基于四元数傅里叶变换和线性正则变换的二维四元数信号采样定理

胡晓晓; 程冬; 高洁欣

doi:10.1631/FITEE.2000499

Your Location：

Home >

Browse articles >

基于四元数傅里叶变换和线性正则变换的二维四元数信号采样定理

常规文章 | Updated：2022-03-28

- 基于四元数傅里叶变换和线性正则变换的二维四元数信号采样定理
  Enhanced Publication
- Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms
- 信息与电子工程前沿（英文） 2022年23卷第3期页码：463-478
- Affiliations：
  
  1.The First Affiliated Hospital of Wenzhou Medical University, Wenzhou Medical University, Wenzhou325000, China
  2.Research Center for Mathematics and Mathematics Education, Beijing Normal University (Zhuhai), Zhuhai519087, China
  3.Department of Mathematics, Faculty of Science and Technology, University of Macau, Macao, China
- Author bio：
  
  ‡Corresponding author
- Funds：
  
  Research Development Foundation of Wenzhou Medical University, China(QTJ18012);Wenzhou Science and Technology Bureau of China(G2020031);Guangdong Basic and Applied Basic Research Foundation of China(2019A1515111185);Science and Technology Development Fund, Macau Special Administrative Region, China(FDCT/085/2018/A2);University of Macau, China(MYRG2019-00039-FST)
- DOI：10.1631/FITEE.2000499
  中图分类号： O236
- 收稿：2020-09-25，
  
  录用：2021-01-18，
  
  纸质出版：2022-03-0
- Accepted：
Scan QR Code
胡晓晓, 程冬, 高洁欣. 基于四元数傅里叶变换和线性正则变换的二维四元数信号采样定理[J]. 信息与电子工程前沿（英文）, 2022,23(3):463-478.

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.
胡晓晓, 程冬, 高洁欣. 基于四元数傅里叶变换和线性正则变换的二维四元数信号采样定理[J]. 信息与电子工程前沿（英文）, 2022,23(3):463-478. DOI： 10.1631/FITEE.2000499.

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. DOI： 10.1631/FITEE.2000499.

摘要

本文主要研究在不同形式四元数傅里叶变换和线性正则变换下有限带宽四元数函数的采样定理。证明了有限带宽四元数函数可通过它们的直接采样或经过微分和希尔伯特变换后的采样重构。此外，讨论了不同形式变换下不同类型采样公式之间的关系。首先，如果四元数函数有限带宽区域是关于原点对称的矩形区域，则不同形式四元数傅里叶变换下四元数采样公式具有相同形式；否则，采样公式是不同的。其次，利用双边四元数傅里叶变换和线性正则变换的关系，得到不同形式四元数线性正则变换下有限带宽四元数函数采样定理。再次，分析了采样公式的截断误差。最后，通过仿真展示采样公式的应用。

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.

关键词

Keywords

references

Alon G , Paran E , 2021 . A quaternionic Nullstellensatz . J Pure Appl Algebr , 225 ( 4 ): 106572 . doi: 10.1016/j.jpaa.2020.106572 http://doi.org/10.1016/j.jpaa.2020.106572

Bahia B , Sacchi MD , 2020 . Widely linear denoising of multicomponent seismic data . Geophys Prospect , 68 ( 2 ): 431 - 445 . doi: 10.1111/1365-2478.12850 http://doi.org/10.1111/1365-2478.12850

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 . doi: 10.1109/78.960432 http://doi.org/10.1109/78.960432

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 . doi: 10.1016/j.jmaa.2014.10.003 http://doi.org/10.1016/j.jmaa.2014.10.003

Cheng D , Kou KI , 2018 . Generalized sampling expansions associated with quaternion Fourier transform . Math Methods Appl Sci , 41 ( 11 ): 4021 - 4032 . doi: 10.1002/mma.4423 http://doi.org/10.1002/mma.4423

Cheng D , Kou KI , 2019 . FFT multichannel interpolation and application to image super-resolution . Signal Process , 162 : 21 - 34 . doi: 10.1016/j.sigpro.2019.03.025 http://doi.org/10.1016/j.sigpro.2019.03.025

Cheng D , Kou KI , 2020 . Multichannel interpolation of nonuniform samples with application to image recovery . J Comput Appl Math , 367 : 112502 . doi: 10.1016/j.cam.2019.112502 http://doi.org/10.1016/j.cam.2019.112502

Ell TA , Le Bihan N , Sangwine SJ , 2014 . Quaternion Fourier Transforms for Signal and Image Processing . John Wiley & Sons , Hoboken, USA .

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 . doi: 10.1109/TSP.2005.851134 http://doi.org/10.1109/TSP.2005.851134

Hitzer E , 2017 . General two-sided quaternion Fourier transform, convolution and Mustard convolution . Adv Appl Clifford Algebr , 27 ( 1 ): 381 - 385 . doi: 10.1007/s00006-016-0684-8 http://doi.org/10.1007/s00006-016-0684-8

Hitzer EMS , 2007 . Quaternion Fourier transform on quaternion fields and generalizations . Adv Appl Clifford Algebr , 17 ( 3 ): 497 - 517 . doi: 10.1007/s00006-007-0037-8 http://doi.org/10.1007/s00006-007-0037-8

Hu XX , Kou KI , 2017 . Quaternion Fourier and linear canonical inversion theorems . Math Methods Appl Sci , 40 ( 7 ): 2421 - 2440 . doi: 10.1002/mma.4148 http://doi.org/10.1002/mma.4148

Hu XX , Kou KI , 2018 . Phase-based edge detection algorithms . Math Methods Appl Sci , 41 ( 11 ): 4148 - 4169 . doi: 10.1002/mma.4567 http://doi.org/10.1002/mma.4567

Jagerman D , 1966 . Bounds for truncation error of the sampling expansion . SIAM J Appl Math , 14 ( 4 ): 714 - 723 . doi: 10.1137/0114060 http://doi.org/10.1137/0114060

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 . doi: 10.1631/FITEE.1500334 http://doi.org/10.1631/FITEE.1500334

Kou KI , Morais J , 2014 . Asymptotic behaviour of the quaternion linear canonical transform and the Bochner-Minlos theorem . Appl Math Comput , 247 : 675 - 688 . doi: 10.1016/j.amc.2014.08.090 http://doi.org/10.1016/j.amc.2014.08.090

Kou KI , Qian T , 2005a . Shannon sampling in the Clifford analysis setting . Z Anal Anwend , 24 ( 4 ): 853 - 870 .

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 . doi: 10.1016/S0252-9602(17)30214-X http://doi.org/10.1016/S0252-9602(17)30214-X

Kou KI , Ou JY , Morais J , 2013 . Uncertainty principle for quaternionic linear canonical transform . Abstr Appl Anal , Article 725952 .

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 . doi: 10.1007/s11045-016-0410-7 http://doi.org/10.1007/s11045-016-0410-7

Lian P , 2021 . Quaternion and fractional Fourier transform in higher dimension . Appl Math Comput , 389 : 125585 . doi: 10.1016/j.amc.2020.125585 http://doi.org/10.1016/j.amc.2020.125585

Marvasti F , 2001 . Nonuniform Sampling: Theory and Practice . Springer Science & Business Media , New York, USA .

Pan WJ , 2000 . Fourier Analysis and Its Applications . Peking University Press , China (in Chinease) .

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 . doi: 10.1109/78.960426 http://doi.org/10.1109/78.960426

Splettstösser W , Stens RL , Wilmes G , 1981 . On approximation by the interpolating series of G. Valiron . Funct Approx Comment Math , 11 : 39 - 56 .

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 .

Zayed AI , 1993 . Advances in Shannon’s Sampling Theory . CRC Press , Boca Raton, USA .

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 . doi: 10.1109/TIP.2016.2567077 http://doi.org/10.1109/TIP.2016.2567077

浏览量

143

Downloads

CSCD

文章被引用时，请邮件提醒。

Submit

工具集

关联资源

暂无数据