A new theoretical derivation of NFFT and its implementation on GPU

Abstract An efficient calculation of NFFT (nonequispaced fast Fourier transforms) is always a challenging task in a variety of application areas, from medical imaging to radio astronomy to chemical simulation. In this article, a new theoretical derivation is proposed for NFFT based on gridding algorithm and new strategies are proposed for the implementation of both forward NFFT and its inverse on both CPU and GPU. The GPU-based version, namely CUNFFT, adopts CUDA (Compute Unified Device Architecture) technology, which supports a fine-grained parallel computing scheme. The approximation errors introduced in the algorithm are discussed with respect to different window functions. Finally, benchmark calculations are executed to illustrate the accuracy and performance of NFFT and CUNFFT. The results show that CUNFFT is not only with high accuracy, but also substantially faster than conventional NFFT on CPU.

[1]  Kurt Jetter,et al.  Algorithms for cardinal interpolation using box splines and radial basis functions , 1991 .

[2]  T. Strohmer,et al.  Efficient numerical methods in non-uniform sampling theory , 1995 .

[3]  Leslie Greengard,et al.  Accelerating the Nonuniform Fast Fourier Transform , 2004, SIAM Rev..

[4]  Jeffrey A. Fessler,et al.  Nonuniform fast Fourier transforms using min-max interpolation , 2003, IEEE Trans. Signal Process..

[5]  J. Benedetto,et al.  Modern Sampling Theory: Mathematics and Applications , 2012 .

[6]  Gabriele Steidl,et al.  Fast Summation at Nonequispaced Knots by NFFT , 2003, SIAM J. Sci. Comput..

[7]  Karsten Fourmont.  Non-Equispaced Fast Fourier Transforms with Applications to Tomography , 2003 .

[8]  Antony Ware,et al.  Fast Approximate Fourier Transforms for Irregularly Spaced Data , 1998, SIAM Rev..

[9]  W. Press,et al.  Fast algorithm for spectral analysis of unevenly sampled data , 1989 .

[10]  Jaan Pelt.  Fast Computation of Trigonometric Sums with Applications to the Frequency Analysis of Astronomical Data , 1997 .

[11]  Jeffrey A. Fessler,et al.  Iterative tomographic image reconstruction using Fourier-based forward and back-projectors , 2004, IEEE Transactions on Medical Imaging.

[12]  Qing Huo Liu,et al.  The Regular Fourier Matrices and Nonuniform Fast Fourier Transforms , 1999, SIAM J. Sci. Comput..

[13]  Ren-Cang Li.  Relative perturbation theory. III. More bounds on eigenvalue variation , 1997 .

[14]  Robert J. Ferguson,et al.  NFFT: Algorithm for irregular sampling , 2010 .

[15]  Laurent Demanet,et al.  Fast Discrete Curvelet Transforms , 2006, Multiscale Model. Simul..

[16]  Gabriele Steidl.  A note on fast Fourier transforms for nonequispaced grids , 1998, Adv. Comput. Math..

[17]  Stefan Kunis,et al.  Stability Results for Scattered Data Interpolation by Trigonometric Polynomials , 2007, SIAM J. Sci. Comput..

[18]  Vladimir Rokhlin,et al.  Fast Fourier Transforms for Nonequispaced Data , 1993, SIAM J. Sci. Comput..

[19]  Dwight G. Nishimura,et al.  Rapid gridding reconstruction with a minimal oversampling ratio , 2005, IEEE Transactions on Medical Imaging.

[20]  Manfred Tasche,et al.  Roundoff Error Analysis for Fast Trigonometric Transforms , 2000 .

[21]  Jianwei Ma,et al.  Combined Complex Ridgelet Shrinkage and Total Variation Minimization , 2006, SIAM J. Sci. Comput..

[22]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[23]  G. Beylkin.  On the Fast Fourier Transform of Functions with Singularities , 1995 .

[24]  A. Macovski,et al.  Selection of a convolution function for Fourier inversion using gridding [computerised tomography application]. , 1991, IEEE transactions on medical imaging.

[25]  A. Duijndam,et al.  Nonuniform fast Fourier transform , 1999 .

[26]  Ami Harten,et al.  Fast multiresolution algorithms for matrix-vector multiplication , 1994 .

[27]  Chris Anderson,et al.  Rapid Computation of the Discrete Fourier Transform , 1996, SIAM J. Sci. Comput..

[28]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[29]  V. Rokhlin,et al.  Fast Fourier Transforms for Nonequispaced Data, II , 1995 .

[30]  Stefan Kunis,et al.  Fast spherical Fourier algorithms , 2003 .

[31]  G. Steidl,et al.  Approximate factorizations of Fourier matrices with nonequispaced knots , 2003 .