Quaternion Fourier and linear canonical inversion theorems

Hu, Xiaoxiao; Kou, Kit Ian

doi:10.1002/mma.4148

Cited by 22 publications

(13 citation statements)

References 38 publications

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“…Higher dimensional extensions of complex analytic functions was considered by several authors . In this paper, we adapted the generalization of analytic functions combining with the QFT and QLCT from 1D to 4D case .…”

Section: Quaternion Analytic Signal and Quaternion Hardy Spacementioning

confidence: 99%

“…Note that it is possible to rewrite the quaternionic function in polar coordinate as in the complex case.…”

Section: Quaternion Analytic Signal and Quaternion Hardy Spacementioning

confidence: 99%

“…Higher dimensional extensions of complex analytic functions was considered by several authors. 15,20,42 In this paper, we adapted the generalization of analytic functions combining with the QFT and QLCT from 1D to 4D case. 15 Let us now review the definitions of the quaternion partial and total Hilbert transforms associated with QFT and QLCT, 15 respectively.…”

Section: Quaternion Analytic Signal and Quaternion Hardy Spacementioning

confidence: 99%

“…where C + ij ∶= {(z 1 , z 2 )|z 1 = x 1 + iy 1 , z 2 = x 2 + jy 2 , y 1 > 0, y 2 > 0.}. Note that it is possible to rewrite the quaternionic function in polar coordinate 20,25,42 as in the complex case.…”

Section: Definition 35 (Quaternion Analytic Signal)mentioning

confidence: 99%

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Phase‐based edge detection algorithms

Kou

2017

Math Methods in App Sciences

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Quaternion analytic signal is regarded as a generalization of analytic signal from 1D to 4D space. It is defined by an original signal with its quaternion partial and total Hilbert transforms. The quaternion analytic signal provides the signal features representation, such as the local amplitude and local phase angle, the latter includes the structural information of the original signal. The aim of the present study is twofold. Firstly, it attempts to analyze the Plemelj‐Sokhotzkis formula associated with quaternion Fourier transform and quaternion linear canonical transform. With these formulae, we show that the quaternion analytic signals are the boundary values of quaternion Hardy functions in the upper half space of 2 complex variables space. Secondly, the quaternion analytic signal can be extended to the quaternion Hardy function in the upper half space of 2 complex variables space. Two novel types of phase‐based edge detectors are proposed, namely, quaternion differential phase angle and quaternion differential phase congruency methods. In terms of peak signal‐to‐noise ratio and structural similarity index measure, comparisons with competing methods on real‐world images consistently show the superiority of the proposed methods.

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Section: Quaternion Analytic Signal and Quaternion Hardy Spacementioning

confidence: 99%

“…Note that it is possible to rewrite the quaternionic function in polar coordinate as in the complex case.…”

Section: Quaternion Analytic Signal and Quaternion Hardy Spacementioning

confidence: 99%

Section: Quaternion Analytic Signal and Quaternion Hardy Spacementioning

confidence: 99%

Section: Definition 35 (Quaternion Analytic Signal)mentioning

confidence: 99%

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Phase‐based edge detection algorithms

Kou

2017

Math Methods in App Sciences

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“…The right-sided quaternion linear canonical transform is obtained by substituting the Fourier kernel with the right-sided QFT kernel in the LCT definition, and so on. Recent works related to some important properties of the QLCT such as Parseval's theorem, reconstruction formula, and component-wise uncertainty principles were also published [18,[23][24][25]. It was found that the properties of the QLCT are extensions of the corresponding version of the QFT with some modifications.…”

Section: Introductionmentioning

confidence: 99%

A Version of Uncertainty Principle for Quaternion Linear Canonical Transform

Bahri

Resnawati

Musdalifah

2018

Abstract and Applied Analysis

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In recent years, the two-dimensional (2D) quaternion Fourier and quaternion linear canonical transforms have been the focus of many research papers. In the present paper, based on the relationship between the quaternion Fourier transform (QFT) and the quaternion linear canonical transform (QLCT), we derive a version of the uncertainty principle associated with the QLCT. We also discuss the generalization of the Hausdorff-Young inequality in the QLCT domain.

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Uncertainty principles associated with quaternion linear canonical transform and their estimates

Kundu

Prasad

2022

Math Methods in App Sciences

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In this paper, the relation between linear canonical transform (LCT) and quaternion linear canonical transform (QLCT) by using orthogonal plan split (OPS) method is studied. Further, several inequalities like sharp Young–Hausdorff inequality and Pitt's inequality are proved. Moreover, entropy uncertainty principle, sharp local uncertainty principle, and Heisenberg's uncertainty principle for the two‐sided QLCT are also discussed.

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Quaternion Fourier and linear canonical inversion theorems

Cited by 22 publications

References 38 publications

Phase‐based edge detection algorithms

Phase‐based edge detection algorithms

A Version of Uncertainty Principle for Quaternion Linear Canonical Transform

Uncertainty principles associated with quaternion linear canonical transform and their estimates

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