二维Bedrosian定理理论及其应用研究

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 bidimensional Bedrosian theorem. This project will mainly focus on 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 study the amplitude-phase analysis methods and image decomposition schedules. The main object of this project is striving to enrich the ensemble of the theory of the Bedrosian theorem and study the applications in amplitude-phase analysis and image decomposition based on the Bedrosian theorems.

Bedrosian定理是数学和信息科学交叉领域的基础性理论，其决定了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，本项目主要针对方向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，并阐释它们对应的物理意义。在此基础上，探讨二维Bedrosian定理应用于图像的幅相分析和图像分解的策略。本项目研究主要目的是力争完善二维Bedrosian定理理论，探讨二维Bedrosian定理的图像时频分析及分解应用。

Bedrosian定理是数学和信息科学交叉领域的基础性理论，其决定了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。项目研究之初二维Bedrosian定理的研究还存在着不少空白，所以项目主要针对方向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定理等进行了不同域内的数学推导及理论证明，并阐释它们对应的物理意义。在此基础上，项目探讨了二维Bedrosian定理应用于图像的幅相分析和图像分解的策略。本项目研究主要是完善了二维Bedrosian定理理论，探讨了二维Bedrosian定理的图像时频分析及分解应用等。