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图学学报

图学学报 ›› 2022, Vol. 43 ›› Issue (6): 1002-1017.DOI: 10.11996/JG.j.2095-302X.2022061002

• 综述 • 上一篇    下一篇

U-系统与 V-系统的理论及应用综述 

  

  1. 1. 江南大学人工智能与计算机学院，江苏 无锡 214122；  2. 澳门科技大学计算机科学与工程学院，澳门 999078；  3. 澳门理工大学应用科学学院，澳门 999078；  4. 广东医科大学信息工程学院，广东 东莞 523808；  5. 北方工业大学理学院，北京 100144
  • 出版日期:2022-12-30 发布日期:2023-01-11

A Survey of theory and applications of U-system and V-system 

  1. 1. School of Artificial Intelligence and Computer Science, Jiangnan University, Wuxi Jiangsu 214122, China; 

    2. School of Computer Science and Engineering, Macau University of Science and Technology, Macau 999078, China; 

    3. Faculty of Applied Sciences, Macao Polytechnic University, Macau 999078, China; 

    4. School of Information Engineering, Guangdong Medical University, Dongguan Guangdong 523808, China; 

    5. College of Sciences, North China University of Technology, Beijing 100044, China

  • Online:2022-12-30 Published:2023-01-11

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摘要：

传统的 Fourier 级数在逼近间断信号时因 Gibbs 现象的干扰，会产生比较大的误差。针对此问 题，国内学者齐东旭教授带领的课题组提出了非连续正交函数系的研究课题，其中 U-系统和 V-系统是两类典 型的非连续完备正交函数系。从数学理论上来说，U-系统和 V-系统分别是对著名的 Walsh 函数和 Haar 函数由 分段常数向分段 k 次多项式进行推广的结果，其最重要的特点是函数系中既有光滑函数又有各个层次的间断函 数。因此，U，V-系统可以处理连续和间断并存的信息，在一定程度上弥补了 Fourier 分析和连续小波的缺憾。 本文从理论与应用 2 个方面对 U，V-系统进行了综述。在理论方面，首先介绍了单变量 U-系统与 V-系统各自 的构造方法，其次介绍三角域上 U，V-系统的构造方法，最后介绍 U，V-系统的主要性质。在应用方面，介绍 了若干具有代表性的应用案例。

关键词: U-系统, V-系统, 正交函数, 非连续, 频谱分析

Abstract:

The traditional Fourier analysis and continuous wavelet method will produce relatively enormous errors due to the interference of Gibbs phenomenon. To solve this problem, Qi Dongxu proposed the research topic of discontinuous orthogonal function systems, among which U-system and V-system are two typical discontinuous complete orthogonal function systems. In terms of the mathematical theory, U-system and V-system are the results of the extension of the well-known Walsh function and Haar function from piecewise constant to piecewise k degree polynomial, respectively. The most important feature of U-system is that there are both smooth functions and discontinuous functions at various levels in the function system. Therefore, U- and V- systems can process both continuous and discontinuous information, making up for the shortcomings of Fourier analysis and continuous wavelet to a certain extent. This paper reviewed U- and V- systems from two aspects: theory and application. Theoretically, firstly, the construction methods of univariate U-system and V-system were introduced, respectively, then the construction methods of V-system on triangular domain were introduced, and finally the main properties of U- and V-systems were introduced. In terms of application, some representative cases of applications were introduced. 

Key words: U-system, V-system, orthogonal function, discontinuity, spectral analysis

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