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.
Highlights
The Fourier transform is a powerful analytical tool and has proved to be invaluable in many disciplines such as physics, mathematics and engineering
If a signal changes sharply in the angular direction such that large values of N2 are needed, a large value of N1 is needed to compensate for the effect of increasing N2 on the grid coverage
For a space-limited function, if there is a lot of energy at the origin in the space domain, a larger value of N1 will be required to ensure that the sampling grid gets as close to the origin as possible in the space domain
Summary
The Fourier transform is a powerful analytical tool and has proved to be invaluable in many disciplines such as physics, mathematics and engineering. While the 2D DFT in polar coordinates was demonstrated to have properties and rules as a standalone transform independent of its relationship to any continuous transform, an obvious application of the proposed discrete transform is to approximate its continuous counterpart. The goal of this second part of this two-part paper series is to propose computationally efficient approaches to the computation of the previously proposed 2D DFT in polar coordinates and to validate its effectiveness to approximate the continuous 2D Fourier transform in polar coordinates. The motivation of this definition and the transform rules (multiplication, convolution, shift etc) are given in the first part of this two-part paper. Sample Matlab code is included in the appendix of the paper
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