Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Discrete Two Dimensional Fourier Transform in Polar Coordinates Part II: Numerical Computation and Approximation of the Continuous Transform

Version 1 : Received: 12 July 2019 / Approved: 16 July 2019 / Online: 16 July 2019 (08:00:00 CEST)

A peer-reviewed article of this Preprint also exists.

Yao X, Baddour N. 2020. Discrete two dimensional Fourier transform in polar coordinates part II: numerical computation and approximation of the continuous transform. PeerJ Computer Science 6:e257 https://doi.org/10.7717/peerj-cs.257 Yao X, Baddour N. 2020. Discrete two dimensional Fourier transform in polar coordinates part II: numerical computation and approximation of the continuous transform. PeerJ Computer Science 6:e257 https://doi.org/10.7717/peerj-cs.257

## 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 (DHT) and inverse DFT sequence can be exploited for efficient code. 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.

## Keywords

Fourier Theory; DFT in polar coordinates; polar coordinates; multidimensional DFT; discrete Hankel transform; discrete Fourier transform; orthogonality

## Subject

Computer Science and Mathematics, Discrete Mathematics and Combinatorics

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