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.

Abstract. We define a notion of Radon Transform for data in an n by n grid. It is based on summation along lines of absolute slope less than 1 (as a function either of x or of y), with values at non-Cartesian locations defined using trigonometric interpolation on a zero-padded grid. The definition is geometrically faithful: the lines exhibit no ‘wraparound effects’. For a special set of lines equispaced in slope (rather than angle), we describe an exact algorithm which uses O(N log N) flops, where N = n2 is the number of pixels. This relies on a discrete projection-slice theorem relating this Radon transform and what we call the Pseudopolar Fourier transform. The Pseudopolar FT evaluates the 2-D Fourier transform on a non-Cartesian pointset, which we call the pseudopolar grid. Fast Pseudopolar FT – the process of rapid exact evaluation of the 2-D Fourier transform at these non-Cartesian grid points – is possible using chirp-Z transforms. This Radon transform is one-to-one and hence invertible on its range; it is rapidly invertible to any degree of desired accuracy using a preconditioned conjugate gradient solver. Empirically, the numerical conditioning is superb; the singular value spread of the preconditioned Radon transform turns out numerically to be less than 10%, and three iterations of the conjugate gradient solver typically suffice for 6 digit accuracy. We also describe a 3-D version of the transform.

Fourier-based reprojection methods have the potential to reduce the computation time in iterative tomographic image reconstruction. Interpolation errors are a limitation of Fourier-based reprojection methods. We apply a min-max interpolation method for the nonuniform fast Fourier transform (NUFFT) to minimize the interpolation errors. Numerical results show that the min-max NUFFT approach provides substantially lower approximation errors in tomographic reprojection and backprojection than conventional interpolation methods.

In this paper we develop a correlation method for the template matching problem in pattern recognition which includes translations, rotations, and dilations in a natural way. The correlation method is implemented using Fourier analysis on the “discrete motion group ” and fast Fourier transform methods. A brief introduction to Fourier methods on the discrete motion group is given and the efficiency of these methods is discussed. Results of the numerical implementation are given for particular examples. c ○ 1999 Academic Press Key Words: pattern analysis; object recognition and indexing.

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.

We propose a fast algorithm for far-field SAR imaging based on a new fast back-projection algorithm developed for tomography. We also modify the algorithm for the near-field scenario. The fast back-projection algorithm for SAR has computational complexity O(N²log N). Compared to traditional FFT-based methods, our new algorithm has potential advantages: the new algorithm does not need frequency-domain interpolation, which becomes complex for the wide-angle case; the new approach is applicable to the near-field scenario, taking into account wavefront curvature; and the back-projection algorithm can be easily adapted to parallel computing architectures. For some scenarios of interest, the computational cost of the new backprojection approach is similar to or less than that for FFT-based algorithms.