TY - JOUR

T1 - Direct inversion of the three-dimensional pseudo-polar fourier transform

AU - Averbuch, Amir

AU - Shabat, Gil

AU - Shkolnisky, Yoel

N1 - Publisher Copyright:
© 2016 Society for Industrial and Applied Mathematics.

PY - 2016

Y1 - 2016

N2 - The pseudo-polar Fourier transform is a specialized nonequally spaced Fourier trans- form, which evaluates the Fourier transform on a near-polar grid known as the pseudo-polar grid. The advantage of the pseudo-polar grid over other nonuniform sampling geometries is that the trans- formation, which samples the Fourier transform on the pseudo-polar grid, can be inverted using a fast and stable algorithm. For other sampling geometries, even if the nonequally spaced Fourier transform can be inverted, the only known algorithms are iterative. The convergence speed of these algorithms and their accuracy are dificult to control, as they depend both on the sampling ge- ometry and on the unknown reconstructed object. In this paper, a direct inversion algorithm for the three-dimensional pseudo-polar Fourier transform is presented. The algorithm is based only on one-dimensional resampling operations and is shown to be significantly faster than existing iterative inversion algorithms.

AB - The pseudo-polar Fourier transform is a specialized nonequally spaced Fourier trans- form, which evaluates the Fourier transform on a near-polar grid known as the pseudo-polar grid. The advantage of the pseudo-polar grid over other nonuniform sampling geometries is that the trans- formation, which samples the Fourier transform on the pseudo-polar grid, can be inverted using a fast and stable algorithm. For other sampling geometries, even if the nonequally spaced Fourier transform can be inverted, the only known algorithms are iterative. The convergence speed of these algorithms and their accuracy are dificult to control, as they depend both on the sampling ge- ometry and on the unknown reconstructed object. In this paper, a direct inversion algorithm for the three-dimensional pseudo-polar Fourier transform is presented. The algorithm is based only on one-dimensional resampling operations and is shown to be significantly faster than existing iterative inversion algorithms.

KW - 3D pseudo-polar Fourier transform

KW - Polar Fourier transform

KW - Radon transform

KW - Toeplitz matrices

KW - Unequally spaced FFT

UR - http://www.scopus.com/inward/record.url?scp=84964859061&partnerID=8YFLogxK

U2 - 10.1137/15M1031916

DO - 10.1137/15M1031916

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AN - SCOPUS:84964859061

SN - 1064-8275

VL - 38

SP - A1100-A1120

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

IS - 2

ER -