Fast and Accurate Polar Fourier Transform

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. 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(N 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 method 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. For those interested primarily in Polar FFT’s, the Pseudo-Polar FFT plays the role of a halfway point – a nearly-Polar system from which conversion to Polar Coordinates uses processes relying purely on 1-D FFT’s and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesian-based unequally-sampled FFT method to ours, both algorithms using a small-support interpolation and no pre-compensating, and show marked advantage to the use of the Pseudo-Polar initial grid.