Summary: Fast and Accurate Polar Fourier Transform
December 1st, 2004
In a wide range of applied problems of 2-D and 3-D imaging a continuous formulation of the problem
places great emphasis on obtaining and manipulating the Fourier transform in polar coordinates. However,
the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic.
It is widely believed that "there is no Polar Fast Fourier Transform (FFT)" and that for practical work,
continuum ideas are at best a source of inspiration rather than a practical source of algorithmic approaches.
In this article we develop a fast high accuracy Polar FFT. For a given two-dimensional signal of size
N × N, the proposed algorithm's complexity is O(N2
log N), just like in a Cartesian 2D-FFT. A special
feature of our approach is that it involves only 1-D equispaced FFT's and 1-D interpolations. A central tool
in our approach is the pseudo-polar FFT, an FFT where the evaluation frequencies lie in an oversampled set
of non-angularly equispaced points. We describe the concept of pseudo-polar domain, including fast forward
and inverse transforms, a quasi-Parseval relation, and provide empirical and theoretical analysis of the Gram