Two-Dimensional Fourier Transforms in Polar Coordinates

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.

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“…16and 17were used to implement the forward and inverse transform. The continuous 2D-FT can be calculated from Baddour (2011) Fðr; cÞ ¼…”

Section: Four-term Sinusoid and Sinc Functionmentioning

confidence: 99%

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

Yao

Baddour

2020

PeerJ Computer Science

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“…In this section, it is assumed that the function is a space limited function, defined in [0, ] rR  . The sampling points are defined as equation (12) in the space domain and (13) in the frequency domain. In the following, a relationship between 2 N , 1 N and the area of the gap in both domains is discussed.…”

Section: Space-limited Functionmentioning

confidence: 99%

“…Moreover, the function can be considered as either an effectively space-limited function or an effectively band-limited function. For the purposes of testing it, it shall be considered as a spacelimited function and equations (12) and (13) will be used to proceed with the forward and inverse transform in sequence.…”

Section: Figure 5 the Original Gaussian Function And Its 2d-fourier Tmentioning

confidence: 99%

“…sinc function in the radial direction and a four-term sinusoid in the angular direction. The continuous 2D-FT can be calculated from[12] theorem for the angular direction, the highest angular frequency in equation (58) results in being at least 31 required to reconstruct the signal. Therefore, at least 31 terms are required to calculate the continuous 2D-FT, which can be written as Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted:…”

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

Yao

Baddour

2019

Preprint

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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 twopaper 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.

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“…It easily expands to multiple dimensions, with all the same rules of the one-dimensional (1D) case carrying into the multiple dimensions. Recent work has developed the complete toolkit for working with the continuous multidimensional Fourier transform in two-dimensional (2D) polar and three-dimensional (3D) spherical polar coordinates [2][3][4]. However, to date no discrete version of the 2D Fourier transform exists in polar coordinates.…”

Section: Introductionmentioning

confidence: 99%

Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules

Baddour

2019

Mathematics

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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 this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two-part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT.

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Two-Dimensional Fourier Transforms in Polar Coordinates

Cited by 36 publications

References 7 publications

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

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

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

Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules

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