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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2024, Vol. 41 ›› Issue (5): 962-972.doi: 10.3969/j.issn.1005-3085.2024.05.012

• • 上一篇    下一篇

便于快速计算的4带小波子带算子的范数

邹庆云1,   王国秋2   

  1. 1. 湖南文理学院数理学院，常德 415000
    2. 湖南师范大学数学与统计学院，长沙  410081
  • 收稿日期:2022-01-12 接受日期:2022-09-13 出版日期:2024-10-15
  • 基金资助:
    国家自然科学基金 (11901194)；湖南省教育厅重点项目 (20A345).

The Norm of Sub-band Operators Associated to Four-band Wavelets with Fast Calculation Structure

ZOU Qingyun1,   WANG Guoqiu2   

  1. 1. Department of Mathematics and Physics Science, Hunan University of Arts and Science, Changde 415000
    2. School of Mathematics and Statistics, Hunan Normal University, Changsha 410081
  • Received:2022-01-12 Accepted:2022-09-13 Online:2024-10-15
  • Supported by:
    The National Natural Science Foundation of China (11901194); the Key Research Foundation of Hunan Provincial Education Department (20A345).

摘要：

多带小波比2带小波具有更丰富的参数空间、更灵活的时频铺叠、提供更好的能量压缩，所以在信号处理、数值分析等领域有着广泛的应用。对子带算子理论进行了研究，建立了优化模型，能够在含有自由参数的紧支撑对称多带双正交小波中挑选子带算子范数最小的小波，为研究适合数字图像处理的小波理论及其快速算法提供参考。首先，给出了双无限维矩阵–子带算子的定义，发展了循环矩阵的理论，得到了便于快速计算的4带双正交小波子带算子的范数。通过构造一个特定的三角函数并计算其最大值，得到该子带算子的范数。然后，建立了范数最小化模型，构造并挑选一类具有快速计算结构的4带双正交小波滤波器。最后，通过实例验证了所得到的结论。

关键词: 多带小波, 循环矩阵, 双正交性, 范数, 对称性

Abstract:

Multi-band wavelets have applied in many areas such as signal processing and numerical analysis due to their richer parameter space to have a more flexible time-frequency tiling, to give better energy compaction than 2-band wavelets. The sub-band operators are studied, an optimization model is built, and the wavelets with the smallest norm of the sub-band operator can be selected from the symmetric multi-band biorthogonal wavelets with free parameters, which can be applied in digital image processing based on wavelet theory and its fast algorithms. Firstly, the sub-band operator which is an infinite-dimensional matrix, is introduced, circular matrix theory is developed, and the norm of a sub-band operator associated to four-band wavelets with fast calculation structure is obtained. It is easy to obtain the norm of the sub-band operator with some structure by construction a function and computing the maximum. Secondly, a model to minimize the norm is built and four-band biorthogonal wavelets filter bands with fast calculation structure are designed. Lastly, an example is provided to illustrate the proposed results.

Key words: multi-band wavelets, circular matrix, biorthogonality, norm, symmetry

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