study employs phantoms to assess the efficiency of acceleration methods at ULF MRI, in which signals are detected by ultra-sensitive superconducting quantum interference devices (SQUIDs). We compare the performance of fast Fourier transform (FFT), generalized auto-calibrating partially parallel acquisitions (GRAPPA), and eigenvector-based SPIRiT (ESPIRiT) in Cartesian sampling, while also evaluating non-uniform FFT (NUFFT), GRAPPA operator gridding, and ESPIRiT in nonCartesian sampling. We design a resolution phantom to investigate the effectiveness of these methods in maintaining image resolution. In Cartesian sampling, GRAPPA and ESPIRiT jointly regularized by total variation and ℓ1-norm (TVJℓ1 -ESPIRiT) methods reconstructed good-quality phantom images with an acceleration factor of R = 2. In contrast, TVJℓ1-ESPIRiT exhibited improved image quality and much less signal loss even for R = 4. In radial sampling, TVJℓ1-ESPIRiT reduced the acquisition time to 1.69 minutes at R = 4, with a respective improvement of 12.26 dB in peak SNR compared to NUFFT. The resolution phantom imaging showed that the reconstructions by PI and CS maintained the original resolution of 2 mm. This study improves the practicality of ULF MRI at microtesla fields by implementing imaging acceleration with PI and CS in different k-space sampling.

Proposal for a Doppler Shift Compensation Method Using Non-Uniform FFT with Pilot Carrier Frequency for OFDM-Based Underwater Acoustic Communication Systems

In this paper, we propose a method that uses the non-uniform Fast Fourier Transform (FFT) to compensate for Doppler frequency shifts in orthogonal frequency division multiplexing (OFDM) based underwater acoustic (UWA) communication systems. To estimate the Doppler frequency shift, the paper proposes using a pilot carrier frequency (PCF) that is identified by transmitting with greater power in the frequency domain compared to other subcarriers. This identifier helps estimate both the Doppler frequency shift and the channel. The paper proposes estimating and compensating for the Doppler frequency shift in two steps: Step 1 performs coarse frequency synchronization, where the Doppler frequency shift is determined by using PCF signal. The Doppler shift correction is obtained by re-sampling the received signal, and then interpolating re-sampled signal. However, the Doppler shift compensation in the step 1 cannot correct the phase distortion of the measured PCF signal. Therefore, performing step 2, known as fine frequency synchronization, is necessary. In this step, the correction for the phase distortion is determined based on the phase difference between the two consecutive OFDM signals in one frame. The remaining Doppler frequency shift is adjusted based on the phase deviation, using the non-uniform FFT. By using the non-uniform FFT, the complexity of the ICI compensation is significantly reduced, and the quality of Doppler shift compensation is improved. The experience results show that the transmitted text will be decoded correctly by using the proposed technique

Simulation of turbulent wind field in multi-spatial dimensions using a novel non-uniform FFT enhanced stochastic wave-based spectral representation method

Simulation of stochastic wind field is necessary for wind-induced vibration analysis of wind-sensitive structures. Stochastic wave-based spectral representation method (SWSRM) is one of the most widely used methods for wind field simulation. However, it may face challenges to simulate turbulent wind field in multi-spatial dimensions due to its large memory consumption and low simulation efficiency. In this study, a novel non-uniform FFT (NUFFT) enhanced SWSRM is proposed. The approach for selecting sampling points of wave number is also established based on adaptive integral method. Numerical experiments are further designed to demonstrate the validity and effectiveness of the enhanced SWSRM. Results show that the proposed method can significantly improve the simulation efficiency and reduce the memory consumption in multi-spatial dimensions. In addition, the simulation points can be arbitrarily selected using NUFFT enhanced SWSRM. More importantly, the proposed method can improve the turbulent spectral accuracy in low-frequency region.

Application of the non-uniform Fourier transform to non-uniformly sampled Fourier transform spectrometers

Resampling by interpolation is the traditional method to process interferograms from non-uniformly sampled Fourier transform spectrometers. The non-uniform fast Fourier transform (NUFFT) is an alternative approach that has been mostly overlooked. With the aid of experiments on a high-resolution interferometer with a variety of optical sources, these two methods are compared. It is found that the NUFFT is comparable to interpolation in spectral profile shape and spectral noise levels and is better in spectral amplitude and computer performance. A significant advantage is also found in the case of under-sampling and noise performance by NUFFT due to the unique non-periodic nature of non-uniform sampling. In addition, a novel implementation of NUFFT is presented and analysed.

Efficient simulation of non-stationary non-homogeneous wind field: Fusion of multi-dimensional interpolation and NUFFT

The stochastic wave approach is a new realization of the spectral representation method and has been widely used for the simulation of stationary non-homogeneous or non-stationary homogeneous wind fields. However, it inevitably creates many unnecessary simulation points due to the invoking of the Fast Fourier Transform (FFT) and suffers from low efficiency in non-stationary non-homogeneous scenarios due to the incapability to accelerate the summation of trigonometric functions. In this study, an efficient approach is developed for the simulation of non-stationary non-homogeneous wind field. Central to this approach is a multi-dimensional interpolation used to decouple the evolutionary spectra accompanied by the application of non-uniform FFT (NUFFT) to expedite the simulation. A surrogate model that accurately approximates the 3D evolutionary spectrum related to height, frequency, and time is established by the 3D reduced Hermite interpolation, which enables a successful separation of the frequency components. Then, the NUFFT is applied to speed up the summation of trigonometric functions over wavenumbers with a well-designed non-uniformly distributed sampling points, which effectively reduces the segments used in the Fourier transform. Since there is no required mapping between the wavenumbers and the locations of simulation points in NUFFT, a significant number of unnecessary simulation points in the FFT scenario is avoided. Moreover, the wind samples at different locations can be obtained by directly altering the query points of NUFFT, thus the efficiency of the enhanced approach is almost independent of the number of simulation points. The numerical example in simulating the non-stationary non-homogeneous wind field of a long-span cable-stayed bridge demonstrates the efficiency of the developed approach and validates the effectiveness of the simulated wind samples in view of evolutionary spectrum and coherence function.

Optimization in the space domain for density compensation with the nonuniform FFT

The non-uniform Discrete Fourier Transform algorithm has shown great utility for reconstructing images from non-uniformly spaced Fourier samples in several imaging modalities. Due to the non-uniform spacing, some correction for the variable density of the samples must be made. Common methods for generating density compensation values are either sub-optimal or only consider a finite set of points in the optimization. This manuscript presents an algorithm for generating density compensation values from a set of Fourier samples that takes into account the point spread function over an entire rectangular region in the image domain. We show that the reconstructed images using the density compensation values of this method are of superior quality when compared to other standard methods. Results are shown with a numerical phantom and with magnetic resonance images of the abdomen and the knee.

Modelling and computation of gravitational attraction, gradient tensors, rotational and horizontal invariants of Asteroid Bennu (101955), Itokawa (25143) and Eros (433) via 2D Non-Uniform FFT

The internal structure and mass distribution of the terrestrial objects are yet unknown. The 2D gravity model with a constant density of the terrestrial objects can shed light on the surficial or textural heterogeneity due to topographic variations of the terrestrial objects. Three different asteroids, which are Bennu (101955), Itokawa (25143) and Eros (433) are modelled in this study. During the modelling phase, a different number of edges, elements, nodes, and faces are used to describe the 3D models of Bennu, Itokawa, and Eros. These 3D models are used in 2D Non-Uniform Fast Fourier Transform (NU-FFT) applications to obtain gravitational attraction with a constant density polyhedron model. Tensor gradients and tensor invariants of the modelled gravity anomaly are calculated. Three major outcomes are interpreted from gradient tensors and tensor invariants. Firstly, textural heterogeneity due to relatively low topography is detected in the central part of Bennu. Secondly, considerably different properties which can be related to surface variations between the two lobes of Itokawa are observed. Lastly, directional surficial heterogeneities were detected in Eros.

Efficient Approximation of Jacobian Matrices Involving a Non-Uniform Fast Fourier Transform (NUFFT).

There is growing interest in learning Fourier domain sampling strategies (particularly for magnetic resonance imaging, MRI) using optimization approaches. For non-Cartesian sampling, the system models typically involve non-uniform fast Fourier transform (NUFFT) operations. Commonly used NUFFT algorithms contain frequency domain interpolation, which is not differentiable with respect to the sampling pattern, complicating the use of gradient methods. This paper describes an efficient and accurate approach for computing approximate gradients involving NUFFTs. Multiple numerical experiments validate the improved accuracy and efficiency of the proposed approximation. As an application to computational imaging, the NUFFT Jacobians were used to optimize non-Cartesian MRI sampling trajectories via data-driven stochastic optimization. Specifically, the sampling patterns were learned with respect to various model-based image reconstruction (MBIR) algorithms. The proposed approach enables sampling optimization for image sizes that are infeasible with standard auto-differentiation methods due to memory limits. The synergistic acquisition and reconstruction design leads to remarkably improved image quality. In fact, we show that model-based image reconstruction methods with suitably optimized imaging parameters can perform nearly as well as CNN-based methods.

Optimized, Parallel Nonuniform Fast Fourier Transform (FFT) for Radiation From Aperiodic Arrays

Optimized, Parallel Nonuniform Fast Fourier Transform (FFT) for Radiation From Aperiodic Arrays