Nonuniform Fast Fourier Transform Research Articles

Noise-Resistant Estimation Algorithm for TDOA, FDOA and Differential Doppler Rate in Passive Sensing

This paper addresses the joint time difference of arrival (TDOA), frequency difference of arrival (FDOA) and differential Doppler rate estimation problem for high-speed maneuvering targets in passive location systems, involving linear range migration (LRM), quadratic range migration (QRM) and linear Doppler frequency migration (LDFM) within observation time. A noise-resistant estimation algorithm based on second-order keystone transform (SKT) and non-uniform fast Fourier transform (NUFFT) is proposed. After QRM correction via SKT, a phase compensation function is constructed to eliminate LRM and estimate the FDOA. Then, NUFFT is used to remove LDFM and realize the joint estimation of TDOA and differential Doppler rate. Comparisons with several relatively new algorithms indicate that the proposed algorithm can obtain a good trade-off between computational cost and estimation performance. Extensive numerical examples, analysis of computational complexity and estimation performance can validate the effectiveness of the proposed method.

Reconstruction of undersampled 3D non-Cartesian image-based navigators for coronary MRA using an unrolled deep learning model.

To rapidly reconstruct undersampled 3D non-Cartesian image-based navigators (iNAVs) using an unrolled deep learning (DL) model, enabling nonrigid motion correction in coronary magnetic resonance angiography (CMRA). An end-to-end unrolled network is trained to reconstruct beat-to-beat 3D iNAVs acquired during a CMRA sequence. The unrolled model incorporates a nonuniform FFT operator in TensorFlow to perform the data-consistency operation, and the regularization term is learned by a convolutional neural network (CNN) based on the proximal gradient descent algorithm. The training set includes 6,000 3D iNAVs acquired from 7 different subjects and 11 scans using a variable-density (VD) cones trajectory. For testing, 3D iNAVs from 4 additional subjects are reconstructed using the unrolled model. To validate reconstruction accuracy, global and localized motion estimates from DL model-based 3D iNAVs are compared with those extracted from 3D iNAVs reconstructed with -ESPIRiT. Then, the high-resolution coronary MRA images motion corrected with autofocusing using the -ESPIRiT and DL model-based 3D iNAVs are assessed for differences. 3D iNAVs reconstructed using the DL model-based approach and conventional -ESPIRiT generate similar global and localized motion estimates and provide equivalent coronary image quality. Reconstruction with the unrolled network completes in a fraction of the time compared to CPU and GPU implementations of -ESPIRiT (20× and 3× speed increases, respectively). We have developed a deep neural network architecture to reconstruct undersampled 3D non-Cartesian VD cones iNAVs. Our approach decreases reconstruction time for 3D iNAVs, while preserving the accuracy of nonrigid motion information offered by them for correction.

A Method for Energy-Efficient Sampling of Analog to Digital Converters

We present PANDA, a data acquisition technique for energy-constrained sensor nodes that reduces the energy per operation required to sample and preprocess analog sensor data. PANDA takes advantage of the energy consumption patterns of commodity microcontrollers by sampling input signals in short bursts followed by long periods of inactivity. This approach reduces the overhead of repetitively transitioning the CPU and analog components in and out of low-power sleep states. This nonuniformly-spaced input data is then fed to a nonuniform FFT algorithm that computes the frequency spectrum. We show that the spectrum computed with the nonuniform FFT is very close to the spectrum that would be computed from uniformly sampled data preprocessed with a conventional FFT. The output of the nonuniform FFT can be filtered or post-processed with conventional frequency domain analysis techniques, and a uniformly resampled output can be constructed with the conventional inverse FFT. We compare the energy consumption patterns of burst-mode sampling to those of conventional uniform sampling in several real sensor nodes. We demonstrate that for reasonably sized input datasets, burst mode sampling and postprocessing consumes more than 17 percent less energy than conventional uniform sampling, including the additional computations required to compute the nonuniform DFT.

Near-Field 3D SAR Imaging Using a Scanning Linear MIMO Array With Arbitrary Topologies

For near-field synthetic aperture radar (SAR) imaging, frequency domain imaging algorithms typically require the array elements to be uniformly arranged to facilitate the application of fast Fourier transform (FFT) operations; this greatly restricts the array types that can be used for practical applications. For arrays with non-uniformly positioned antennas, no general and efficient imaging algorithm is available yet. To address this issue, an effective algorithm for efficient three-dimensional (3D) SAR imaging with arbitrary linear multi-input-multi-output (MIMO) array topologies is proposed. Specifically, non-uniform FFT (NUFFT) technology is utilized to tackle the non-uniform sampling issues in both the spatial and the wavenumber domain. Despite the intuitive idea, to achieve satisfied imaging, some issues are non-trivial and should be carefully addressed. Both simulations and experiments show the superiority of the proposed algorithm on computational complexity while maintain the same high imaging precision compared with the back-projection algorithm (BPA). Moreover, we suggest that the concept of NUFFT-based imaging can also be generalized to other imaging regimes with arbitrary array configurations.

Malliavin--Mancino Estimators Implemented with Nonuniform Fast Fourier Transforms

We implement and test kernel averaging Non-Uniform Fast Fourier Transform (NUFFT) methods to enhance the performance of correlation and covariance estimation on asynchronously sampled event-data using the Malliavin-Mancino Fourier estimator. The methods are benchmarked for Dirichlet and Fej\'{e}r Fourier basis kernels. We consider test cases formed from Geometric Brownian motions to replicate synchronous and asynchronous data for benchmarking purposes. We consider three standard averaging kernels to convolve the event-data for synchronisation via over-sampling for use with the Fast Fourier Transform (FFT): the Gaussian kernel, the Kaiser-Bessel kernel, and the exponential of semi-circle kernel. First, this allows us to demonstrate the performance of the estimator with different combinations of basis kernels and averaging kernels. Second, we investigate and compare the impact of the averaging scales explicit in each averaging kernel and its relationship between the time-scale averaging implicit in the Malliavin-Mancino estimator. Third, we demonstrate the relationship between time-scale averaging based on the number of Fourier coefficients used in the estimator to a theoretical model of the Epps effect. We briefly demonstrate the methods on Trade-and-Quote (TAQ) data from the Johannesburg Stock Exchange to make an initial visualisation of the correlation dynamics for various time-scales under market microstructure.

Fast coherent integration method for moving target detection with random PRI variation

This Letter addresses the coherent integration problem for the moving target detection in a pulse repetition interval (PRI)-staggered radar system. Due to the random sampling, the complex coupling effects between range and azimuth will severely degrade the target parameter estimation and coherent integration performance. To deal with this issue, a fast coherent integration detection method with random azimuth sampling is proposed, where range migration is firstly compensated by applying the range-frequency reversal transform. Then, the random phase fluctuation caused by the staggered PRI can be efficiently removed by applying the non-uniform FFT technique. Finally, a moving target can be well focused. Simulated results are provided to validate the effectiveness of the proposed algorithm.

Radar Coherent Detection for Maneuvering Target Based on Product-Scaled Integrated Cubic Phase Function

This paper considers the long-time coherent detection problem for maneuvering targets with jerk motion. A novel method based on product-scaled integrated cubic phase function (PSICPF) is proposed. The main strategy of PSICPF is to estimate target’s motion parameters along the slow time for each range frequency cell. In order to eliminate the coupling terms between range frequency and slow time, the scaled nonuniform fast Fourier transform (SNUFFT) is newly defined in the integrated cubic phase function (ICPF). Then, the product operation is employed to coherently synthesize the estimation results, improve the antinoise performance, and suppress the cross terms. Finally, coherent integration is achieved via keystone transform (KT) and fold factor searching. Analysis demonstrates that the SNUFFT has the same computational complexity with nonuniform fast Fourier transform (NUFFT), and thus the PSICPF could be efficiently implemented via complex multiplications, the fast Fourier transform (FFT), and NUFFT. Detailed comparisons with other representative methods in computational cost, motion parameter estimation performance, and detection ability indicate that the PSICPF could achieve a good balance between the computational cost and detection ability. Simulations and real data processing results are presented to verify the effectiveness of the proposed method.

NUFFT for the Efficient Spectral Domain MoM Analysis of a Wide Variety of Multilayered Periodic Structures

In this paper the Method of Moments (MoM) in the spectral domain is used for the analysis of multilayered structures containing periodic arrays of either patches or apertures. The patches and apertures may have many different geometries including complex surfaces limited by two parallel lines and two arbitrary curves, circular and elliptic rings, circular and elliptic arcs, and circular and elliptic sectors. Basis functions accounting for edge singularities are used in the approximation of the electric/magnetic current density on the patches/apertures, which enables a fast convergence of MoM with respect to the number of basis functions. Since the 2-D Fourier transforms of the basis functions cannot be obtained in closed-form, these Fourier transforms are efficiently computed by means of the nonuniform fast Fourier transform (NUFFT) algorithm. Results have been obtained for frequency-selective surfaces (FSSs), and for the elements used in the design of both reflectarray and metasurface antennas. The results obtained indicate that the software based on the NUFFT is only 15% slower than the standard spectral domain MoM software used for structures in which the 2-D Fourier transform of the basis functions is analytical, and between 50 and 80 times faster than CST.

Efficient Algorithm for SAR Refocusing of Ground Fast-Maneuvering Targets

The synthetic aperture radar (SAR) image of moving targets will defocus due to the unknown motion parameters. For fast-maneuvering targets, the range cell migration (RCM), Doppler frequency migration and Doppler ambiguity are complex problems. As a result, focusing of fast-maneuvering targets is difficult. In this work, an efficient SAR refocusing algorithm is proposed for fast-maneuvering targets. The proposed algorithm mainly contains three steps. Firstly, the RCM is corrected using sequence reversing, matrix complex multiplication and an improved second-order RCM correction function. Secondly, a 1D scaled Fourier transform is introduced to estimate the remaining chirp rate. Thirdly, a matched filter based on the estimated chirp rate is proposed to focus the maneuvering target in the range–azimuth time domain. The proposed method is computationally efficient because it can be implemented by the fast Fourier transform (FFT), inverse FFT and non-uniform FFT. A new deramp function is proposed to further address the serious problem of Doppler ambiguity. A spurious peak recognition procedure is proposed on the basis of the cross-term analysis. Simulated and real data processing results demonstrate the validity of the proposed target focusing algorithm and spurious peak recognition procedure.

Coherent integration method for random pulse repetition interval radar based on non‐uniform keystone transform and non‐uniform fast Fourier transform

In order to achieve weak target detection, the range migration and Doppler frequency spread induced by the target motion within the coherent processing interval should be compensated to focus the target energy in a certain range-Doppler cell. This article proposes a new integration method based on non-uniform Keystone transform and non-uniform fast Fourier transform (NUFFT) for random pulse repetition interval (PRI) radar. In this method, non-uniform Keystone transform is applied to correct the linear range migration. After that, the acceleration is estimated by dechirp process and the Doppler frequency spread is compensated. Finally, the target energy is accumulated coherently via NUFFT. Compared with the random PRI generalised Radon-Fourier transform (RPRI-GRFT) and the moving target detection method for random PRI radar, the proposed algorithm achieves a good balance between the detection performance and the computational cost. Simulation results are provided to demonstrate the effectiveness of the proposed method.

Feasibility Study of Passive Bistatic Radar Based on Phased Array Radar Signals

This paper presents the concept of a passive bistatic radar (PBR) system using existing phased array radar (PAR) as the source of illumination. Different from PBR based on common civil illuminators of opportunity, we develop an experimental PBR system using an high-power air surveillance PAR with abundant signal modulation forms as the transmitter. After the introduction of the PBR system and PAR signals, it can be concluded that the agility of the waveform parameters of PAR signal brings two problems to the signal processing of the PBR systems, which are not discussed in conventional PBR systems. The first problem is the time and frequency synchronization of the system, so we propose a direct wave parameter estimation method based on template matching to estimate the parameters of the transmitted signal in real time to achieve time and frequency synchronization of the system. The second problem is the coherent integration for moving target detection and weak target detection, so we propose a coherent integration method based on Radon–Nonuniform Fast Fourier Transform (Radon-NUFFT) to deal with the problems introduced by the agile waveform parameters. Preliminary results from the field experiment demonstrate the feasibility of the PBR system based on PAR signals, and the effectiveness of the proposed methods is verified.

Accelerated near-field algorithm of sparse apertures by non-uniform fast Fourier transform.

We present an accelerated algorithm for calculating the near-field of non-uniform sparse apertures with non-uniform fast Fourier transform (NUFFT). The distances of the adjacent units in non-uniform sparse apertures are unequal and larger than half a wavelength. The near-field of apertures can be calculated by the angular spectrum method and the convolution methods, and according to the different convolution kernels, the convolution methods can be divided as the Fresnel kernel convolution and the Rayleigh-Sommerfeld kernel convolution. The Fresnel kernel is the approximation of the Rayleigh-Sommerfeld kernel in the far regions of the near-field zone. In uniform apertures, the three methods can be accelerated by fast Fourier transform (FFT). However, FFT should be replaced by NUFFT for non-uniform sparse apertures. The simulation results reveal that the Rayleigh-Sommerfeld convolution with NUFFT (RS-NUFFT) can be applied to all aperture sizes, distributions and near-field distances. After investigating the error sources in RS-NUFFT, the techniques (padding zeros for apertures, increasing sampling rate for the convolution kernel) are developed for increasing the calculation accuracy of RS-NUFFT.

Optimized Nonuniform FFTs and Their Application to Array Factor Computation

We deal with developing an optimized approach for implementing nonuniform fast Fourier transform (NUFFT) algorithms under a general and new perspective for 1-D transformations. The computations of nonequispaced results, nonequispaced data, and Type-3 nonuniform discrete Fourier transforms are tackled in a unified way. They exploit “uniformly sampled” exponentials to interpolate the “nonuniformly sampled” ones involved in the nonuniform discrete Fourier transforms (NUFDTs), so as to enable the use of standard fast Fourier transforms, and an optimized window. The computational costs and the memory requirements are analyzed, and their convenient performance is assessed also by comparing them with other approaches in the literature. Numerical results demonstrate that the method is more accurate and does not introduce any additional computational or memory burden. The computation of the window functions amounts to that of a Legendre polynomial expansion, i.e., a simple polynomial evaluation. This is convenient in terms of computational burden and of the proper arrangement of the calculations. A case study of electromagnetic interest has been carried out by applying the developed NUFFTs to the radiation of linear regular or irregular arrays onto a set of regular or irregular spectral points. Guidelines for multidimensional extension of the proposed approach are also presented.

Fast and High-Resolution Three-Dimensional Hybrid-Domain Photoacoustic Imaging Incorporating Analytical-Focused Transducer Beam Amplitude.

Recently, many reconstruction methods have been developed to improve the lateral resolution of acoustic-resolution photoacoustic microscopy (ARPAM) in out-of-focus regions. Though these methods enhance image resolution to some extent, they require advanced computational hardware and large computational time, especially for three-dimensional (3-D) cases. However, some methods do not consider the finite size of a transducer, while others employ numerical discretization to build a focused transducer model that is less efficient and accurate. To overcome these problems, we propose a 3-D ARPAM imaging reconstruction method with high precision, high efficiency, and low memory cost. It inherits the framework of model-based reconstructions and incorporates the forward acoustic model in the hybrid domain. This hybrid-domain acoustic model promotes an analytical solution to establish a focused transducer model. Furthermore, the non-uniform fast Fourier transform (NUFFT) and deconvolution methods are introduced to reduce the required computational time and memory volume for 3-D reconstructions. According to the experimental results reconstructed by the proposed method, the lateral resolution of an ARPAM image recorded by a 20-MHz focused transducer (NA 0.393) can reach 88.39 [Formula: see text]. This resolution exceeds the diffraction limitation of the focused transducer ( [Formula: see text]). When reconstructing a 3-D image with 200×200×150 pixels, the proposed method takes only 8.15 s using a laptop loaded with Intel Core i7-8550U CPU at 1.8 GHz and 1.06-GB memory.

A GRAPPA algorithm for arbitrary 2D/3D non-Cartesian sampling trajectories with rapid calibration.

GRAPPA is a popular reconstruction method for Cartesian parallel imaging, but is not easily extended to non-Cartesian sampling. We introduce a general and practical GRAPPA algorithm for arbitrary non-Cartesian imaging. We formulate a general GRAPPA reconstruction by associating a unique kernel with each unsampled k-space location with a distinct constellation, that is, local sampling pattern. We calibrate these generalized kernels using the Fourier transform phase shift property applied to fully gridded or separately acquired Cartesian Autocalibration signal (ACS) data. To handle the resulting large number of different kernels, we introduce a fast calibration algorithm based on nonuniform FFT (NUFFT) and adoption of circulant ACS boundary conditions. We applied our method to retrospectively under-sampled rotated stack-of-stars/spirals in vivo datasets, and to a prospectively under-sampled rotated stack-of-spirals functional MRI acquisition with a finger-tapping task. We reconstructed all datasets without performing any trajectory-specific manual adaptation of the method. For the retrospectively under-sampled experiments, our method achieved image quality (i.e., error and g-factor maps) comparable to conjugate gradient SENSE (cg-SENSE) and SPIRiT. Functional activation maps obtained from our method were in good agreement with those obtained using cg-SENSE, but required a shorter total reconstruction time (for the whole time-series): 3minutes (proposed) vs 15minutes (cg-SENSE). This paper introduces a general 3D non-Cartesian GRAPPA that is fast enough for practical use on today's computers. It is a direct generalization of original GRAPPA to non-Cartesian scenarios. The method should be particularly useful in dynamic imaging where a large number of frames are reconstructed from a single set of ACS data.

A unified framework for oscillatory integral transforms: When to use NUFFT or butterfly factorization?

This paper concerns the fast evaluation of the matvec g=Kf for K∈CN×N, which is the discretization of an oscillatory integral transform g(x)=∫K(x,ξ)f(ξ)dξ with a kernel function K(x,ξ)=α(x,ξ)e2πiΦ(x,ξ), where α(x,ξ) is a smooth amplitude function, and Φ(x,ξ) is a piecewise smooth phase function with O(1) discontinuous points in x and ξ. A unified framework is proposed to compute Kf with O(Nlog⁡N) time and memory complexity via the non-uniform fast Fourier transform (NUFFT) or the butterfly factorization (BF), together with an O(N) fast algorithm to determine whether NUFFT or BF is more suitable. This framework works for two cases: 1) explicit formulas for the amplitude and phase functions are known; 2) only indirect access of the amplitude and phase functions are available. Especially in the case of indirect access, our main contributions are: 1) an O(Nlog⁡N) algorithm for recovering the amplitude and phase functions is proposed based on a new low-rank matrix recovery algorithm; 2) a new stable and nearly optimal BF with amplitude and phase functions in a form of a low-rank factorization (IBF-MAT1) is proposed to evaluate the matvec Kf. Numerical results are provided to demonstrate the effectiveness of the proposed framework. 1IBF-MAT means interpolative butterfly factorization based on low-rank matrix information instead of explicit formulas used in the IBF in [22].

Micromagnetic simulator for complex granular systems based on Voronoi tessellation

Abstract A micromagnetic code based on Voronoi tessellation and the non-uniform fast Fourier transform (NUFFT) method is presented. The code is capable of efficiently and accurately simulating magnetization dynamics in large and structurally complex granular systems, such as multilayer granular media used for perpendicular magnetic recording, bit patterned media, granular nanowires, and read heads. In these systems the granular microstructure and distributions in grain and interface properties play an important role in device performance. The presented Voronoi simulator allows comprehensive studies to be performed as it accounts for the detailed granular microstructure and distributions that characterize true systems. Simulation time is greatly reduced by a non-uniform fast Fourier transform algorithm and implementation on graphics processing units (GPUs). Simulations of conventional magnetic recording, heat-assisted magnetization reversal, domain wall dynamics in granular nanowires, and particulate tape recording are presented.

Tissue sodium concentration and sodium T1 mapping of the human brain at 3 T using a Variable Flip Angle method

PurposeThe state-of-the-art method to quantify sodium concentrations in vivo consists in a fully relaxed 3D spin-density (SD) weighted acquisition. Nevertheless, most sodium MRI clinical studies use short-TR SD acquisitions to reduce acquisition durations. We present a clinically viable implementation of the Variable Flip Angle (VFA) method for robust and clinically viable quantification of total sodium concentration (TSC) and longitudinal relaxation rates in vivo in human brain at 3 T. MethodsTwo non-Cartesian steady-state spoiled ultrashort echo time (UTE) scans, performed at optimized flip angles, repetition time and pulse length determined under specific absorption rate constraints, are used to simultaneously compute T1 and total sodium concentration (TSC) maps using the VFA method. Images are reconstructed using the non-uniform Fast Fourier Transform algorithm and TSC maps are corrected for possible inhomogeneity of coil transmission and reception profiles. Fractioned acquisitions are used to correct for potential patient motion. TSC quantifications obtained using the VFA method are validated at first in comparison with a fully-relaxed SD acquisition in a calibration phantom. The robustness of similar VFA acquisitions are compared to the short-TR SD approach in vivo on seven healthy volunteers. ResultsThe VFA method resulted in consistent TSC and T1 estimates across our cohort of healthy subjects, with mean TSC of 38.1 ± 5.0 mmol/L and T1 of 39.2 ± 4.4 ms. These results are in agreement with previously reported values in literature TSC estimations and with the predictions of a 2-compartment model. However, the short-TR SD acquisition systematically underestimated the sodium concentration with a mean TSC of 31 ± 4.5 mmol/L. ConclusionThe VFA method can be applied successfully to image sodium at 3 T in about 20 min and provides robust and intrinsically T1-corrected TSC maps.

A Parallel Nonuniform Fast Fourier Transform Library Based on an “Exponential of Semicircle" Kernel

The nonuniform fast Fourier transform (NUFFT) generalizes the FFT to off-grid data. Its many applications include image reconstruction, data analysis, and the numerical solution of differential equ...

Fast High-Order Integral Equation Methods for Solving Boundary Value Problems of Two Dimensional Heat Equation in Complex Geometry

Efficient high-order integral equation methods have been developed for solving boundary value problems of the heat equation in complex geometry in two dimensions. First, the classical heat potential theory is applied to convert such problems to Volterra integral equations of the second kind via the heat layer potentials, where the unknowns are only on the space–time boundary. However, the heat layer potentials contain convolution integrals in both space and time whose direct evaluation requires $$O(N_S^2N_T^2)$$ work and $$O(N_SN_T)$$ storage, where $$N_S$$ is the total number of discretization points on the spatial boundary and $$N_T$$ is the total number of time steps. In order to evaluate the heat layer potentials accurately and efficiently, they are split into two parts—the local part containing the temporal integration from $$t-\delta $$ to t and the history part containing the temporal integration from 0 to $$t-\delta $$ . The local part can be dealt with efficiently using conventional fast multipole type algorithms. For problems with complex stationary geometry, efficient separated sum-of-exponentials approximations are constructed for the heat kernel and used for the evaluation of the history part. Here all local and history kernels are compressed only once. The resulting algorithm is very efficient with quasilinear complexity in both space and time for both interior and exterior problems. For problems with complex moving geometry, the spectral Fourier approximation is applied for the heat kernel and nonuniform FFT is used to speed up the evaluation of the history part of heat layer potentials. The performance of both algorithms is demonstrated with several numerical examples.