二维Bedrosian定理理论及其应用研究

Study on Bi-Dimensional Bedrosian Principle and Applications

Bedrosian定理是数学和信息科学交叉领域的基础性理论,它决定了Hilbert变换的结果形式,对于信号幅相及时频分析均具有重要的理论意义和应用价值。目前二维Bedrosian定理的研究还存在着不少空白,本文主要针对方向Bedrosian定理、交叉象Bedrosian定理、单象Bedrosian定理、二象Bedrosian定理、四元Bedrosian定理以及单基解析Bedrosian定理等进行不同域内的数学推导及理论证明的分析介绍,并阐释了它们对应的物理意义。在此基础上,介绍了二维Bedrosian定理应用于图像的幅相分析和图像分解的策略。最后,在二维Bedrosian定理理论、二维Bedrosian定理的图像时频分析及分解应用等综述基础上,系统性地对未来的工作进行了展望。

The Bedrosian theorem is the crossed elementary theory in mathematics and information fields, which determines the results of Hilbert transform and plays an important role in amplitude and phase analysis for signal processing. However, there have been many blank aspects in bidimen-sional Bedrosian theorem. This paper will mainly focus on the review of the theoretical proofs and physical sense explanation for the partial Bedrosian theorem, the cross-orthant Bedrosian theorem, the single-orthant Bedrosian theorem, the bi-orthant Bedrosian theorem, the quaternion Bedrosian theorem and the monogenic Bedrosian theorem in different domains. Based on these Bedrosian theorems, we will review the amplitude-phase analysis methods and image decomposition schedules. The main object of this paper is striving to review the ensemble of the theory of the Bedrosian theorem and study the applications in amplitude-phase analysis and image decomposi-tion based on the Bedrosian theorems.