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SIAM J. SCI. COMPUT. c 2008 Society for Industrial and Applied Mathematics
Vol. 30, No. 2, pp. 764784
A FRAMEWORK FOR DISCRETE INTEGRAL
TRANSFORMATIONS ITHE PSEUDOPOLAR
FOURIER TRANSFORM
A. AVERBUCH, R. R. COIFMAN, D. L. DONOHO§, M. ISRAELI¶, AND
Y. SHKOLNISKY
In memory of Moshe Israeli 19402007
Abstract. 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 concentricsquares 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 onedimensional operations, and uses no approximate interpolations.
We prove that the ppFT is invertible and develop two algorithms for its inversion: iterative and direct,
