CLC number: O236
On-line Access: 2024-08-27
Received: 2023-10-17
Revision Accepted: 2024-05-08
Crosschecked: 2021-01-18
Cited: 0
Clicked: 6025
Citations: Bibtex RefMan EndNote GB/T7714
Xiaoxiao HU, Dong CHENG, Kit Ian KOU. Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms[J]. Frontiers of Information Technology & Electronic Engineering, 2022, 23(3): 463-478.
@article{title="Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms",
author="Xiaoxiao HU, Dong CHENG, Kit Ian KOU",
journal="Frontiers of Information Technology & Electronic Engineering",
volume="23",
number="3",
pages="463-478",
year="2022",
publisher="Zhejiang University Press & Springer",
doi="10.1631/FITEE.2000499"
}
%0 Journal Article
%T Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms
%A Xiaoxiao HU
%A Dong CHENG
%A Kit Ian KOU
%J Frontiers of Information Technology & Electronic Engineering
%V 23
%N 3
%P 463-478
%@ 2095-9184
%D 2022
%I Zhejiang University Press & Springer
%DOI 10.1631/FITEE.2000499
TY - JOUR
T1 - Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms
A1 - Xiaoxiao HU
A1 - Dong CHENG
A1 - Kit Ian KOU
J0 - Frontiers of Information Technology & Electronic Engineering
VL - 23
IS - 3
SP - 463
EP - 478
%@ 2095-9184
Y1 - 2022
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/FITEE.2000499
Abstract: The main purpose of this paper is to study different types of sampling formulas of quaternionic functions, which are bandlimited under various quaternion Fourier and linear canonical transforms. We show that the quaternionic bandlimited functions can be reconstructed from their samples as well as the samples of their derivatives and Hilbert transforms. In addition, the relationships among different types of sampling formulas under various transforms are discussed. First, if the quaternionic function is bandlimited to a rectangle that is symmetric about the origin, then the sampling formulas under various quaternion Fourier transforms are identical. If this rectangle is not symmetric about the origin, then the sampling formulas under various quaternion Fourier transforms are different from each other. Second, using the relationship between the two-sided quaternion Fourier transform and the linear canonical transform, we derive sampling formulas under various quaternion linear canonical transforms. Third, truncation errors of these sampling formulas are estimated. Finally, some simulations are provided to show how the sampling formulas can be used in applications.
[1]Alon G, Paran E, 2021. A quaternionic Nullstellensatz. J Pure Appl Algebr, 225(4):106572.
[2]Bahia B, Sacchi MD, 2020. Widely linear denoising of multicomponent seismic data. Geophys Prospect, 68(2):431-445.
[3]Bulow T, Sommer G, 2001. Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case. IEEE Trans Signal Process, 49(11):2844-2852.
[4]Chen LP, Kou KI, Liu MS, 2015. Pitt’s inequality and the uncertainty principle associated with the quaternion Fourier transform. J Math Anal Appl, 423(1):681-700.
[5]Cheng D, Kou KI, 2018. Generalized sampling expansions associated with quaternion Fourier transform. Math Methods Appl Sci, 41(11):4021-4032.
[6]Cheng D, Kou KI, 2019. FFT multichannel interpolation and application to image super-resolution. Signal Process, 162:21-34.
[7]Cheng D, Kou KI, 2020. Multichannel interpolation of nonuniform samples with application to image recovery. J Comput Appl Math, 367:112502.
[8]Ell TA, Le Bihan N, Sangwine SJ, 2014. Quaternion Fourier Transforms for Signal and Image Processing. John Wiley & Sons, Hoboken, USA.
[9]Hahn SL, Snopek KM, 2005. Wigner distributions and ambiguity functions of 2-D quaternionic and monogenic signals. IEEE Trans Signal Process, 53(8):3111-3128.
[10]Hitzer E, 2017. General two-sided quaternion Fourier transform, convolution and Mustard convolution. Adv Appl Clifford Algebr, 27(1):381-385.
[11]Hitzer EMS, 2007. Quaternion Fourier transform on quaternion fields and generalizations. Adv Appl Clifford Algebr, 17(3):497-517.
[12]Hu XX, Kou KI, 2017. Quaternion Fourier and linear canonical inversion theorems. Math Methods Appl Sci, 40(7):2421-2440.
[13]Hu XX, Kou KI, 2018. Phase-based edge detection algorithms. Math Methods Appl Sci, 41(11):4148-4169.
[14]Jagerman D, 1966. Bounds for truncation error of the sampling expansion. SIAM J Appl Math, 14(4):714-723.
[15]Jiang MD, Li Y, Liu W, 2016. Properties of a general quaternion-valued gradient operator and its applications to signal processing. Front Inform Technol Electron Eng, 17(2):83-95.
[16]Kou KI, Morais J, 2014. Asymptotic behaviour of the quaternion linear canonical transform and the Bochner-Minlos theorem. Appl Math Comput, 247:675-688.
[17]Kou KI, Qian T, 2005a. Shannon sampling in the Clifford analysis setting. Z Anal Anwend, 24(4):853-870.
[18]Kou KI, Qian T, 2005b. Shannon sampling and estimation of band-limited functions in the several complex variables setting. Acta Math Sci, 25(4):741-754.
[19]Kou KI, Ou JY, Morais J, 2013. Uncertainty principle for quaternionic linear canonical transform. Abstr Appl Anal, Article 725952.
[20]Kou KI, Liu MS, Morais JP, et al., 2017. Envelope detection using generalized analytic signal in 2D QLCT domains. Multidim Syst Signal Process, 28(4):1343-1366.
[21]Lian P, 2021. Quaternion and fractional Fourier transform in higher dimension. Appl Math Comput, 389:125585.
[22]Marvasti F, 2001. Nonuniform Sampling: Theory and Practice. Springer Science & Business Media, New York, USA.
[23]Pan WJ, 2000. Fourier Analysis and Its Applications. Peking University Press, China (in Chinease).
[24]Pei SC, Ding JJ, Chang JH, 2001. Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT. IEEE Trans Signal Process, 49(11):2783-2797.
[25]Splettstösser W, Stens RL, Wilmes G, 1981. On approximation by the interpolating series of G. Valiron. Funct Approx Comment Math, 11:39-56.
[26]Yao K, Thomas JB, 1966. On truncation error bounds for sampling representations of band-limited signals. IEEE Trans Aerosp Electron Syst, AES-2(6):640-647.
[27]Zayed AI, 1993. Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton, USA.
[28]Zou CM, Kou KI, Wang YL, 2016. Quaternion collaborative and sparse representation with application to color face recognition. IEEE Trans Image Process, 25(7):3287-3302.
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