Non-uniform Fast Fourier Transform Algorithm Research Articles

Fast numerical simulation of 2D gravity anomaly based on nonuniform fast Fourier transform in mixed space-wavenumber domain

Wavenumber domain gravity modeling is typically based on the fast Fourier transform (FFT), requiring uniform sampling in both space and wavenumber domains and grid expansion to guarantee the accuracy. In this paper, we develop an efficient and highly accurate 2D forward modeling algorithm based on the nonuniform fast Fourier transform (NUFFT) in the hybrid space and wavenumber domain. To retain the accuracy of space domain methods, the Fourier transform is carried out only in the horizontal direction based on NUFFT and no transform is applied in the vertical direction (make the discretization in the vertical direction flexible). Thus, the original gravity modeling problem is transformed into a sequence of independent space domain 1D integral equations, which can be solved in parallel efficiently. For the 1D integral in the space domain, the density is represented by quadratic shape function and the independent equations can be solved analytically. The application of the NUFFT algorithm allows for the nonuniform discretization of a model and improves calculation efficiency while ensuring calculation accuracy. Based on a 2D prism model, the accuracy of the proposed algorithm is verified and its numerical performance in terms of efficiency and accuracy is compared with the standard FFT and Gauss-FFT methods, indicating that at the same accuracy our algorithm is most efficient. Then, three more complex models and one BP density model are designed to further demonstrate the good numerical performance of the proposed algorithm.

Numerical simulation of laser propagation in ocean turbulence with the nonuniform fast Fourier transform algorithm

Our work studies the spatial distribution of ocean optical turbulence through numerical simulation. We used the ocean optical turbulence phase screen created by the ocean power spectrum to study the effect of varying the refractive index of water via a laser beam propagating through ocean optical turbulence. Because the intensity of ocean optical turbulence is much higher than that of atmospheric turbulence, when simulating the ocean phase screen, traditional methods based on the fast Fourier transform (FFT) algorithm introduce enormous errors in the low-frequency band. Some interpolation algorithms commonly used in atmospheric turbulence simulation can reduce these errors. However, their calculation complexity is always high, so the computational speed is slow. We combine a nonuniform sampling method based on variations in the ocean turbulent power spectrum and the nonuniform FFT algorithm to generate phase screens of ocean optical turbulence. The proposed algorithm can solve the deficiency of the long runtime of the traditional algorithm with even better accuracy. In addition to the traditional phase structure function, we measured the far-field distribution of the laser transmitting through water turbulence caused by temperature contrasts, and the experimental results have outstanding agreement with the simulated far-field image from our ocean phase screen. To the best of our knowledge, this is the first work to verify the accuracy of the phase screen by conducting far-field image experiments.

Fourier-domain modeling of gravity effects caused by a vertical polyhedral prism, with application to a water reservoir storage process

We have developed a new Fourier-domain algorithm for the modeling of gravity effects caused by a vertical polyhedral prism. Simplified and more compact Fourier-domain expressions are derived for the vertical gravity anomaly on a 2D plane either above, in the middle of, or below the vertical polyhedral prism. A new Fourier-domain algorithm, which combines a low-order Gauss fast Fourier transform (FFT) algorithm and a low-order nonuniform FFT algorithm, to permit polar sampling near the zero wave vector, is introduced. To validate the numerical efficiency of the new algorithm, we carry out a series of synthetic tests. Numerical results show that the new algorithm is superior in numerical accuracy and efficiency compared to a pure Gauss-FFT algorithm, and both Fourier methods run much faster than traditional space-domain analytical solutions. We then apply the Fourier forward algorithms to the prediction of vertical component of gravity at many survey points, as a function of the water level of a reservoir located in Xianlin, Hangzhou, China. The reservoir geometry is approximated by a single vertical polyhedral prism. Water storage corresponds to increased heights of the polyhedral prism. Detailed maps of vertical gravity value changes corresponding to a water level rise of 5 and 10 m, with spatial resolution [Formula: see text], on the local topography covering a study area of approximately [Formula: see text] around the water reservoir, are calculated using the new algorithm. Then, the gravity values are classified according to the accuracy of the gravimeter, providing a reference of optimal site selection for a proposed cold-atom gravimetry survey to be carried out in the near future.

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.

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.

A Novel Nonuniform Fast Fourier Transform Algorithm and Its Application to Aperiodic Arrays

A simple and effective nonuniform fast Fourier transform algorithm is presented. The algorithm is based on the interpolation of the exponential function in the Fourier transform using efficient truncated expansion algorithms for band-limited functions. Examples regarding the far-field evaluation of nonuniformly spaced antenna arrays are provided, and the behavior of the approximation error is analyzed, to confirm the effectiveness of the method.

An improved semi-Lagrangian time splitting spectral method for the semi-classical Schrödinger equation with vector potentials using NUFFT

In this paper, we propose a new time splitting Fourier spectral method for the semi-classical Schrödinger equation with vector potentials. Compared with the results in [21], our method achieves spectral accuracy in space by interpolating the Fourier series via the NonUniform Fast Fourier Transform (NUFFT) algorithm in the convection step. The NUFFT algorithm helps maintain high spatial accuracy of Fourier method, and at the same time improve the efficiency from O(N2) (of direct computation) to O(Nlog⁡N) operations, where N is the total number of grid points. The kinetic step and potential step are solved by analytical solution with pseudo-spectral approximation, and, therefore, we obtain spectral accuracy in space for the whole method. We prove that the method is unconditionally stable, and we show improved error estimates for both the wave function and physical observables, which agree with the results in [3] for vanishing potential cases and are superior to those in [21]. Extensive one and two dimensional numerical studies are presented to verify the properties of the proposed method, and simulations of 3D problems are demonstrated to show its potential for future practical applications.

Mean square optimal NUFFT approximation for efficient non-Cartesian MRI reconstruction

The fast evaluation of the discrete Fourier transform of an image at non-uniform sampling locations is key to efficient iterative non-Cartesian MRI reconstruction algorithms. Current non-uniform fast Fourier transform (NUFFT) approximations rely on the interpolation of oversampled uniform Fourier samples. The main challenge is high memory demand due to oversampling, especially when multidimensional datasets are involved. The main focus of this work is to design an NUFFT algorithm with minimal memory demands. Specifically, we introduce an analytical expression for the expected mean square error in the NUFFT approximation based on our earlier work. We then introduce an iterative algorithm to design the interpolator and scale factors. Experimental comparisons show that the proposed optimized NUFFT scheme provides considerably lower approximation errors than the previous designs [1] that rely on worst case error metrics. The improved approximations are also seen to considerably reduce the errors and artifacts in non-Cartesian MRI reconstruction.

Selection of convolution kernel in non-uniform fast Fourier transform for Fourier domain optical coherence tomography

Gridding based non-uniform fast Fourier transform (NUFFT) has recently been shown as an efficient method of processing non-linearly sampled data from Fourier-domain optical coherence tomography (FD-OCT). This method requires selecting design parameters, such as kernel function type, oversampling ratio and kernel width, to balance between computational complexity and accuracy. The Kaiser-Bessel (KB) and Gaussian kernels have been used independently on the NUFFT algorithm for FD-OCT. This paper compares the reconstruction error and speed for the optimization of these design parameters and justifies particular kernel choice for FD-OCT applications. It is found that for on-the-fly computation of the kernel function, the simpler Gaussian function offers a better accuracy-speed tradeoff. The KB kernel, however, is a better choice in the pre-computed kernel mode of NUFFT, in which the processing speed is no longer dependent on the kernel function type. Finally, the algorithm is used to reconstruct in-vivo images of a human finger at a camera limited 50k A-line/s.

Improved Accuracy Factors for the Nonuniform Fast Fourier Transform (NUFFT) Algorithm

Based on the regular Fourier matrix, a new set of accuracy factors is proposed for the nonuniform fast Fourier transform algorithm to improve the accuracy of transformed data. It shows that the proposed factors can reduce the errors by three to more than ten times with almost the same number of arithmetic operations. Numerical examples are shown for the applications in computational electromagnetics.

Fast Fourier Transform for Discontinuous Functions

In computational electromagnetics and other areas of computational science and engineering, Fourier transforms of discontinuous functions are often required. We present a fast algorithm for the evaluation of the Fourier transform of piecewise smooth functions with uniformly or nonuniformly sampled data by using a double interpolation procedure combined with the fast Fourier transform (FFT) algorithm. We call this the discontinuous FFT algorithm. For N sample points, the complexity of the algorithm is O(/spl nu/Np+/spl nu/Nlog(N)) where p is the interpolation order and /spl nu/ is the oversampling factor. The method also provides a new nonuniform FFT algorithm for continuous functions. Numerical experiments demonstrate the high efficiency and accuracy of this discontinuous FFT algorithm.