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

Bahri, Mawardi; Ashino, Ryuichi

doi:10.1155/2019/1062979

Cited by 10 publications

(8 citation statements)

References 36 publications

(41 reference statements)

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“…We will review several integral transformations pertaining to LCT [1,2] and the QLCT [31,32]. Definition 1.…”

Section: Linear Canonical Integral Transformmentioning

confidence: 99%

“…In recent years, with the continuous progress of mathematical theory, researchers have begun to extend the concept of integral transforms to the field of quaternion algebra. This extension has led to new theoretical frameworks, such as the quaternion Fourier transform (QFT) [27,28], the quaternion fractional Fourier transform (QFRFT) [29,30], the quaternion linear canonical transform (QLCT) [31][32][33][34], and the quaternion offset linear canonical transform (QOLCT) [7,35,36]. These theoretical frameworks provide new methods and tools for processing and analyzing quaternion signals.…”

Section: Introductionmentioning

confidence: 99%

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Weighted Convolution for Quaternion Linear Canonical Cosine Transform and Its Application

Wang,

Feng

2024

Axioms

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Convolution plays a pivotal role in the domains of signal processing and optics. This paper primarily focuses on studying the weighted convolution for quaternion linear canonical cosine transform (QLCcT) and its application in multiplicative filter analysis. Firstly, we propose QLCcT by combining quaternion algebra with linear canonical cosine transform (LCcT), which extends LCcT to Hamiltonian quaternion algebra. Secondly, we introduce weighted convolution and correlation operations for QLCcT, accompanied by their corresponding theorems. We also explore the properties of QLCcT. Thirdly, we utilize these proposed convolution structures to analyze multiplicative filter models that offer lower computational complexity compared to existing methods based on quaternion linear canonical transform (QLCT). Additionally, we discuss the rationale behind studying such transforms using quaternion functions as an illustrative example.

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“…We will review several integral transformations pertaining to LCT [1,2] and the QLCT [31,32]. Definition 1.…”

Section: Linear Canonical Integral Transformmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Weighted Convolution for Quaternion Linear Canonical Cosine Transform and Its Application

Wang,

Feng

2024

Axioms

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“…The Heisenberg uncertainty principles of the QLCT have been studied in [8,11,12]. Based on the Heisenberg uncertainty principles of the QLCT, we can obtain the Heisenberg uncertainty principle of the RBiQLCT.…”

Section: Heisenberg Uncertainty Principle For the Biqlctmentioning

confidence: 99%

“…Some useful properties of the QLCT such as linearity, reconstruction formula, continuity, boundedness, and positivity inversion formula are established in [6,7]. Mawardi et al [8] studied the convolution and correlation theorems and uncertainty principle of the one-sided QLCT. Li et al [9] studied the convolution, correlation, and product theorems for the QLCT.…”

Section: Introductionmentioning

confidence: 99%

Theories and applications associated with biquaternion linear canonical transform

Gao

2023

Math Methods in App Sciences

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The quaternion linear canonical transform (QLCT) has been widely used in color image processing. Biquaternion is a more generalized algebra of quaternion, which has attracted scholars' research interest in recent years. In this paper, a new transform is proposed called the biquaternion linear canonical transforms (BiQLCTs). Due to the noncommutativity of biquaternion algebra multiplication, there are three different types of the BiQLCTs: Left‐sided BiQLCT, right‐sided BiQLCT, and two‐side BiQLCT. The transforms are the extension of the complex linear canonical transforms. Then, the relationships between the three kinds of transforms are obtained. Next, based on the right‐side biquaternion linear canonical transform (RBiQLCT), some general properties of this transform are proved. Moreover, the convolution and correlation theorems of the RBiQLCT are studied. As an application, according to the convolution operator and convolution theorem, the biquaternion linear time‐invariant system is analyzed. Finally, the Heisenberg uncertainty principle for the RBiQLCT is established.

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“…The quaternion number system was first described by Hamilton as a generalization of complex numbers [13,14]. In recent years, researchers have extended integral transforms into the quaternion algebra domain, leading to the development of theoretical frameworks such as quaternion Fourier transform (QFT) [15][16][17][18], quaternion fractional Fourier transform (QFRFT) [19], quaternion windowed fractional Fourier transform (QWFRFT) [20][21][22][23][24], quaternion linear canonical transform (QLCT) [25,26], and quaternion offset linear canonical transform [27]. Several important properties of QLCT have been investigated, including linearity, time shift, modulation, reconstruction formula, boundedness , and uncertainty principles in [28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications

Yang,

Feng,

Wang

et al. 2024

Mathematics

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The quaternion windowed linear canonical transform is a tool for processing multidimensional data and enhancing the quality and efficiency of signal and image processing; however, it has disadvantages due to the noncommutativity of quaternion multiplication. In contrast, reduced biquaternions, as a special case of four-dimensional algebra, possess unique advantages in computation because they satisfy the multiplicative exchange rule. This paper proposes the reduced biquaternion windowed linear canonical transform (RBWLCT) by combining the reduced biquaternion signal and the windowed linear canonical transform that has computational efficiency thanks to the commutative property. Firstly, we introduce the concept of a RBWLCT, which can extract the time local features of an image and has the advantages of both time-frequency analysis and feature extraction; moreover, we also provide some fundamental properties. Secondly, we propose convolution and correlation operations for RBWLCT along with their corresponding generalized convolution, correlation, and product theorems. Thirdly, we present a fast algorithm for RBWLCT and analyze its computational complexity based on two dimensional Fourier transform (2D FTs). Finally, simulations and examples are provided to demonstrate that the proposed transform effectively captures the local RBWLCT-frequency components with enhanced degrees of freedom and exhibits significant concentrations.

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

Cited by 10 publications

References 36 publications

Weighted Convolution for Quaternion Linear Canonical Cosine Transform and Its Application

Weighted Convolution for Quaternion Linear Canonical Cosine Transform and Its Application

Theories and applications associated with biquaternion linear canonical transform

Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications

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