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Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications

Author

Listed:
  • Hehe Yang

    (School of Mathematics and Computer Science, Yanan University, Yanan 716000, China)

  • Qiang Feng

    (School of Mathematics and Computer Science, Yanan University, Yanan 716000, China)

  • Xiaoxia Wang

    (School of Mathematics and Computer Science, Yanan University, Yanan 716000, China)

  • Didar Urynbassarova

    (National Engineering Academy of the Republic of Kazakhstan, Almaty 050000, Kazakhstan)

  • Aajaz A. Teali

    (Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University, Jalandhar 144411, Punjab, India)

Abstract

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.

Suggested Citation

  • Hehe Yang & Qiang Feng & Xiaoxia Wang & Didar Urynbassarova & Aajaz A. Teali, 2024. "Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications," Mathematics, MDPI, vol. 12(5), pages 1-21, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:5:p:743-:d:1349760
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    References listed on IDEAS

    as
    1. Kit Ian Kou & Jian-Yu Ou & Joao Morais, 2013. "On Uncertainty Principle for Quaternionic Linear Canonical Transform," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-14, April.
    2. Lei Huang & Ke Zhang & Yi Chai & Shuiqing Xu, 2017. "Computation of the Short-Time Linear Canonical Transform with Dual Window," Mathematical Problems in Engineering, Hindawi, vol. 2017, pages 1-8, September.
    3. Mawardi Bahri & Ryuichi Ashino, 2016. "A Simplified Proof of Uncertainty Principle for Quaternion Linear Canonical Transform," Abstract and Applied Analysis, Hindawi, vol. 2016, pages 1-11, January.
    4. Didar Urynbassarova & Aajaz A. Teali, 2023. "Convolution, Correlation, and Uncertainty Principles for the Quaternion Offset Linear Canonical Transform," Mathematics, MDPI, vol. 11(9), pages 1-24, May.
    Full references (including those not matched with items on IDEAS)

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