Nonuniform Fast Fourier Transform Research Articles

Fast phase-only synthesis of conformal reflectarrays

Phase-only synthesis (POS) techniques represent an important tool for reflectarray design. Indeed, although based on approximate scattering models, for planar antennas, they are numerically effective thanks to the use of fast Fourier transform (FFT) routines. Conformal reflectarrays are becoming of interest as they reduce the frequency dependence of the feed/elements path length. Although synthesis algorithms can be extended from planar to conformal geometries, improving computational efficiency is not trivial. We present a POS, power pattern synthesis technique of conformal, shaped-beam reflectarrays, based on the p-series approach, on non-uniform FFTs, and on fast implementations of the FFT and of matrix–vector and vector–vector multiplications. The approach allows modulating computational burden and accuracy, depending on the needs. The method faces crucial issues concerning the reliability of the solution, by adopting a multi-step strategy. At first, the unknown phase shifts are represented by few degrees of freedom and an efficient and effective global optimisation algorithm is applied. Then, assuming as a starting point the obtained solution, all the reflectarray degrees of freedom are taken into account. The computational performance of the approach and its capability in synthesising shaped-beam patterns are discussed against parabolic and faceted reflectarrays, also considering the coverage of an operative mission.

A new reconstruction approach in diffraction ultrasound tomography: Combine transmission mode with reflection mode tomography

Diffraction Ultrasound Tomography based on Fourier scattering theorem is studied. In this paper, we reveal the relation between the transmission mode (TMDT) and the reflection mode diffraction tomography (RMDT). Different from the transmission mode ultrasound tomography which lose the details of the picture, We combine the forward scatter field data and the backward data with the higher frequency information. The iterative reconstruction method with forward non-uniform fast Fourier transform (NUFFT) is used in the proposed algorithm. The simulation result indicates that the algorithm proposed in this paper is obviously improve the reconstruction quality than the TMDT or the RMDT.

A Reconstruction Algorithm for Photoacoustic Imaging Based on the Nonuniform FFT

Fourier reconstruction algorithms significantly outperform conventional backprojection algorithms in terms of computation time. In photoacoustic imaging, these methods require interpolation in the Fourier space domain, which creates artifacts in reconstructed images. We propose a novel reconstruction algorithm that applies the one-dimensional nonuniform fast Fourier transform to photoacoustic imaging. It is shown theoretically and numerically that our algorithm avoids artifacts while preserving the computational effectiveness of Fourier reconstruction.

A geometric nonuniform fast Fourier transform

An efficient algorithm is presented for the computation of Fourier coefficients of piecewise-polynomial densities on flat geometric objects in arbitrary dimension and codimension. Applications range from standard nonuniform FFTs of scattered point data, through line and surface potentials in two and three dimensions, to volumetric transforms in three dimensions. Input densities are smoothed with a B-spline kernel, sampled on a uniform grid, and transformed by a standard FFT, and the resulting coefficients are unsmoothed by division. Any specified accuracy can be achieved, and numerical experiments demonstrate the efficiency of the algorithm for a gallery of realistic examples.

Optimized Least-Square Nonuniform Fast Fourier Transform

The main focus of this paper is to derive a memory efficient approximation to the nonuniform Fourier transform of a support limited sequence. We show that the standard nonuniform fast Fourier transform (NUFFT) scheme is a shift invariant approximation of the exact Fourier transform. Based on the theory of shift-invariant representations, we derive an exact expression for the worst-case mean square approximation error. Using this metric, we evaluate the optimal scale-factors and the interpolator that provides the least approximation error. We also derive the upper-bound for the error component due to the lookup tablebased evaluation of the interpolator; we use this metric to ensure that this component is not the dominant one. Theoretical and experimental comparisons with standard NUFFT schemes clearly demonstrate the significant improvement in accuracy over conventional schemes, especially when the size of the uniform fast Fourier transform (FFT) is small. Since the memory requirement of the algorithm is dependent on the size of the uniform FFT, the proposed developments can lead to iterative signal reconstruction algorithms with significantly lower memory demands.

NUFFT-Based Iterative Reconstruction Algorithm for Synthetic Aperture Imaging Radiometers

Synthetic aperture imaging radiometers (SAIRs) are powerful sensors for high-resolution observation of the Earth at low microwave frequencies. However, the future application of this new technology is limited by its large number of element antennas and receiving channels. In order to further reduce the complexity of the whole system, the circular and rotating array configurations have been proposed. Their disadvantages lie primarily in more complicated image reconstruction, because the conventional fast Fourier transform (FFT) method could not be used directly for image reconstruction. This letter introduces an iterative method for the reconstruction of radiometric brightness temperature maps from nonuniform visibility function samples. This method combines the conjugate gradient algorithm with min-max nonuniform FFT approach, which has been shown to provide considerably improved image quality relative to the traditional gridding method. Simulations for SAIRs were performed and showed better image quality in comparison to the traditional frequency interpolation method.

Least-square NUFFT methods applied to 2-D and 3-D radially encoded MR image reconstruction.

Radially encoded MRI has gained increasing attention due to its motion insensitivity and reduced artifacts. However, because its samples are collected nonuniformly in the k-space, multidimensional (especially 3-D) radially sampled MRI image reconstruction is challenging. The objective of this paper is to develop a reconstruction technique in high dimensions with on-the-fly kernel calculation. It implements general multidimensional nonuniform fast Fourier transform (NUFFT) algorithms and incorporates them into a k-space image reconstruction framework. The method is then applied to reconstruct from the radially encoded k-space data, although the method is applicable to any non-Cartesian patterns. Performance comparisons are made against the conventional Kaiser-Bessel (KB) gridding method for 2-D and 3-D radially encoded computer-simulated phantoms and physically scanned phantoms. The results show that the NUFFT reconstruction method has better accuracy-efficiency tradeoff than the KB gridding method when the kernel weights are calculated on the fly. It is found that for a particular conventional kernel function, using its corresponding deapodization function as a scaling factor in the NUFFT framework has the potential to improve accuracy. In particular, when a cosine scaling factor is used, the NUFFT method is faster than KB gridding method since a closed-form solution is available and is less computationally expensive than the KB kernel (KB griding requires computation of Bessel functions). The NUFFT method has been successfully applied to 2-D and 3-D in vivo studies on small animals.

A NEW EFFICIENT FDTD TIME-TO-FREQUENCY-DOMAIN CONVERSION ALGORITHM

The time-to-frequency-domain conversion is often required in many applications of the finite-difference time-domain (FDTD) method. This paper presents a new FDTD time-to-frequency- domain conversion algorithm based on the optimization of nonuniform fast Fourier transform (NUFFT) with several redundancy-reduction techniques. The proposed algorithm can perform the FDTD conversion at multiple desired frequencies without the limitation of uniformly spaced frequencies in the fast Fourier transform (FFT). In addition, with a very low storage cost, the algorithm can be much more efficient than other FDTD conversion techniques if a moderate number of frequencies or more are of interest. This algorithm is very useful for some FDTD applications.

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.

The Chebyshev fast Gauss and nonuniform fast Fourier transforms and their application to the evaluation of distributed heat potentials

We present a method for the fast and accurate computation of distributed heat potentials in two dimensions. The distributed source is assumed to be given in terms of piecewise space–time Chebyshev polynomials. We discretize uniformly in time, whereas in space the polynomials are defined on the leaf nodes of a quadtree. The quadtree can vary at each time step. We combine a product integration rule with fast algorithms (fast heat potentials, nonuniform FFT, fast Gauss transform) to obtain a high-order accurate method with optimal complexity. If N is the number of time steps, M is the maximum number of leaf nodes over all the time steps and the input contains a qth-order polynomial representation of f, then, our method requires O ( q 3 MN log M ) work to evaluate the heat potential at arbitrary MN space–time target locations. The overall convergence rate of the method is of order q. We present numerical experiments for q = 4, 8, and 16, and we verify the theoretical convergence rate of the method. When the solution is sufficiently smooth, the 16th-order variant results in significant computational savings, even in the case in which we require only a few digits of accuracy.

Reconstruction of Seismic Data with Least Squares Inversion Based on Nonuniform Fast Fourier Transform

AbstractReconstruction of uneven seismic data is a crucial step of the analyzing and processing of the data. In this paper, a least squares inversion based on non‐uniform fast Fourier transform is proposed, which can reconstruct both uneven and alias seismic data. The algorithm of non‐uniform fast Fourier transform is used to do Fourier transform of uneven data. An adaptive DFT weighted norm regularization constraint is used during the reconstruction. The inverse problem is solved by the preconditioned conjugate gradient method, which makes the solutions stable and convergence quick. Experimental results on theoretical model and raw uneven seismic data show that the proposed method is fast, efficient and applicable.

On NUFFT-based gridding for non-Cartesian MRI

For MRI with non-Cartesian sampling, the conventional approach to reconstructing images is to use the gridding method with a Kaiser–Bessel (KB) interpolation kernel. Recently, Sha et al. [L. Sha, H. Guo, A.W. Song, An improved gridding method for spiral MRI using nonuniform fast Fourier transform, J. Magn. Reson. 162(2) (2003) 250–258] proposed an alternative method based on a nonuniform FFT (NUFFT) with least-squares (LS) design of the interpolation coefficients. They described this LS_NUFFT method as shift variant and reported that it yielded smaller reconstruction approximation errors than the conventional shift-invariant KB approach. This paper analyzes the LS_NUFFT approach in detail. We show that when one accounts for a certain linear phase factor, the core of the LS_NUFFT interpolator is in fact real and shift invariant. Furthermore, we find that the KB approach yields smaller errors than the original LS_NUFFT approach. We show that optimizing certain scaling factors can lead to a somewhat improved LS_NUFFT approach, but the high computation cost seems to outweigh the modest reduction in reconstruction error. We conclude that the standard KB approach, with appropriate parameters as described in the literature, remains the practical method of choice for gridding reconstruction in MRI.

Fast Computation of Fourier Integral Operators

We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation, general hyperbolic equations, and curvilinear tomography. The problem is to numerically evaluate a so-called Fourier integral operator (FIO) of the form $\int e^{2\pi i \Phi(x,\xi)} a(x,\xi) \hat{f}(\xi) \d\xi$ at points given on a Cartesian grid. Here, $\xi$ is a frequency variable, $\hat f(\xi)$ is the Fourier transform of the input $f$, $a(x,\xi)$ is an amplitude, and $\Phi(x,\xi)$ is a phase function, which is typically as large as $|\xi|$; hence the integral is highly oscillatory. Because a FIO is a dense matrix, a naive matrix vector product with an input given on a Cartesian grid of size $N$ by $N$ would require $O(N^4)$ operations. This paper develops a new numerical algorithm which requires $O(N^{2.5} \log N)$ operations and as low as $O(\sqrt{N})$ in storage space (the constants in front of these estimates are small). It operates by localizing the integral over polar wedges with small angular aperture in the frequency plane. On each wedge, the algorithm factorizes the kernel $e^{2 \pi i \Phi(x,\xi)} a(x,\xi)$ into two components: (1) a diffeomorphism which is handled by means of a nonuniform FFT and (2) a residual factor which is handled by numerical separation of the spatial and frequency variables. The key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size. Several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology. We also discuss the potential of our ideas for various applications such as reflection seismology.

Two-dimensional and three-dimensional NUFFT migration method for landmine detection using ground-penetrating Radar

Ground-penetrating radar (GPR) has been widely used for landmine detection due to its high signal-to-noise ratio (SNR) and superior ability to image nonmetallic landmines. Processing GPR data to obtain better target images and to assist further object detection has been an active research area. Phase-shift migration is a widely used method; however, its wavenumber space is nonuniformly sampled because of the nonlinear relationship between the uniform frequency samples and the wavenumbers. Conventional methods use linear interpolation to obtain uniform wavenumber samples and compute the fast Fourier transform (FFT). This paper develops two- and three-dimensional migration methods that process GPR data to obtain images close to the actual target geometries using a nonuniform fast Fourier transform (NUFFT) algorithm. The proposed method is first compared to the conventional migration approaches on simulated data and then applied to landmine field data sets. Results suggest that the NUFFT migration method is useful in focusing images, estimating landmine structure, and retaining relatively high signal-to-noise ratio in the migrated data. The processed data sets are then fed to the normalized energy and least-mean-square-based anomaly detectors. Receiver operating characteristic curves of data sets processed by different migration methods are compared. The NUFFT migration shows potential improvements on both classifiers with a reduced false alarm rate at most probabilities of detection.

Fourier-based forward and back-projectors in iterative fan-beam tomographic image reconstruction

Fourier-based forward and back-projection methods can reduce computation in iterative tomographic image reconstruction. Recently, an optimized nonuniform fast Fourier transform (NUFFT) approach was shown to yield accurate parallel-beam projections. In this paper, we extend the NUFFT approach to describe an O (N2 log N) projector/backprojector pair for fan-beam transmission tomography. Simulations and experiments with real CT data show that fan-beam Fourier-based forward and back-projection methods can reduce computation for iterative reconstruction while still providing accuracy comparable to their O (N3) space-based counterparts.

Improving Non-Cartesian MRI Reconstruction through Discontinuity Subtraction

Non-Cartesian sampling is widely used for fast magnetic resonance imaging (MRI). Accurate and fast image reconstruction from non-Cartesian k-space data becomes a challenge and gains a lot of attention. Images provided by conventional direct reconstruction methods usually bear ringing, streaking, and other leakage artifacts caused by discontinuous structures. In this paper, we tackle these problems by analyzing the principal point spread function (PSF) of non-Cartesian reconstruction and propose a leakage reduction reconstruction scheme based on discontinuity subtraction. Data fidelity in k-space is enforced during each iteration. Multidimensional nonuniform fast Fourier transform (NUFFT) algorithms are utilized to simulate the k-space samples as well as to reconstruct images. The proposed method is compared to the direct reconstruction method on computer-simulated phantoms and physical scans. Non-Cartesian sampling trajectories including 2D spiral, 2D and 3D radial trajectories are studied. The proposed method is found useful on reducing artifacts due to high image discontinuities. It also improves the quality of images reconstructed from undersampled data.

Toeplitz-based iterative image reconstruction for MRI with correction for magnetic field inhomogeneity

In some types of magnetic resonance (MR) imaging, particularly functional brain scans, the conventional Fourier model for the measurements is inaccurate. Magnetic field inhomogeneities, which are caused by imperfect main fields and by magnetic susceptibility variations, induce distortions in images that are reconstructed by conventional Fourier methods. These artifacts hamper the use of functional MR imaging (fMRI) in brain regions near air/tissue interfaces. Recently, iterative methods that combine the conjugate gradient (CG) algorithm with nonuniform FFT (NUFFT) operations have been shown to provide considerably improved image quality relative to the conjugate-phase method. However, for non-Cartesian k-space trajectories, each CG-NUFFT iteration requires numerous k-space interpolations; these are operations that are computationally expensive and poorly suited to fast hardware implementations. This paper proposes a faster iterative approach to field-corrected MR image reconstruction based on the CG algorithm and certain Toeplitz matrices. This CG-Toeplitz approach requires k-space interpolations only for the initial iteration; thereafter, only fast Fourier transforms (FFTs) are required. Simulation results show that the proposed CG-Toeplitz approach produces equivalent image quality as the CG-NUFFT method with significantly reduced computation time.

The type 3 nonuniform FFT and its applications

The nonequispaced or nonuniform fast Fourier transform (NUFFT) arises in a variety of application areas, including imaging processing and the numerical solution of partial differential equations. In its most general form, it takes as input an irregular sampling of a function and seeks to compute its Fourier transform at a nonuniform sampling of frequency locations. This is sometimes referred to as the NUFFT of type 3. Like the fast Fourier transform, the amount of work required is of the order O( N log N), where N denotes the number of sampling points in both the physical and spectral domains. In this short note, we present the essential ideas underlying the algorithm in simple terms. We also illustrate its utility with application to problems in magnetic resonance imagin and heat flow.

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.

An Efficient Analysis of Shielded Single and Multiple Coupled Microstrip Lines With the Nonuniform Fast Fourier Transform (NUFFT) Technique

A nonuniform fast Fourier transform (NUFFT) technique is incorporated into the spectral-domain approach for the analysis of shielded single and multiple coupled microstrip lines. Each of the spectral-domain Green's functions is decomposed into an asymptotic part and a remaining part. At the interface of layered dielectrics with conducting strips, the product of a basis function and an associated Green's function constitutes an expansion E-field. The inverse Fourier transform (IFT) of the expansion E-field is its spatial distribution all over the interface. We take this advantage to match the final boundary conditions on all the conducting strips simultaneously. As a result, if all the strips are at one interface, the number of operations required in this method is proportional to N/sub /spl lscr//, but not to N/sub /spl lscr///sup 2/, where N/sub /spl lscr// is the number of the strips. The IFT of the asymptotic part of each expansion E-field can be obtained analytically, and that of the remaining part can be quickly processed by the NUFFT. The Gauss-Chebyshev quadrature is used to accelerate the computations of the integrals resulted from the Galerkin's procedure. The proposed method is also applied to investigate the dispersion characteristics of coupled lines with finite metallization thickness and of coupled lines at different levels. A convergence analysis of the results is presented and a comparison of used CPU time is discussed.