Peer Review #3 of "Discrete two dimensional Fourier transform in polar coordinates part II: numerical computation and approximation of the continuous transform (v0.2)"

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

10.20944/preprints201907.0189.v1 ◽

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.

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

Mathematics ◽

10.3390/math8111928 ◽

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.

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

10.20944/preprints201907.0151.v1 ◽

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.

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

Mathematics ◽

10.3390/math7080698 ◽

2019 ◽

Vol 7 (8) ◽

pp. 698 ◽

Cited By ~ 1

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.

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Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100133916 ◽

1990 ◽

Vol 48 (2) ◽

pp. 64-65

Author(s):

L. Reimer ◽

R. Oelgeklaus

Keyword(s):

Fourier Transform ◽

Energy Loss ◽

Electron Energy ◽

Rotational Symmetry ◽

Mean Free Path ◽

Single Scattering ◽

Specimen Thickness ◽

Two Dimensional ◽

Image Position ◽

Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

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