A new kind of convolution, correlation and product theorems related to quaternion linear canonical transform

Li, Zhenwei; Gao, Wen‐Biao; Li, Bing‐Zhao

doi:10.1007/s11760-020-01728-x

Cited by 11 publications

(6 citation statements)

References 26 publications

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“…By repeating this process n − 1 additional times, we obtain (20). Using a similar argument as Equation (20), we can obtain the proof for Equation (21). By Riesz's interpolation theorem, we obtain the Hausdorff-Young inequality (see [25]), that is…”

Section: Definitionmentioning

confidence: 91%

“…This theorem is inspired by the work of Hitzer [19] who established the convolution theorem associated with general two-sided quaternion Fourier transform. Paper [20,21] proposed similar work for convolution theorem, but different type of the quaternion linear canonical transform which established the convolution theorem for the two-sided quaternion linear canonical transform. We finally derive correlation theorem of continuous quaternion signals associated with the general two-sided quaternion linear canonical transform.…”

Section: Introductionmentioning

confidence: 99%

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A Variation on Inequality for Quaternion Fourier Transform, Modified Convolution and Correlation Theorems for General Quaternion Linear Canonical Transform

Bahri

Karim

2022

Symmetry

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The quaternion linear canonical transform is an important tool in applied mathematics and it is closely related to the quaternion Fourier transform. In this work, using a symmetric form of the two-sided quaternion Fourier transform (QFT), we first derive a variation on the Heisenberg-type uncertainty principle related to this transformation. We then consider the general two-sided quaternion linear canonical transform. It may be considered as an extension of the two-sided quaternion linear canonical transform. Based on an orthogonal plane split, we develop the convolution theorem that associated with the general two-sided quaternion linear canonical transform and then derive its correlation theorem. We finally discuss how to apply general two-sided quaternion linear canonical transform to study the generalized swept-frequency filters.

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Section: Definitionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

A Variation on Inequality for Quaternion Fourier Transform, Modified Convolution and Correlation Theorems for General Quaternion Linear Canonical Transform

Bahri

Karim

2022

Symmetry

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“…The linear canonical transform (LCT) is a three-free-parameter class of linear integral transforms, which encompasses a number of well-known unitary transformations as well as signal processing and optics-related mathematical operations, for example, the Fourier transform, the fractional Fourier transform, the Fresnel transform and the scaling operations [3][4][5][6][7][8][9][10]. Due to the extra degrees of freedom and simple geometrical manifestation, the LCT [30] is more flexible than other transforms and is as such suitable as well as powerful tool for investigating deep problems in science and engineering. The application areas for LCT have been growing over the last two decades at a very rapid rate and it has been applied in a number of fields including optics, quantum physics, time-frequency analysis, filter design, phase reconstruction, pattern recognition, radar analysis, holographic three-dimensional television and many more [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

Generalized sampling expansion for the quaternion linear canonical transform

Siddiqui,

Bing-Zhao,

Samad

2024

SIViP

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The theory of quaternions has gained firm ground in recent times and is being widely explored, with the field of signal and image processing being no exception. However, many important aspects of quaternionic signals are yet to be explored, particularly the formulation of Generalized Sampling Expansions (GSE). In the present article, our aim is to formulate the GSE in the realm of a one-dimensional quaternion linear canonical transform (QLCT). To facilitate the intent, we construct a set of quaternion-valued filter functions which are used to construct a system of equations determining the synthesis functions for the process of reconstruction. Besides, as a special case, another sampling formula involving the derivatives of the quaternionic signal is also obtained in the sequel. Since derivatives contain information about the edges and curves appearing in images, therefore, by invoking the higher degrees of freedom pertaining to QLCT, the obtained sampling formula can be used to develop new image scaling algorithms with state-of-the-art flexibility. As an endorsement of the obtained results, an example with simulations demonstrating the signal reconstruction is presented at the end.

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“…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. Hu et al [10] proposed various types of convolution formulas associated with QLCT and discussed the applications in the integral equations and the design of multiplicative filters.…”

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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A new kind of convolution, correlation and product theorems related to quaternion linear canonical transform

Cited by 11 publications

References 26 publications

A Variation on Inequality for Quaternion Fourier Transform, Modified Convolution and Correlation Theorems for General Quaternion Linear Canonical Transform

A Variation on Inequality for Quaternion Fourier Transform, Modified Convolution and Correlation Theorems for General Quaternion Linear Canonical Transform

Generalized sampling expansion for the quaternion linear canonical transform

Theories and applications associated with biquaternion linear canonical transform

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