Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle

Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle

A definition of the two-dimensional quaternion linear canonical transform (QLCT) is proposed. The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the definition of the classical linear canonical transform (LCT). Several usef...

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Main Authors: Mawardi Bahri, Ryuichi Ashino
Format: Article
Language:English
Published: Hindawi Limited 2019-01-01
Series:Journal of Mathematics
Online Access:http://dx.doi.org/10.1155/2019/1062979
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spelling doaj-021b67feb3444afebcbcdbf59061f1602020-11-25T01:36:37ZengHindawi LimitedJournal of Mathematics2314-46292314-47852019-01-01201910.1155/2019/10629791062979Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty PrincipleMawardi Bahri0Ryuichi Ashino1Department of Mathematics, Universitas Hasanuddin, Tamalanrea, Makassar, IndonesiaMathematics and Informatics, Osaka Kyoiku University, Osaka 582-8582, JapanA definition of the two-dimensional quaternion linear canonical transform (QLCT) is proposed. The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the definition of the classical linear canonical transform (LCT). Several useful properties of the QLCT are obtained from the properties of the QLCT kernel. Based on the convolutions and correlations of the LCT and QFT, convolution and correlation theorems associated with the QLCT are studied. An uncertainty principle for the QLCT is established. It is shown that the localization of a quaternion-valued function and the localization of the QLCT are inversely proportional and that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.http://dx.doi.org/10.1155/2019/1062979
collection DOAJ
language English
format Article
sources DOAJ
author Mawardi Bahri
Ryuichi Ashino
spellingShingle Mawardi Bahri
Ryuichi Ashino
Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle
Journal of Mathematics
author_facet Mawardi Bahri
Ryuichi Ashino
author_sort Mawardi Bahri
title Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle
title_short Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle
title_full Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle
title_fullStr Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle
title_full_unstemmed Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle
title_sort two-dimensional quaternion linear canonical transform: properties, convolution, correlation, and uncertainty principle
publisher Hindawi Limited
series Journal of Mathematics
issn 2314-4629
2314-4785
publishDate 2019-01-01
description A definition of the two-dimensional quaternion linear canonical transform (QLCT) is proposed. The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the definition of the classical linear canonical transform (LCT). Several useful properties of the QLCT are obtained from the properties of the QLCT kernel. Based on the convolutions and correlations of the LCT and QFT, convolution and correlation theorems associated with the QLCT are studied. An uncertainty principle for the QLCT is established. It is shown that the localization of a quaternion-valued function and the localization of the QLCT are inversely proportional and that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.
url http://dx.doi.org/10.1155/2019/1062979
work_keys_str_mv AT mawardibahri twodimensionalquaternionlinearcanonicaltransformpropertiesconvolutioncorrelationanduncertaintyprinciple
AT ryuichiashino twodimensionalquaternionlinearcanonicaltransformpropertiesconvolutioncorrelationanduncertaintyprinciple
_version_ 1725062000943300608

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