TY - JOUR

T1 - A framework for discrete integral transformations I - The pseudopolar Fourier transform

AU - Averbuch, A.

AU - Coifman, R. R.

AU - Donoho, D. L.

AU - Israeli, M.

AU - Shkolnisky, Y.

PY - 2007

Y1 - 2007

N2 - The Fourier transform of a continuous function, evaluated at frequencies expressed in polar coordinates, is an important conceptual tool for understanding physical continuum phenomena. An analogous tool, suitable for computations on discrete grids, could be very useful; however, no exact analogue exists in the discrete case. In this paper we present the notion of pseudopolar grid (pp grid) and the pseudopolar Fourier transform (ppFT), which evaluates the discrete Fourier transform at points of the pp grid. The pp grid is a type of concentric-squares grid in which the radial density of squares is twice as high as usual. The pp grid consists of equally spaced samples along rays, where different rays are equally spaced in slope rather than angle. We develop a fast algorithm for the ppFT, with the same complexity order as the Cartesian fast Fourier transform; the algorithm is stable, invertible, requires only one-dimensional operations, and uses no approximate interpolations. We prove that the ppFT is invertible and develop two algorithms for its inversion: iterative and direct, both with complexity O(n2 log n), where n × n is the size of the reconstructed image. The iterative algorithm applies conjugate gradients to the Gram operator of the ppFT. Since the transform is ill-conditioned, we introduce a preconditioner, which significantly accelerates the convergence. The direct inversion algorithm utilizes the special frequency domain structure of the transform in two steps. First, it resamples the pp grid to a Cartesian frequency grid and then recovers the image from the Cartesian frequency grid.

AB - The Fourier transform of a continuous function, evaluated at frequencies expressed in polar coordinates, is an important conceptual tool for understanding physical continuum phenomena. An analogous tool, suitable for computations on discrete grids, could be very useful; however, no exact analogue exists in the discrete case. In this paper we present the notion of pseudopolar grid (pp grid) and the pseudopolar Fourier transform (ppFT), which evaluates the discrete Fourier transform at points of the pp grid. The pp grid is a type of concentric-squares grid in which the radial density of squares is twice as high as usual. The pp grid consists of equally spaced samples along rays, where different rays are equally spaced in slope rather than angle. We develop a fast algorithm for the ppFT, with the same complexity order as the Cartesian fast Fourier transform; the algorithm is stable, invertible, requires only one-dimensional operations, and uses no approximate interpolations. We prove that the ppFT is invertible and develop two algorithms for its inversion: iterative and direct, both with complexity O(n2 log n), where n × n is the size of the reconstructed image. The iterative algorithm applies conjugate gradients to the Gram operator of the ppFT. Since the transform is ill-conditioned, we introduce a preconditioner, which significantly accelerates the convergence. The direct inversion algorithm utilizes the special frequency domain structure of the transform in two steps. First, it resamples the pp grid to a Cartesian frequency grid and then recovers the image from the Cartesian frequency grid.

KW - Concentric-squares grid

KW - Fractional Fourier transform

KW - Linogram

KW - Polar Fourier transform

KW - Pseudopolar Fourier transform

KW - Unequally spaced FFT

UR - http://www.scopus.com/inward/record.url?scp=44949095124&partnerID=8YFLogxK

U2 - 10.1137/060650283

DO - 10.1137/060650283

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AN - SCOPUS:44949095124

SN - 1064-8275

VL - 30

SP - 764

EP - 784

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

IS - 2

ER -