 
Summary: THE PSEUDOPOLAR FFT AND ITS APPLICATIONS
AMIR AVERBUCH \Lambda , RONALD COIFMAN y , DAVID DONOHO z , MOSHE ISRAELI x , AND
JOHAN WALD ' EN 
Abstract. We present a two and threedimensional Fast Fourier Transform, where the result in
the 2D dimensional case is contained in coordinates that are ``close to polar,'' while in the 3D case
they are ``close to spherical.'' We call these transforms the pseudopolar FFT and pseudospherical
FFT, repsectively. The move from Cartesian coordinatesto polar coordinates is algebraically accurate
and no interpolation is needed. We give examples of applications to computer tomography.
Key words. Fast Fourier Transform, Radon transform, computer tomography, Fractional
Fourier Transform
AMS subject classifications. 44A12, 65R10, 65T20, 92C55
1. Introduction. There are several applications where we would like to know
the Fourier transform of a function, in polar coordinates. One important application
is if we want to do computer tomography. The Fourier slice theorem establishes
a relationship between the Fourier transform in polar coordinates and the Radon
transform of the function. Another example is when we analyze wave front sets,
where we are interested in ``cutting out cones'' of the Fourier transform of a function.
Ideally, we would like to have a version of the Fast Fourier Transform (FFT) that
produces a result in polar coordinates. As points in polar coordinates are unequally
spaced (compared with our original Cartesian data), one approach would be to use
