Nonuniform Fast Fourier Transform Research Articles

Modifications on Multichannel Reconstruction Algorithm for SAR Processing Based on Periodic Nonuniform Sampling Theory and Nonuniform Fast Fourier Transform

The next generation of space-borne synthetic aperture radar (SAR) systems will emphasize on high-resolution and wide-swath imaging. For these design goals, multichannel technology in azimuth is a promising candidate and can provide global monitoring capacity for the continuous observation of a highly dynamic and rapidly changing world with high spatial resolution and short repeat intervals. In the multichannel SAR system, the nonuniform azimuth signal needs to be reconstructed when the pulse repetition frequency is different from a specific one. A reconstruction algorithm based on periodic nonuniform sampling theory has been proposed in current literature, but it is computationally rather expensive. In this paper, nonuniform fast Fourier transform is employed to reduce the computation load of the original algorithm from $O(N^2) $ to $O(N\log N) $ , where $N$ is azimuth sample number. Furthermore, a modification is implemented by taking the out-of-band energy into account to suppress the azimuth ambiguity when the signal is not band-limited, which is actually the case of the azimuth modulation of SAR with a real antenna pattern. The computation and reconstruction performances validate the proposed method’s effectiveness in multichannel signal processing and its suppression effect on azimuth ambiguity.

Compressed sensing reconstruction of cardiac cine MRI using golden angle spiral trajectories

In dynamic cardiac cine Magnetic Resonance Imaging (MRI), the spatiotemporal resolution is limited by the low imaging speed. Compressed sensing (CS) theory has been applied to improve the imaging speed and thus the spatiotemporal resolution. The purpose of this paper is to improve CS reconstruction of under sampled data by exploiting spatiotemporal sparsity and efficient spiral trajectories.We extend k–t sparse algorithm to spiral trajectories to achieve high spatio temporal resolutions in cardiac cine imaging. We have exploited spatiotemporal sparsity of cardiac cine MRI by applying a 2D+time wavelet-Fourier transform. For efficient coverage of k-space, we have used a modified version of multi shot (interleaved) spirals trajectories. In order to reduce incoherent aliasing artifact, we use different random undersampling pattern for each temporal frame. Finally, we have used nonuniform fast Fourier transform (NUFFT) algorithm to reconstruct the image from the non-uniformly acquired samples.The proposed approach was tested in simulated and cardiac cine MRI data. Results show that higher acceleration factors with improved image quality can be obtained with the proposed approach in comparison to the existing state-of-the-art method.The flexibility of the introduced method should allow it to be used not only for the challenging case of cardiac imaging, but also for other patient motion where the patient moves or breathes during acquisition.

ISAR Imaging of Nonuniformly Rotating Target Based on a Fast Parameter Estimation Algorithm of Cubic Phase Signal

In inverse synthetic aperture radar (ISAR) imaging of nonuniformly rotating targets, such as highly maneuvering airplanes and ships fluctuating with oceanic waves, azimuth echoes have to be modeled as cubic phase signals (CPSs) after the range migration compensation and the translational-induced phase error correction. For the CPS model, the chirp rate and the quadratic chirp rate, which deteriorate the azimuth focusing quality due to the Doppler frequency shift, need to be estimated with a parameter estimation algorithm. In this paper, by employing the proposed generalized scaled Fourier transform (GSCFT) and the nonuniform fast Fourier transform (NUFFT), a fast parameter estimation algorithm is presented and utilized in the ISAR imaging of the nonuniformly rotating target. Compared to the scaled Fourier transform-based algorithm, advantages of the fast parameter estimation algorithm include the following: 1) the computational cost is lower due to the utilization of the NUFFT, and 2) the GSCFT has a wider applicability in ISAR imaging applications. The CPS model and the algorithm implementation are verified with the real radar data of a ship target. In addition, the cross-term, which plays an important role in correlation algorithms, is analyzed for the fast parameter estimation algorithm. Through simulations of the synthetic data and the real radar data, we verify the effectiveness of the fast parameter estimation algorithm and the corresponding ISAR imaging algorithm.

ISAR Imaging for Fluctuating Ships Based on a Fast Bilinear Parameter Estimation Algorithm

For inverse synthetic aperture radar (ISAR) imaging of ships fluctuating with oceanic waves, azimuth echoes of a range cell have to be modeled as cubic phase signals (CPSs) after the range alignment and the phase adjustment. In ISAR imaging based on the CPS model, the chirp rate and the quadratic chirp rate are identified as causes of the target image defocus and need to be estimated with an effective algorithm. In this paper, a fast bilinear parameter estimation algorithm is proposed and applied in ISAR imaging for fluctuating ships by employing the cubic phase bilinear function, the nonuniform fast Fourier transform (NUFFT), and the parameter space switching method. Compared to two existing representative parameter estimation algorithms for the CPS, the advantages of this proposed estimation algorithm are: 1) the computational cost is lower due to the NUFFT and the parameter space switching method and 2) the bilinearity and the energy accumulation operation guarantee a higher anti-noise performance and a better suppression on cross-terms. Through simulations on synthetic models and the real radar data, we verify the effectiveness of this fast bilinear parameter estimation algorithm and the corresponding ISAR imaging algorithm.

Three-dimensional dictionary-learning reconstruction of (23)Na MRI data.

To reduce noise and artifacts in (23)Na MRI with a Compressed Sensing reconstruction and a learned dictionary as sparsifying transform. A three-dimensional dictionary-learning compressed sensing reconstruction algorithm (3D-DLCS) for the reconstruction of undersampled 3D radial (23)Na data is presented. The dictionary used as the sparsifying transform is learned with a K-singular-value-decomposition (K-SVD) algorithm. The reconstruction parameters are optimized on simulated data, and the quality of the reconstructions is assessed with peak signal-to-noise ratio (PSNR) and structural similarity (SSIM). The performance of the algorithm is evaluated in phantom and in vivo (23)Na MRI data of seven volunteers and compared with nonuniform fast Fourier transform (NUFFT) and other Compressed Sensing reconstructions. The reconstructions of simulated data have maximal PSNR and SSIM for an undersampling factor (USF) of 10 with numbers of averages equal to the USF. For 10-fold undersampling, the PSNR is increased by 5.1 dB compared with the NUFFT reconstruction, and the SSIM by 24%. These results are confirmed by phantom and in vivo (23)Na measurements in the volunteers that show markedly reduced noise and undersampling artifacts in the case of 3D-DLCS reconstructions. The 3D-DLCS algorithm enables precise reconstruction of undersampled (23)Na MRI data with markedly reduced noise and artifact levels compared with NUFFT reconstruction. Small structures are well preserved.

Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT

We present efficient and accurate numerical methods for computing the ground state and dynamics of the nonlinear Schrödinger equation (NLSE) with nonlocal interactions based on a fast and accurate evaluation of the long-range interactions via the nonuniform fast Fourier transform (NUFFT). We begin with a review of the fast and accurate NUFFT based method in [29] for nonlocal interactions where the singularity of the Fourier symbol of the interaction kernel at the origin can be canceled by switching to spherical or polar coordinates. We then extend the method to compute other nonlocal interactions whose Fourier symbols have stronger singularity at the origin that cannot be canceled by the coordinate transform. Many of these interactions do not decay at infinity in the physical space, which adds another layer of complexity since it is more difficult to impose the correct artificial boundary conditions for the truncated bounded computational domain. The performance of our method against other existing methods is illustrated numerically, with particular attention on the effect of the size of the computational domain in the physical space. Finally, to study the ground state and dynamics of the NLSE, we propose efficient and accurate numerical methods by combining the NUFFT method for potential evaluation with the normalized gradient flow using backward Euler Fourier pseudospectral discretization and time-splitting Fourier pseudospectral method, respectively. Extensive numerical comparisons are carried out between these methods and other existing methods for computing the ground state and dynamics of the NLSE with various nonlocal interactions. Numerical results show that our scheme performs much better than those existing methods in terms of both accuracy and efficiency.

Off-axis angular spectrum method with variable sampling interval

We proposed a novel off-axis angular spectrum method (ASM) for simulating free space wave propagation with a large shifted destination plane. The off-axis numerical simulation took wave propagation between a parallel source and a destination plane, but a destination plane was shifted from a source plane. The shifted angular spectrum method was proposed for diffraction simulation with a shifted destination plane and satisfied the Nyquist condition for sampling by limiting a bandwidth of a propagation field to avoid an aliasing error due to under sampling. However, the effective sampling number of the shifted ASM decreased when the shifted distance of the destination plane was large which caused a numerical error in the diffraction simulation. To compensate for the decrease of an effective sampling number for the large shifted destination plane, we used a variable sampling interval in a Fourier space to maintain the same effective sampling number independent of the shifted distance of the destination plane. As a result, our proposed off-axis ASM with a variable sampling interval can produce simulation results with high accuracy for nearly every shifted distance of a destination plane when an off-axis angle is less than 75°. We compared the performances of the off-axis ASM using the Chirp Z transform and non-uniform FFT for implementing a variable spatial frequency in a Fourier space.

A Fast Radial Scanned Near-Field 3-D SAR Imaging System and the Reconstruction Method

This paper presents a near-field 3-D synthetic aperture radar (SAR) imaging system for which the 2-D aperture is radially scanned. Compared to the current SAR imaging systems, the proposed system has several advantages such as quick data collection, full 360° inspection of target, and simple image formation processing. However, in radial scan, the samples do not fall on a Cartesian grid, which prevents us from using the fast Fourier transform (FFT) to form SAR image without calling for interpolation. In this paper, the 2-D nonuniform FFT (NUFFT) is used for dealing with the problem. After 2-D NUFFT of the radial sampled data, the 3-D reflectivity image can be efficiently reconstructed by using the 3-D version of the range migration algorithm. The Stolt mapping has been implemented implicitly by another 1-D NUFFT to reduce the artifacts caused by the conventional interpolation processing. In addition, to alleviate the data sampling burden, a compressed 2-D slow-time sampling strategy is also discussed. Finally, the proposed Rad-SAR system and the imaging method are demonstrated using near-field wideband simulation data.

An Efficient Calculation of the Far Field Radiated by Non-Uniformly Sampled Planar Fields Complying Nyquist Theorem

An efficient and accurate method to calculate the far field radiated by non-uniformly sampled planar apertures using the non-uniform fast Fourier transform (NUFFT) is shown. The sampled fields in the planar aperture have to comply the Nyquist theorem. The method takes into account the amplitude of the unit cell radiation pattern, which allows to compute more accurately the copolar and crosspolar components of the far field. Due to the generality of the NUFFT, it also allows to overcome some limitations of the FFT when computing the radiated fields of reflectarrays with periodicity larger than half a wavelength. Because the NUFFT is precision-dependent, a discussion of how its accuracy can affect the computed radiated fields is carried out. Numerical examples are provided to show the accuracy and performance of the NUFFT with regard to the FFT and direct evaluation of the fields. Finally, a study of computing times comparing the FFT, NUFFT and direct evaluation is presented.

An Accurate Conformal Fourier Transform Method for 3D Discontinuous Functions

Fourier transform of discontinuous functions are often encountered in computational electromagnetics and other areas. In this work, a highly accurate, fast conformal Fourier transform (CFT) algorithm is proposed to evaluate the finite Fourier transform of 3D discontinuous functions. A curved tetrahedron mesh combined with curvilinear coordinate transform, instead of the Cartesian grid, is adopted to flexibly model an arbitrary shape of the discontinuity boundary. This enables us to take full advantages of high order interpolation and Gaussian quadrature methods to achieve highly accurate Fourier integration results with a low sampling density. The 3D nonuniform fast Fourier transform (NUFFT) helps to keep the complexity of the proposed algorithm to that similar to the traditional 3D FFT algorithm. Therefore, the proposed CFT algorithm can achieve order of magnitude higher accuracy than 3D FFT with lower sampling density and similar computation time. The convergence is proved and verified.

Multi-Static Multi-Frequency Image Reconstruction Based on Non-Uniform FFT

Multi-Static Multi-Frequency Image Reconstruction Based on Non-Uniform FFT

ISAR Imaging of Targets With Complex Motion Based on the Chirp Rate–Quadratic Chirp Rate Distribution

In inverse synthetic aperture radar (ISAR) imaging of targets with complex motion such as fluctuating ships with oceanic waves and high maneuvering airplanes, the azimuth echo signals can be modeled as cubic phase signals (CPSs) after the migration compensation. The chirp rate (CR) and the quadratic chirp rate (QCR) are two important physical quantities of the CPS, which deteriorate the azimuth focusing quality due to the Doppler frequency shift. With these two quantities, other parameters can be estimated by using the fast Fourier transform (FFT). Therefore, the CPS can be uniquely determined by both CR and QCR. In this paper, based on the proposed generalized keystone transform and the parametric instantaneous autocorrelation function, a novel distribution of the CPS, known as the CR–QCR distribution (CRQCRD), is presented and applied in a newly proposed ISAR imaging algorithm for targets with complex motion. The CRQCRD is simple and only requires the FFT and the nonuniform FFT (NUFFT). Owing to the application of the NUFFT, the computational cost is saved, and the searching procedure is unnecessary for the nonuniformly spaced signal. Compared to other four representative methods for CPSs, the CRQCRD, which can acquire higher antinoise performance and no error propagation, is searching-free and more suitable for the situation of multitargets. Several simulation examples, analyses of the antinoise performance, and ISAR images validate the effectiveness of the CRQCRD and the corresponding ISAR imaging algorithm.

Exploring Irregular Time Series Through the Non-Uniform Fourier Transform

One of the fundamental shortcoming of the popular analysis tools for time series is that they require the data to be taken at uniform time intervals. However, the real-world time series, such as those from financial markets, are mostly from irregular time intervals. It is a common practice to resample the irregular time series into a regular one, but, there are significant limitations on this practice. For example, if one is to resample the trading activities on a stock into hourly series, then the time series can only last through the trading day because there usually is no trading in the night. In this work, we directly explore the dynamics of irregular time series through a tool known as Non-Uniform Fast Fourier Transform (NUFFT). To illustrate its effectiveness, we apply NUFFT on the trading records of natural gas futures contracts for the last seven years. Results accurately capture well-known structural features in the trading records, such as weekly and daily cycles, and at the same time also reveal unknown or unexplored features, such as the presence of multiple power laws. In particular, we observe a new power law in the Fourier spectra in recent years.

Scaled diffraction calculation between tilted planes using nonuniform fast Fourier transform.

A method of diffraction calculation between tilted planes with variable sampling rates is proposed. The proposed method is based on the Fourier spectrum rotation from a tilted plane to a parallel plane. The nonuniform fast Fourier transformation (NUFFT) is used to calculate the nonuniform sampled Fourier spectrum on the tilted plane with variable sampling rates, which overcomes the sampling restriction of FFT in the conventional method. Both of the computer simulation and the optical experiment shows the feasibility of our method in calculating the hologram of polygon-based object with scalable size, which can be considered as an important application in the holographic three-dimensional display.

A comparative study of compressed sensing approaches for 3-D synthetic aperture radar image reconstruction

This paper investigates two compressed sensing (CS) approaches that can be used to reconstruct 3-D synthetic aperture radar (SAR) images with undersampled measurements. Combining CS with the range migration algorithm (RMA), using either Stolt transform or non-uniform fast Fourier transform (NUFFT), yields two different approaches: Stolt-CS and NUFFT-CS. These approaches can decrease the load of data acquisition while recovering satisfactory 3-D SAR images through l1-norm optimization. A simulated image is used as the ground truth to facilitate the comparative study. The 2-D structured similarity (SSIM) index is extended to 3-D to assess the quality of the reconstructed images. Both the simulation and the experimental reconstruction results demonstrate that the Stolt-CS contributes little to image quality improvement or computational complexity reduction due to the inaccuracy of the Stolt transform. In contrast, the NUFFT-CS achieves a good tradeoff between the reconstruction quality and the computational costs.

GPU-accelerated non-uniform fast Fourier transform-based compressive sensing spectral domain optical coherence tomography.

We implemented the graphics processing unit (GPU) accelerated compressive sensing (CS) non-uniform in k-space spectral domain optical coherence tomography (SD OCT). Kaiser-Bessel (KB) function and Gaussian function are used independently as the convolution kernel in the gridding-based non-uniform fast Fourier transform (NUFFT) algorithm with different oversampling ratios and kernel widths. Our implementation is compared with the GPU-accelerated modified non-uniform discrete Fourier transform (MNUDFT) matrix-based CS SD OCT and the GPU-accelerated fast Fourier transform (FFT)-based CS SD OCT. It was found that our implementation has comparable performance to the GPU-accelerated MNUDFT-based CS SD OCT in terms of image quality while providing more than 5 times speed enhancement. When compared to the GPU-accelerated FFT based-CS SD OCT, it shows smaller background noise and less side lobes while eliminating the need for the cumbersome k-space grid filling and the k-linear calibration procedure. Finally, we demonstrated that by using a conventional desktop computer architecture having three GPUs, real-time B-mode imaging can be obtained in excess of 30 fps for the GPU-accelerated NUFFT based CS SD OCT with frame size 2048(axial) × 1,000(lateral).

Polar format algorithm based on fast Gaussian grid non‐uniform fast Fourier transform for spotlight synthetic aperture radar imaging

Conventional polar format algorithm (PFA) and its variations normally assume an ideal imaging geometry of a linear flight path, which cannot counteract the non-uniformity of samples existing in the circumstance of a highly non-linear aperture. This scenario may occur because of various factors such as rugged topography, atmospheric turbulence and intended manoeuvres. To address this challenge, this study develops a novel PFA with aperture accommodation for spotlight synthetic aperture radar (SAR) imaging. The proposed algorithm provides a new adaption of the fast Gaussian grid (FGG) non-uniform fast Fourier transform (NUFFT) scheme to two-dimensional (2D) SAR imaging of 3D scene, whose main idea is to convolve the non-uniform samples onto a uniform grid with a Gaussian kernel and then exploit the efficient 2D FFT for image reconstruction. In virtue of the 2D FGG NUFFT, a favourable tradeoff between reconstruction precision and computational efficiency is gained. Simulations illustrate the superiority of the proposed algorithm over the conventional non-uniform Sinc interpolation method. Also, the public real Gotcha dataset validates the effectiveness of the authors method.

SU‐E‐I‐41: Non‐Cartesian MR Image Reconstruction with Integrated Gradient Non‐Linearity Correction

Purpose:Nonlinearities in the spatial encoding gradients of MRI systems cause geometric distortion in images. Typically, this is retrospectively corrected via image‐domain interpolation (a.k.a., “gradwarp”) albeit with a loss of spatial resolution. For non‐Cartesian MRI, the latter problem is exaggerated by noise and undersampling artifact. In this study, we describe a novel correction strategy that accounts for gradient nonlinearities during — rather than after — non‐Cartesian MRI reconstruction, and demonstrate that this approach mitigates the resolution loss that can occur with standard methods.Methods:To test the proposed method, the American College of Radiology (ACR) quality control phantom was scanned on at 1.5 T (General Electric, v16.0, “zoom” gradient) using a 1.6x undersampled 3D non‐ Cartesian Shells trajectory (GRE, FOV=24 cm3, 120 shells, 16552 shots, 512 readout, matrix=2403). Image reconstruction was first performed via standard k‐space density‐compensated gridding and retrospectively corrected via cubic spline interpolation. Image reconstruction was then separately performed using a k‐space and image‐domain densitycompensated type‐3 non‐uniform fast Fourier transform (NUFFT), which provides a direct mapping between non‐Cartesian k‐space samples and warped image space voxel locations. Thus, no separate distortion correction procedure is needed for the proposed approach. The gradient distortion field was determined using vendor provided calibration data.Results:Phantom scan results show that both processing approaches successfully correct geometric distortion. However, visual inspection of the ACR phantom spatial resolution inserts shows that the proposed strategy preserves the resolution of the nominal (uncorrected) reconstruction while “gradwarp” imparts marked spatial blurring (especially for the 1.0 and 1.1 mm inserts) and thus resolution loss.Conclusion:We' ve presented a novel reconstruction strategy for non‐Cartesian MRI that corrects for gradient nonlinearities during — rather than after — reconstruction, and thus better preserves image resolution than traditional interpolation‐based methods. This approach is expected to be especially advantageous when imaging with non‐standard magnet geometries.NIH RR018898; NIH EB10065

Non-uniform FFT for the finite element computation of the micromagnetic scalar potential

We present a quasi-linearly scaling, first order polynomial finite element method for the solution of the magnetostatic open boundary problem by splitting the magnetic scalar potential. The potential is determined by solving a Dirichlet problem and evaluation of the single layer potential by a fast approximation technique based on Fourier approximation of the kernel function. The latter approximation leads to a generalization of the well-known convolution theorem used in finite difference methods. We address it by a non-uniform FFT approach. Overall, our method scales O(M+N+NlogN) for N nodes and M surface triangles. We confirm our approach by several numerical tests.

Edge Detection from Non-Uniform Fourier Data Using the Convolutional Gridding Algorithm

Detecting edges in images from a finite sampling of Fourier data is important in a variety of applications. For example, internal edge information can be used to identify tissue boundaries of the brain in a magnetic resonance imaging (MRI) scan, which is an essential part of clinical diagnosis. Likewise, it can also be used to identify targets from synthetic aperture radar data. Edge information is also critical in determining regions of smoothness so that high resolution reconstruction algorithms, i.e. those that do not smear over the internal boundaries of an image, can be applied. In some applications, such as MRI, the sampling patterns may be designed to oversample the low frequency while more sparsely sampling the high frequency modes. This type of non-uniform sampling creates additional difficulties in processing the image. In particular, there is no fast reconstruction algorithm, since the FFT is not applicable. However, interpolating such highly non-uniform Fourier data to the uniform coefficients (so that the FFT can be employed) may introduce large errors in the high frequency modes, which is especially problematic for edge detection. Convolutional gridding, also referred to as the non-uniform FFT, is a forward method that uses a convolution process to obtain uniform Fourier data so that the FFT can be directly applied to recover the underlying image. Carefully chosen parameters ensure that the algorithm retains accuracy in the high frequency coefficients. Similarly, the convolutional gridding edge detection algorithm developed in this paper provides an efficient and robust way to calculate edges. We demonstrate our technique in one and two dimensional examples.