Discrete two dimensional Fourier transform in polar coordinates part II: numerical computation and approximation of the continuous transform

Abstract

The theory of the continuous two-dimensional (2D) Fourier Transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In the first part of this two-paper series, we proposed and evaluated the theory of the 2D Discrete Fourier Transform (DFT) in polar coordinates. The theory of the actual manipulated quantities was shown, including the standard set of shift, modulation, multiplication, and convolution rules. In this second part of the series, we address the computational aspects of the 2D DFT in polar coordinates. Specifically, we demonstrate how the decomposition of the 2D DFT as a DFT, Discrete Hankel Transform and inverse DFT sequence can be exploited for coding. We also demonstrate how the proposed 2D DFT can be used to approximate the continuous forward and inverse Fourier transform in polar coordinates in the same manner that the 1D DFT can be used to approximate its continuous counterpart.

Spectral Solver for Cauchy Problems in Polar Coordinates Using Discrete Hankel Transforms

“…From Eqn. (18) and the convergence rate of Fourier-Bessel series for functions satisfying the homogeneous Dirichlet condition, a q,N N −5/2 , we obtain the general error of DHTs, E D (q = 0, N ) N −3/2 , and…”

“…(4) defined for the DHTs are not evenly spaced and vary with the angular mode q. Instead of using Baddour and Yao's unevenly sampled grid [17,18], which causes unwanted artifacts in the center of the domain, we choose an evenly sampled grid that is finer in the radial direction before performing the azimuthal FFT (hereafter referred to as the FFT grid), as visualized in Figure 1a. We then numerically interpolate each radial function f q on the corresponding DHT grid {r q,i } i=1,...,N to obtain {f q (r q,i )} i=1,...,N , as shown in Figure 1b.…”