A Framework for Discrete Integral Transformations I-The Pseudopolar Fourier Transform

Abstract. Computing the Fourier transform of a function in polar coordinates is an important building block in many scientific disciplines and numerical schemes. In this paper we present the pseudo-polar Fourier transform that samples the Fourier transform on the pseudo-polar grid, also known as the concentric squares grid. The pseudo-polar grid consists of equally spaced samples along rays, where different rays are equally spaced and not equally angled. The pseudo-polar Fourier transform is shown to be fast (the same complexity as the FFT), stable, invertible, requires only 1D operations, and uses no interpolations. We prove that the pseudo-polar Fourier transform is invertible and develop two algorithms for its inversion: iterative and direct, both with complexity O(n logn), where n × n is the size of the reconstructed image. The iterative algorithm is based on the application of the conjugate-gradient method to the Gram operator of the pseudo-polar Fourier transform. 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 pseudo-polar grid to a Cartesian frequency grid, and then, recovers the image from the Cartesian frequency grid.