The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformly-spaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the min-max approach provides substantially lower approximation errors than conventional interpolation methods. The min-max criterion is also useful for optimizing the parameters of interpolation kernels such as the Kaiser-Bessel function.

The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain. When the sampling is uniform, the fast Fourier transform (FFT) allows this calculation to be computed in O(N log N) operations rather than O(N 2) operations. Unfortunately, when the sampling is nonuniform, the FFT does not apply. Over the last few years, a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform FFTs (NUFFTs). These rely on a mixture of interpolation and the judicious use of the FFT on an oversampled grid [A. Dutt and V. Rokhlin, SIAM J. Sci. Comput., 14 (1993), pp. 1368–1383]. In this paper, we observe that one of the standard interpolation or “gridding ” schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights. This is of particular value in two- and threedimensional settings, saving either 10dN in storage in d dimensions or a factor of about 5–10 in CPUtime (independent of dimension).

In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given two-dimensional signal of size N × N, the proposed algorithm’s complexity is O(N^2 log N), just like in a Cartesian 2D-FFT. A special feature of our approach is that it involves only 1D equispaced FFT’s and 1D interpolations. A central tool in our method is the pseudo-Polar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudo-Polar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT’s, the pseudo-Polar FFT plays the role of a halfway point—a nearly-Polar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT’s and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesian-based unequally-sampled FFT method to ours, both algorithms using a small-support interpolation and no pre-compensating, and show marked advantage to the use of the pseudo-Polar initial grid.

NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and its variants, as well as a general guideline for using the library. Numerical examples for a number of applications are given.

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 ∫ e2πiΦ(x,ξ) a(x, ξ) ˆ f(ξ)dξ at points given on a Cartesian grid. Here, ξ is a frequency variable, ˆ f(ξ) is the Fourier transform of the input f, a(x, ξ) isan amplitude, and Φ(x, ξ) is a phase function, which is typically as large as |ξ|; 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 ( √ 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 e2πiΦ(x,ξ) a(x, ξ) 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.

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In this article we develop a fast high accuracy Polar FFT. For a given two-dimensional signal of size N N , the proposed algorithm's complexity is O(N log N), just like in a Cartesian 2D-FFT. A special feature of our approach is that it involves only 1-D equispaced FFT's and 1D interpolations. A central tool in our approach is the pseudo-polar FFT, an FFT where the evaluation frequencies lie in an oversampled set of non-angularly equispaced points. The pseudo-polar FFT plays the role of a halfway point -- a nearly-polar system from which conversion to Polar Coordinates uses processes relying purely on interpolation operations. We describe the conversion process, and compare accuracy results obtained by unequallysampled FFT methods to ours and show marked advantage to our approach.

MRI rosette k-space trajectory MRI Application ky kx 2 Ignoring lots of things:

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In this paper, we outline two parameter estimation schemes for discerning canonical shape parameters from bistatic and monostatic radar data collected over a non-linear flight path. We present initial results characterizing both optimal and sub-optimal estimation performance for location and size estimation of plates, dihedrals, and trihedrals. Index Terms — bistatic SAR, ATR, canonical shapes, CRB 1.