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New Sampling Expansion Related to Derivatives in Quaternion Fourier Transform Domain

Author

Listed:
  • Siddiqui Saima

    (School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
    Current address: Beijing Key Laboratory on MCAACI, Beijing 100081, China.
    These authors contributed equally to this work.)

  • Bingzhao Li

    (School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
    Current address: Beijing Key Laboratory on MCAACI, Beijing 100081, China.
    These authors contributed equally to this work.)

  • Samad Muhammad Adnan

    (School of Automation, Beijing Institute of Technology, Beijing 100081, China
    These authors contributed equally to this work.)

Abstract

The theory of quaternions has gained a 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 Fourier transform. We have designed quaternion Fourier filters to reconstruct the signal, using the signal and its derivative. Since derivatives contain information about the edges and curves appearing in images, therefore, such a sampling formula is of substantial importance for image processing, particularly in image super-resolution procedures. Moreover, the presented sampling expansion can be applied in the field of image enhancement, color image processing, image restoration and compression and filtering, etc. Finally, an illustrative example is presented to demonstrate the efficacy of the proposed technique with vivid simulations in MATLAB.

Suggested Citation

  • Siddiqui Saima & Bingzhao Li & Samad Muhammad Adnan, 2022. "New Sampling Expansion Related to Derivatives in Quaternion Fourier Transform Domain," Mathematics, MDPI, vol. 10(8), pages 1-12, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:8:p:1217-:d:789268
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    References listed on IDEAS

    as
    1. De Bie, H. & De Schepper, N. & Ell, T.A. & Rubrecht, K. & Sangwine, S.J., 2015. "Connecting spatial and frequency domains for the quaternion Fourier transform," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 581-593.
    2. Zhen-Wei Li & Wen-Biao Gao & Bing-Zhao Li, 2020. "The Solvability of a Class of Convolution Equations Associated with 2D FRFT," Mathematics, MDPI, vol. 8(11), pages 1-12, November.
    3. Mawardi Bahri & Ryuichi Ashino & Rémi Vaillancourt, 2013. "Convolution Theorems for Quaternion Fourier Transform: Properties and Applications," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-10, November.
    4. Bing-Zhao Li & Tian-Zhou Xu, 2012. "Sampling in the Linear Canonical Transform Domain," Mathematical Problems in Engineering, Hindawi, vol. 2012, pages 1-13, July.
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    Cited by:

    1. Johnny Rodríguez-Maldonado & Cornelio Posadas-Castillo & Ernesto Zambrano-Serrano, 2022. "Alternative Method to Estimate the Fourier Expansions and Its Rate of Change," Mathematics, MDPI, vol. 10(20), pages 1-12, October.

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