Nonuniform Fast Fourier Transform Research Articles

An Inverse Synthetic Aperture Ladar Imaging Algorithm of Maneuvering Target Based on Integral Cubic Phase Function-Fractional Fourier Transform

When imaging maneuvering targets with inverse synthetic aperture ladar (ISAL), dispersion and Doppler frequency time-variation exist in the range and cross-range echo signal, respectively. To solve this problem, an ISAL imaging algorithm based on integral cubic phase function-fractional Fourier transform (ICPF-FRFT) is proposed in this paper. The accurate ISAL echo signal model is established for a space maneuvering target that quickly approximates the uniform acceleration motion. On this basis, the chirp rate of the echo signal is quickly estimated by using the ICPF algorithm, which uses the non-uniform fast Fourier transform (NUFFT) method for fast calculations. At the best rotation angle, the range compression is realized by FRFT and the range dispersion is eliminated. After motion compensation, separation imaging of strong and weak scattering points is realized by using ICPF-FRFT and CLEAN technique and the azimuth defocusing problem is solved. The effectiveness of the proposed method is verified by a simulation experiment of an aircraft scattering point model and real data.

Group Sparse Basis Pursuit Denoising Reconstruction Algorithm for Polarimetric Through-the-Wall Radar Imaging

Polarimetric through-the-wall radar imaging (TWRI) system has the enhancing performance in the detection, imaging, and classification of concealed targets behind the wall. We propose a group sparse basis pursuit denoising- (BPDN-) based imaging approach for polarimetric TWRI system in this paper. The proposed imaging method combines the spectral projection gradient L1-norm (SPGL1) algorithm with the nonuniform fast Fourier transform (NUFFT) technique to implement the imaging reconstruction of observed scene. The experimental results have demonstrated that compared to the existing compressive sensing- (CS-) based imaging algorithms, the proposed NUFFT-based SPGL1 algorithm can significantly reduce the required computer memory and achieve the improved imaging reconstruction performance with the high computational efficiency.

Rapid dynamic contrast-enhanced MRI for small animals at 7T using 3D ultra-short echo time and golden-angle radial sparse parallel MRI.

To develop a rapid dynamic contrast-enhanced MRI method with high spatial and temporal resolution for small-animal imaging at 7 Tesla. An ultra-short echo time (UTE) pulse sequence using a 3D golden-angle radial sampling was implemented to achieve isotropic spatial resolution with flexible temporal resolution. Continuously acquired radial spokes were grouped into subsets for image reconstruction using a multicoil compressed sensing approach (Golden-angle RAdial Sparse Parallel; GRASP). The proposed 3D-UTE-GRASP method with high temporal and spatial resolutions was tested using 7 mice with GL261 intracranial glioma models. Iterative reconstruction with different temporal resolutions and regularization factors λ showed that, in all cases, the cost function decreased to less than 2.5% of its starting value within 20 iterations. The difference between the time-intensity curves of 3D-UTE-GRASP and nonuniform fast Fourier transform (NUFFT) images was minimal when λ was 1% of the maximum signal intensity of the initial NUFFT images. The 3D isotropic images were used to generate pharmacokinetic parameter maps to show the detailed images of the tumor characteristics in 3D and also to show longitudinal changes during tumor growth. This feasibility study demonstrated that the proposed 3D-UTE-GRASP method can be used for effective measurement of the 3D spatial heterogeneity of tumor pharmacokinetic parameters.

Fast and Accurate Simulation Technique for Large Irregular Arrays

A fast full-wave simulation technique is presented for the analysis of large irregular planar arrays of identical 3-D metallic antennas. The solution method relies on the Macro Basis Functions (MBF) approach and an interpolatory technique to compute the interactions between MBFs. The Harmonic-polynomial (HARP) model is established for the near-field interactions in a modified system of coordinates. For extremely large arrays made of complex antennas, two approaches assuming a limited radius of influence for mutual coupling are considered: one is based on a sparse-matrix LU decomposition and the other one on a tessellation of the array in the form of overlapping sub-arrays. The computation of all embedded element patterns is sped up with the help of the non-uniform FFT algorithm. Extensive validations are shown for arrays of log-periodic antennas envisaged for the low-frequency SKA (Square Kilometer Array) radio-telescope. The analysis of SKA stations with such a large number of elements has not been treated yet in the literature. Validations include comparison with results obtained with commercial software and with experiments. The proposed method is particularly well suited to array synthesis, in which several orders of magnitude can be saved in terms of computation time.

Efficient type‐4 and type‐5 non‐uniform FFT methods in the one‐dimensional case

The so-called non-uniform fast Fourier transform (NFFT) is a family of algorithms for efficiently computing the Fourier transform of finite-length signals, whenever the time or frequency grid is non-uniformly spaced. Among the five usual NFFT types, types 4 and 5 involve an inversion problem, and this makes them the most intensive computationally. The usual efficient methods for these last types are either based on a fast multipole (FM) or on an iterative conjugate gradient (CG) method. The purpose of this study is to present efficient methods for these type-4 and type-5 NFFTs in the one-dimensional case that just require three NFFTs of types 1 or 2 plus some additional fast Fourier transforms (FFTs). Fundamentally, they are based on exploiting the Lagrange formula structure. The proposed methods roughly provide a factor-ten improvement on the FM and CG alternatives in computational burden. The study includes several numerical examples in double precision, in which the proposed and the Gaussian elimination, CG and FM methods are compared, both in terms of round-off error and computational burden.

Modified slanted-edge method for camera modulation transfer function measurement using nonuniform fast Fourier transform technique

ISO 12233 slanted-edge method experiences errors using fast Fourier transform (FFT) in the camera modulation transfer function (MTF) measurement due to tilt angle errors in the knife-edge resulting in nonuniform sampling of the edge spread function (ESF). In order to resolve this problem, a modified slanted-edge method using nonuniform fast Fourier transform (NUFFT) for camera MTF measurement is proposed. Theoretical simulations for images with noise at a different nonuniform sampling rate of ESF are performed using the proposed modified slanted-edge method. It is shown that the proposed method successfully eliminates the error due to the nonuniform sampling of the ESF. An experimental setup for camera MTF measurement is established to verify the accuracy of the proposed method. The experiment results show that under different nonuniform sampling rates of ESF, the proposed modified slanted-edge method has improved accuracy for the camera MTF measurement compared to the ISO 12233 slanted-edge method.

A Nonuniform Fast Fourier Transform Based on Low Rank Approximation

By viewing the nonuniform discrete Fourier transform (NUDFT) as a perturbed version of a uniform discrete Fourier transform, we propose a fast and quasi-optimal algorithm for computing the NUDFT based on the fast Fourier transform (FFT). Our key observation is that an NUDFT and DFT matrix divided entry by entry is often well approximated by a low rank matrix, allowing us to express a NUDFT matrix as a sum of diagonally scaled DFT matrices. Our algorithm is simple to implement, automatically adapts to any working precision, and is competitive with state-of-the-art algorithms. In the fully uniform case, our algorithm is essentially the FFT. We also describe quasi-optimal algorithms for the inverse NUDFT and two-dimensional NUDFTs.

Parameterized two-dimensional representation for multicomponent cubic phase signals and its application in ISAR imaging of fluctuating ship

The cubic phase signal (CPS) can be determined by two physical quantities, the chirp rate and quadratic chirp rate. With this characteristic, this paper proposes the parameterized two-dimensional representation (PTDR), which can be seen as an extension of the chirp rate-quadratic chirp rate distribution (CRQCRD). The goal of the PTDR is to accurately estimate parameters of multicomponent CPSs under low signal-to-noise ratios (SNRs). The PTDR is simple and can be implemented by using the complex multiplication, fast Fourier transform (FFT) and nonuniform FFT (NUFFT). In this paper, the computational cost, resolution, cross term suppression and anti-noise performance are analyzed for the PTDR. With several synthetic examples, comparisons with other five algorithms are performed and demonstrate that the PTDR can achieve higher resolution, cross term suppression and anti-noise performance with a moderate computational cost. Finally, the PTDR is employed to reconstruct inverse synthetic aperture radar (ISAR) image of the fluctuating ship and the real radar data simulations validate the effectiveness of the PTDR-based ISAR imaging algorithm.

ICON-MIC: Implementing a CPU/MIC Collaboration Parallel Framework for ICON on Tianhe-2 Supercomputer.

Electron tomography (ET) is an important technique for studying the three-dimensional structures of the biological ultrastructure. Recently, ET has reached sub-nanometer resolution for investigating the native and conformational dynamics of macromolecular complexes by combining with the sub-tomogram averaging approach. Due to the limited sampling angles, ET reconstruction typically suffers from the "missing wedge" problem. Using a validation procedure, iterative compressed-sensing optimized nonuniform fast Fourier transform (NUFFT) reconstruction (ICON) demonstrates its power in restoring validated missing information for a low-signal-to-noise ratio biological ET dataset. However, the huge computational demand has become a bottleneck for the application of ICON. In this work, we implemented a parallel acceleration technology ICON-many integrated core (MIC) on Xeon Phi cards to address the huge computational demand of ICON. During this step, we parallelize the element-wise matrix operations and use the efficient summation of a matrix to reduce the cost of matrix computation. We also developed parallel versions of NUFFT on MIC to achieve a high acceleration of ICON by using more efficient fast Fourier transform (FFT) calculation. We then proposed a hybrid task allocation strategy (two-level load balancing) to improve the overall performance of ICON-MIC by making full use of the idle resources on Tianhe-2 supercomputer. Experimental results using two different datasets show that ICON-MIC has high accuracy in biological specimens under different noise levels and a significant acceleration, up to 13.3 × , compared with the CPU version. Further, ICON-MIC has good scalability efficiency and overall performance on Tianhe-2 supercomputer.

Efficient and noise-resistant parameter estimation method for maneuvering targets in stepped frequency radar

In this paper, a novel efficient parameter estimation method is proposed for maneuvering targets based on the stepped frequency radar. Since the envelope migration can be represented by the exponential phase variance with respect to range frequency, the received echoes are first converted into the range frequency domain and modeled as cubic phase (CP) signals. Then, a new correlation transform, referred to as recursive delay correlation transform (RDCT), is defined and applied to estimate the parameters of maneuvering targets. Since RDCT is simple and only requires the fast Fourier transform (FFT) and the nonuniform FFT (NUFFT), the searching operation is unnecessary and the computational cost is significantly reduced. Compared to the other four representative methods, the proposed RDCT-based method has no error propagation and can greatly improve the anti-noise performance on the premise of retaining the computational efficiency, achieving a good balance between the computational complexity and estimation performance. Theoretical analyses of anti-noise and anti-aliasing performance, and several simulation results verify the effectiveness of the proposed parameter estimation method.

Fast computation of far-field pulse-echo PSF of arbitrary arrays for large sparse 2-D ultrasound array design

Large wideband two-dimensional (2-D) arrays are essential for high-resolution three-dimensional (3-D) ultrasound imaging. Since the tremendous element number of a full sampled large 2-D array is not affordable in any practical 3-D ultrasound imaging system, it is necessary to reduce the element number through sparse 2-D array design. Sparse array design requires that both the positions and weights of the array elements should be arbitrarily alterable. Hence a proper evaluation tool that can deal with arbitrary array is integral to optimizing the array structure and apodization function. It is known that pulse-echo point spread function (PSF) has been a common tool used to evaluate the performance of wideband arrays in ultrasound imaging all along, which also plays an important role in wideband ultrasound simulations. However, so far the conventional ultrasound simulation tools can only calculate pulse-echo PSF of arbitrary wideband arrays in the time domain because of the existence of nonuniform nodes in the spatial impulse response expressions, which obstructs their application of FFT to do fast computation of the time-domain convolutions. As a result, ultra-high time consumption of pulse-echo PSF computation of a large arbitrary wideband array hampers it to be taken as the evaluation tool by any stochastic optimization methods which need massive iterations in designing large sparse 2-D arrays. This paper aims to make available the pulse-echo PSF tool in designing large sparse 2-D arrays by proposing a fast computation method of far-field pulse-echo PSFs of arbitrary wideband arrays. In the paper, fast computation of wideband spatial impulse responses of a 2-D array is first realized in frequency domain by employing the nonuniform fast Fourier transform (NUFFT), under the point source assumption in far-field. On the basis of that, fast computation of time-domain convolutions is made possible by using FFT. In addition, a short inverse FFT (IFFT) is applied in recovering the time-domain envelopes rather than the detailed waveforms of beam pulses to extract the pulse-echo PSF, which further accelerates the computation. Compared with the computation speed of the time domain method, i.e. Field II, the proposed method achieves an improvement of three orders of magnitude with comparable accuracy for a 100×100 wideband 2-D array. The proposed method makes it possible for applying stochastic optimization methods to design arbitrary large wideband 2-D arrays using pulse-echo PSF as the evaluation tool.

Microwave Imaging From Sparse Measurements for Near-Field Synthetic Aperture Radar

This paper reports the experimental studies for four image reconstruction methods from sparse measurement using wideband microwave synthetic aperture radar systems. The four methods include two denoising methods using zero filling (ZF) and nonuniform fast Fourier transform (NUFFT), and two compressed sensing (CS) methods using the orthogonal matching pursuit and the conjugate gradient algorithms. The specimens under test (SUTs) consist of a tray of small rocks with different densities with/without one piece wrapped in an aluminum foil. The raw measurements of the SUTs are randomly undersampled in the spatial domain, and the images are reconstructed from the measurements of 10%–60% sparse-sampling rates. The results show that the CS method achieves good image quality with as low as 30% sparse-sampling rate, while ZF and NUFFT require 50% to obtain acceptable quality. An enhanced Otsu’s method is also proposed to detect the foiled rock from sparse reconstructions, which improves detection performance for the sparse-sampling rate of 5%–15%. The reduction of spatial measurement leads to reduced cost or reduced measurement time.

A Novel Through‐the‐Wall Imaging Algorithm Combined with Phase Shift Migration and NUFFT

This paper proposes a novel Through wall imaging (TWI) algorithm combined with phase compensation and Nonuniform fast Fourier transform (NUFFT). The wall effect is cancelled with phase compensation, which is implemented by multiplying the conjugated wall transmissivity with the received echoes. The NUFFT technique is implemented to calculate the imaging results. The proposed method has two merits: The phase compensation avoids a time-consuming calculation of the incidences angles; the image is reconstructed more accurately and efficiently than the conventional Interpolation-fast Fourier transform (IPFFT) method. The new method is implemented on 2D synthetic-generated data and experimental data. The results demonstrate that our method generates high quality focused images and achieves target geometrical identification within a very short time.

3D MR fingerprinting with accelerated stack-of-spirals and hybrid sliding-window and GRAPPA reconstruction

PurposeWhole-brain high-resolution quantitative imaging is extremely encoding intensive, and its rapid and robust acquisition remains a challenge. Here we present a 3D MR fingerprinting (MRF) acquisition with a hybrid sliding-window (SW) and GRAPPA reconstruction strategy to obtain high-resolution T1, T2 and proton density (PD) maps with whole brain coverage in a clinically feasible timeframe. Methods3D MRF data were acquired using a highly under-sampled stack-of-spirals trajectory with a steady-state precession (FISP) sequence. For data reconstruction, kx-ky under-sampling was mitigated using SW combination along the temporal axis. Non-uniform fast Fourier transform (NUFFT) was then applied to create Cartesian k-space data that are fully-sampled in the in-plane direction, and Cartesian GRAPPA was performed to resolve kz under-sampling to create an alias-free SW dataset. T1, T2 and PD maps were then obtained using dictionary matching. ResultsPhantom study demonstrated that the proposed 3D-MRF acquisition/reconstruction method is able to produce quantitative maps that are consistent with conventional quantification techniques. Retrospectively under-sampled in vivo acquisition revealed that SW + GRAPPA substantially improves quantification accuracy over the current state-of-the-art accelerated 3D MRF. Prospectively under-sampled in vivo study showed that whole brain T1, T2 and PD maps with 1 mm3 resolution could be obtained in 7.5 min. Conclusions3D MRF stack-of-spirals acquisition with hybrid SW + GRAPPA reconstruction may provide a feasible approach for rapid, high-resolution quantitative whole-brain imaging.

Technical Note: Evaluation of pre-reconstruction interpolation methods for iterative reconstruction of radial k-space data.

To evaluate the use of three different pre-reconstruction interpolation methods to convert non-Cartesian k-space data to Cartesian samples such that iterative reconstructions can be performed more simply and more rapidly. Phantom as well as cardiac perfusion radial datasets were reconstructed by four different methods. Three of the methods used pre-reconstruction interpolation once followed by a fast Fourier transform (FFT) at each iteration. The methods were: bilinear interpolation of nearest-neighbor points (BINN), 3-point interpolation, and a multi-coil interpolator called GRAPPA Operator Gridding (GROG). The fourth method performed a full non-Uniform FFT (NUFFT) at each iteration. An iterative reconstruction with spatiotemporal total variation constraints was used with each method. Differences in the images were quantified and compared. The GROG multicoil interpolation, the 3-point interpolation, and the NUFFT-at-each-iteration approaches produced high quality images compared to BINN, with the GROG-derived images having the fewest streaks among the three preinterpolation approaches. However, all reconstruction methods produced approximately equal results when applied to perfusion quantitation tasks. Pre-reconstruction interpolation gave approximately an 83% reduction in reconstruction time. Image quality suffers little from using a pre-reconstruction interpolation approach compared to the more accurate NUFFT-based approach. GROG-based pre-reconstruction interpolation appears to offer the best compromise by using multicoil information to perform the interpolation to Cartesian sample points prior to image reconstruction. Speed gains depend on the implementation and relatively standard optimizations on a MATLAB platform result in preinterpolation speedups of ~ 6 compared to using NUFFT at every iteration, reducing the reconstruction time from around 42 min to 7 min.

Memory-Optimized Re-Gridding Architecture for Non-Uniform Fast Fourier Transform

This paper proposes a power-efficient and memory-optimized FPGA-based solution for the memory and compute-intense re-gridding process used in implementing Non-uniform Fast Fourier Transform (NuFFT) algorithm. Re-gridding refers to mapping non-equispaced sampled data onto a uniform grid using an interpolation kernel function. Re-gridding is the most time-consuming step in the entire NuFFT computation. The proposed solution is based on better utilization of FPGA resources and minimizing the number of accesses to the external memory. We demonstrate high performance over a wide range of configurations and data-sizes. This paper targets a generic solution to arbitrary sampling trajectories and gives trajectory specific solutions for some well-known trajectories in NuFFT applications, such as magnetic resonance imaging and synthetic aperture radar. Compared with existing solutions, throughput is improved by over 9.6 when compared with the existing FPGA-based techniques, whereas computational power efficiency (in terms of MFLOPS/Watt) is improved by over 15 times. Compared with GPU-based technique, 9.59 times higher MFLOPS per watts are achieved. The proposed architecture is implemented using hardware description language as well as high-level synthesis (HLS)-based OpenCL framework and the comparison is reported. Hence, this paper also serves as a reference for the comparison of HLS-based solutions against optimized HDLs. Accuracy of the re-gridding process is also reported in terms of signal-to-noise ratio.

A novel nonlocal potential solver based on nonuniform FFT for efficient simulation of the Davey−Stewartson equations

We propose an efficient and accurate solver for the nonlocal potential in the Davey−Stewartson equations using nonuniform FFT (NUFFT). A discontinuity in the Fourier transform of the nonlocal potential causes accuracy locking if the potential is solved by standard FFT with periodic boundary conditions on a truncated domain. Using the fact that the discontinuity disappears in polar coordinates, we reformulate the potential integral and split it into high and low frequency parts. The high frequency part can be approximated by the standard FFT method, while the low frequency part is evaluated with a high order Gauss quadrature accelerated by nonuniform FFT. The NUFFT solver has O(N log N) complexity, where N is the total number of discretization points, and achieves higher accuracy than standard FFT solver, which makes its use in simulations very attractive. Extensive numerical results show the efficiency and accuracy of the proposed new method.

ISAR Imaging of Nonuniformly Rotating Target Based on the Multicomponent CPS Model Under Low SNR Environment

The inverse synthetic aperture radar (ISAR) imaging for nonuniformly rotating target has always been a challenging task due to the time-varying Doppler parameter, especially in the low signal-to-noise ratio (SNR) environment. In this paper, a novel ISAR imaging algorithm for nonuniformly rotating targets in the low SNR environment based on the parameter estimation approach is presented. First, the received signal in this work in a range bin is modeled as a multicomponent cubic phase signal (CPS) after motion compensation. Two approaches, namely coherently integrated modified cubic phase function (CIMCPF) and coherently integrated modified high-order ambiguous function (CIMHAF), are proposed to, respectively, estimate the second-order and the third-order coefficients in the CPS. Thanks to the coherent integrations developed in both CIMCPF and CIMHAF, they demonstrate excellent low SNR performance. Moreover, to efficiently implement the proposed approach, the nonuniform fast Fourier transform (NUFFT) is utilized in this work. Due to the usage of the NUFFT, the computational cost is reduced, and the search procedure also dispenses with the nonuniformly spaced signal. Finally, CIMCPF and CIMHAF are applied to produce ISAR image for a maneuvering target based on the CPS model. Several numerical examples, analyses of the noise tolerance performance for the proposed approaches, and ISAR imaging results demonstrate the superior performance of the proposed method.

An Efficient Boundary Integral Scheme for the MBO Threshold Dynamics Method via the NUFFT

The MBO threshold dynamics method consists of two steps. The first step solves a pure initial value problem of the heat equation with the initial data being the indicator function of some bounded domain. In the second step, the new sharp interface is generated via thresholding either by some prescribed solution value or by volume preserving. We propose an efficient boundary integral scheme for simulating the threshold dynamics via the nonuniform fast Fourier transform (NUFFT). The first step is carried out by evaluating a boundary integral via the NUFFT, and the second step is performed applying a root-finding algorithm along the normal directions of a discrete set of points at the interface. Unlike most existing methods where volume discretization is needed for the whole computational domain, our scheme requires the discretization of physical space only in a small neighborhood of the interface and thus is meshfree. The algorithm is spectrally accurate in space for smooth interfaces and has \(O(N\log N)\) complexity, where N is the total number of discrete points near the interface when the time step \(\Delta t\) is not too small. The performance of the algorithm is illustrated via several numerical examples in both two and three dimensions.

Efficient parameter estimation method for maneuvering targets in discrete randomly-modulated radar

This paper proposes an efficient parameter estimation method for maneuvering targets based on the discrete randomly-modulated (DRM) radar. To correct the range cell migration, the range frequency spectrum of the received echo is evenly divided into two parts, and a synthetic aligned signal is constructed by multiplying one with the conjugate of the other. Then, the aligned azimuth signal is modeled as a nonuniform linear frequency modulated (Nu-LFM) signal and a two-dimensional nonuniform symmetric correlation distribution (2D-NUSCD) is applied to uniquely present the Nu-LFM signal in the 2D frequency domain. The 2D-NUSCD is simple and only requires complex multiplication and nonuniform fast Fourier transform (NUFFT). Owing to the application of NUFFT, the searching processing is unnecessary for the nonuniformly spaced signal and the computational cost is reduced greatly. Furthermore, theoretical analysis illustrates that the proposed method is appropriate for multitargets and is an asymptotically efficient approach with good anti-noise performance. Compared to other representative methods, the proposed method significantly improves the computational efficiency on the premise of retaining the estimation performance. It is suitable for both DRM and determinate radar systems, providing broader application scope. Simulation results verify the effectiveness of the proposed method.