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

2020 ◽  
Vol 6 ◽  
pp. e257
Author(s):  
Xueyang Yao ◽  
Natalie Baddour
Keyword(s):  
Fourier Transform ◽  
Numerical Computation ◽  
Discrete Fourier Transform ◽  
Inverse Fourier Transform ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Computational Aspects ◽  
Discrete Counterpart ◽  
Standard Set

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.

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

2019 ◽  
Author(s):  
Xueyang Yao ◽  
Natalie Baddour
Keyword(s):  
Fourier Transform ◽  
Numerical Computation ◽  
Inverse Fourier Transform ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Computational Aspects ◽  
Discrete Counterpart ◽  
Efficient Code ◽  
Standard Set

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.

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

2019 ◽  
Author(s):  
Natalie Baddour
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Discretization Schemes ◽  
Discrete Counterpart ◽  
Standard Set ◽  
Discrete Theory

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.

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

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 698 ◽  
Author(s):  
Baddour
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Discretization Schemes ◽  
Discrete Counterpart ◽  
Standard Set ◽  
Discrete Theory

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.

scholarly journals The Solvability of a Class of Convolution Equations Associated with 2D FRFT

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1928
Author(s):  
Zhen-Wei Li ◽  
Wen-Biao Gao ◽  
Bing-Zhao Li
Keyword(s):  
Fourier Transform ◽  
Convolution Operator ◽  
Fractional Fourier Transform ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Spatial Shift ◽  
Fractional Hankel Transform ◽  
The Relationship ◽  
Convolution Equations

In this paper, the solvability of a class of convolution equations is discussed by using two-dimensional (2D) fractional Fourier transform (FRFT) in polar coordinates. Firstly, we generalize the 2D FRFT to the polar coordinates setting. The relationship between 2D FRFT and fractional Hankel transform (FRHT) is derived. Secondly, the spatial shift and multiplication theorems for 2D FRFT are proposed by using this relationship. Thirdly, in order to analyze the solvability of the convolution equations, a novel convolution operator for 2D FRFT is proposed, and the corresponding convolution theorem is investigated. Finally, based on the proposed theorems, the solvability of the convolution equations is studied.

Numerical Computation of the Discrete 2D Fourier Transform in Polar Coordinates

2018 ◽  
Author(s):  
Xueyang Yao ◽  
Natalie Baddour
Keyword(s):  
Fourier Transform ◽  
Numerical Simulations ◽  
Discrete Fourier Transform ◽  
Good Accuracy ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Good Precision ◽  
Numerical Implementation ◽  
Inverse Transform ◽  
Accuracy And Precision

The discrete Fourier transform in Cartesian coordinates has proven to be invaluable in many disciplines. However, in application such as photoacoustics and tomography, a discrete 2D-Fourier transform in polar coordinates is needed. In this paper, a discrete 2D-Fourier transform in polar coordinates is presented. It is shown that numerical implementation is best achieved by interpreting the transform as a 1D-discrete Fourier transform (DFT), a 1D-discrete Hankel transform (DHT) and a 1D-discrete inverse transform (IDFT) in sequence. The transform is tested by numerical simulations with respect to accuracy and precision for computation of the continuous 2D transform at specific discrete points. It was found that both the forward and inverse transform showed good accuracy to approximate the continuous Fourier transform. Moreover, good precision results were obtained, which indicate that the proposed transform itself does not add much error.

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