Nonuniform Fast Fourier Research Articles

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.

High-Precision Imaging Algorithm for Highly Squinted SAR With 3D Acceleration

In conventional synthetic aperture radar (SAR) processing algorithms, it is generally assumed that the platform performs a uniform linear motion. However, if there is a three-dimensional acceleration during the flight, the motion trajectory will greatly increase the image processing complexity, especially in highly squinted scenarios. This paper proposes a high-precision imaging algorithm for highly squinted SAR with 3D acceleration. Firstly, the range history is extended by the Taylor series, and then a two-dimensional non-uniform fast Fourier transform (2D-NUFFT) is presented to focus the SAR image in range velocity domain. Moreover, as terrain matching is required in practice, the range-velocity image is further transformed into range-azimuth domain. The effectiveness of the proposed algorithm for highly squinted SAR imaging is verified with both simulation data for multi-point targets and experimental data for surface targets.

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...

An ISAR Imaging Algorithm for Nonuniformly Rotating Targets With Low SNR Based on Modified Bilinear Parameter Estimation of Cubic Phase Signal

For nonuniformly rotating target in the low signal-to-noise ratio (SNR) environment, the inverse synthetic aperture radar (ISAR) imaging is a challenging task due to the Doppler spread induced by time-varying rotation. In this paper, an effective modified bilinear parameter estimation algorithm is proposed and applied to reconstruct the ISAR image of nonuniformly rotating targets with low SNR. First, azimuth echoes of a range cell are modeled as multicomponent cubic phase signals (multi-CPSs) after range alignment and phase adjustment. Then, to estimate the parameter of the CPS (the chirp rate and the quadratic chirp rate, which are identified as causes of the target image defocus), a novel integrated parametric cubic phase function (IPCPF) and a reversing Wigner–Ville distribution processing are developed. Compared to other four representative algorithms, the bilinearity of the IPCPF guarantees a higher antinoise performance and a better suppression on cross terms. Moreover, by utilizing the nonuniform fast Fourier transform and the generalized scaled Fourier transform, the brute-force searching is eliminated and the computational cost is reduced. Finally, a new cross-range scaling method based on a regression analysis is proposed for the ISAR imaging. With the synthetic data and the real radar data, several simulation examples and ISAR imaging results demonstrate the effectiveness and the superiority of the proposed ISAR imaging algorithm.

Trajectory optimized NUFFT: Faster non-Cartesian MRI reconstruction through prior knowledge and parallel architectures.

PurposeThe non‐uniform fast Fourier transform (NUFFT) involves interpolation of non‐uniformly sampled Fourier data onto a Cartesian grid, an interpolation that is slowed by complex, non‐local data access patterns. A faster NUFFT would increase the clinical relevance of the plethora of advanced non‐Cartesian acquisition methods.MethodsHere we customize the NUFFT procedure for a radial trajectory and GPU architecture to eliminate the bottlenecks encountered when allowing for arbitrary trajectories and hardware. We call the result TRON, for TRajectory Optimized NUFFT. We benchmark the speed and accuracy TRON on a Shepp‐Logan phantom and on whole‐body continuous golden‐angle radial MRI.ResultsTRON was 6–30× faster than the closest competitor, depending on test data set, and was the most accurate code tested.ConclusionsSpecialization of the NUFFT algorithm for a particular trajectory yielded significant speed gains. TRON can be easily extended to other trajectories, such as spiral and PROPELLER. TRON can be downloaded at http://github.com/davidssmith/TRON.

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 Fourier transforms of piecewise polynomials

We introduce an efficient algorithm for evaluating the Fourier transform of piecewise-polynomial data on d-dimensional simplices in D-dimensional Euclidean space RD. It generalizes butterfly algorithms for pointwise (d=0) nonuniform fast Fourier transforms, with new dimensional recurrences for exponential-polynomial moments. Error analysis and numerical comparisons with direct evaluation validate the efficiency and accuracy of the algorithm.

Quadratic Frequency Modulation Signals Parameter Estimation Based on Two-Dimensional Product Modified Parameterized Chirp Rate-Quadratic Chirp Rate Distribution.

In an inverse synthetic aperture radar (ISAR) imaging system for targets with complex motion, the azimuth echo signals of the target are always modeled as multicomponent quadratic frequency modulation (QFM) signals. The chirp rate (CR) and quadratic chirp rate (QCR) estimation of QFM signals is very important to solve the ISAR image defocus problem. For multicomponent QFM (multi-QFM) signals, the conventional QR and QCR estimation algorithms suffer from the cross-term and poor anti-noise ability. This paper proposes a novel estimation algorithm called a two-dimensional product modified parameterized chirp rate-quadratic chirp rate distribution (2D-PMPCRD) for QFM signals parameter estimation. The 2D-PMPCRD employs a multi-scale parametric symmetric self-correlation function and modified nonuniform fast Fourier transform-Fast Fourier transform to transform the signals into the chirp rate-quadratic chirp rate (CR-QCR) domains. It can greatly suppress the cross-terms while strengthening the auto-terms by multiplying different CR-QCR domains with different scale factors. Compared with high order ambiguity function-integrated cubic phase function and modified Lv’s distribution, the simulation results verify that the 2D-PMPCRD acquires higher anti-noise performance and obtains better cross-terms suppression performance for multi-QFM signals with reasonable computation cost.

Efficient Implementation of GPR Data Inversion in Case of Spatially Varying Antenna Polarizations

Ground penetrating radar imaging from the data acquired with arbitrarily oriented dipole-like antennas is considered. To take into account variations of antenna orientations resulting in spatial rotation of antenna radiation patterns and polarizations of transmitted fields, the full-wave method that accounts for the near-, intermediate-, and far-field contributions to the radiation patterns is applied for image reconstruction, which is formulated as a linear inversion problem. Two approaches, namely, an interpolation-based method and a nonuniform fast Fourier transform-based method, are suggested to efficiently implement the full-wave method by computing exact Green’s functions. The effectiveness and accuracy of the method proposed have been verified via both numerical simulations and experimental measurements, and significant improvement of the reconstructed image quality compared with the traditional scalar-wave-based migration algorithms is demonstrated. The results can be directly utilized by forward-looking microwave imaging sensors such as installed at tunnel boring machine or can be used for the observation matrix computation in regularization-based inversion algorithms.

Python Non-Uniform Fast Fourier Transform (PyNUFFT): An Accelerated Non-Cartesian MRI Package on a Heterogeneous Platform (CPU/GPU)

A Python non-uniform fast Fourier transform (PyNUFFT) package has been developed to accelerate multidimensional non-Cartesian image reconstruction on heterogeneous platforms. Since scientific computing with Python encompasses a mature and integrated environment, the time efficiency of the NUFFT algorithm has been a major obstacle to real-time non-Cartesian image reconstruction with Python. The current PyNUFFT software enables multi-dimensional NUFFT accelerated on a heterogeneous platform, which yields an efficient solution to many non-Cartesian imaging problems. The PyNUFFT also provides several solvers, including the conjugate gradient method, ℓ1 total variation regularized ordinary least square (L1TV-OLS), and ℓ1 total variation regularized least absolute deviation (L1TV-LAD). Metaprogramming libraries have been employed to accelerate PyNUFFT. The PyNUFFT package has been tested on multi-core central processing units (CPUs) and graphic processing units (GPUs), with acceleration factors of 6.3–9.5× on a 32-thread CPU platform and 5.4–13× on a GPU.

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.

Hopped‐frequency waveform design for range sidelobe suppression in spectral congestion

By properly transmitting pulses through the clear frequency bands, the hopped-frequency (HopF) waveform can achieve a large synthetic bandwidth while avoiding the complexity of wideband hardware as well as the mutual interference to/from adjacent RF users. To optimally design the HopF waveform with low range sidelobes under given spectral constraints, we propose a two-step-based approach in this paper. In the first step, a deterministic approach is presented to produce a rough approximation of the optimum solution to address the non-convexity of the waveform design. In the second step, gradient method is applied to locally refine the solution obtained in the first step. The proposed design method is efficient as the main computation in the first step lies in solving a convex optimisation which can be achieved in polynomial time and the iterative operations in the second step can be implemented using non-uniform fast Fourier transform. Experimental evidence suggests that the proposed method can produce near optimal solution in remarkably little time.

An Efficient Algorithm for MIMO Cylindrical Millimeter-Wave Holographic 3-D Imaging

A novel cylindrical millimeter-wave holography regime using a multi-input-multi-output (MIMO) array is first introduced. More importantly, an accurate and fast 3-D imaging algorithm is developed for this regime. Although many efficient imaging algorithms for synthetic aperture radar (SAR) or circular SAR have been developed, they can hardly be used for this new regime due to the compound effects of a spherical wavefront, MIMO geometry, and cylindrical observation aperture. In this paper, we develop an efficient algorithm mainly based on the ideas of a spherical wave decomposition, circular convolution, and nonuniform fast Fourier transform. The proposed algorithm achieves similar imaging quality with the golden-standard back-projection algorithm (BPA) while costs much less time. Compared with the BPA, the proposed algorithm belongs to a kind of the frequency-domain method built upon range migration techniques. Detailed derivations are first presented. Then, several important issues including resolutions, sampling criteria, and computational complexities are analyzed. Finally, a series of simulations and experiments are carried out, and all the results verify the effectiveness of the proposed algorithm on both accuracy and efficiency.

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.

Accelerating near-field 3D imaging approach for joint high-resolution imaging and phase error correction

The computational complexity and memory requirements of large-scale data seriously affect the application of compressed sensing (CS) in near-field three-dimensional (3-D) imaging system. In addition, as influenced by the measurement environment, the error in echo phase results in imaging defocusing. This paper proposes a CS near-field 3-D imaging approach based on nonuniform fast Fourier transform and phase error correction. It applies the fast Gaussian gridding nonuniform fast Fourier transform technique and Separable Surrogate Functionals with only matrix and vector multiplied to accelerate imaging speed and reduce memory requirements; it adopts the phase error correction technique to realize highly-focused imaging; in addition, a sparse observation approach based on Logistic sequence is proposed in this paper for easy availability of engineering realization for CS imaging. As indicated by numerical analysis and actual measurement in anechoic chamber, the approach proposed in this paper, compared with traditional imaging approaches, has the following advantages: accurate high resolution 3-D image of target can be obtained by applying small amount of observation data (10%); the computational complexity falls from O(LN) to O(3N) and memory occupation quantity drops from O(LN) to O(N); it can effectively perform highly-focused imaging for echo signal with phase error; the measurement matrix designed has better non-coherence and easy availability for engineering realization.

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.

Dimension-Factorized Range Migration Algorithm for Regularly Distributed Array Imaging.

The two-dimensional planar MIMO array is a popular approach for millimeter wave imaging applications. As a promising practical alternative, sparse MIMO arrays have been devised to reduce the number of antenna elements and transmitting/receiving channels with predictable and acceptable loss in image quality. In this paper, a high precision three-dimensional imaging algorithm is proposed for MIMO arrays of the regularly distributed type, especially the sparse varieties. Termed the Dimension-Factorized Range Migration Algorithm, the new imaging approach factorizes the conventional MIMO Range Migration Algorithm into multiple operations across the sparse dimensions. The thinner the sparse dimensions of the array, the more efficient the new algorithm will be. Advantages of the proposed approach are demonstrated by comparison with the conventional MIMO Range Migration Algorithm and its non-uniform fast Fourier transform based variant in terms of all the important characteristics of the approaches, especially the anti-noise capability. The computation cost is analyzed as well to evaluate the efficiency quantitatively.