几何相关流形上的样条函数与小波函数
Splines and Wavelets on Geophysically Relevant Manifolds
摘要：
对单位球$\mathbb{S}^{2}$的分析在地震学、天气预报、天体物理学、信号分析、晶体学、计算机视觉、计算机断层扫描、神经科学和统计学中有许多应用。在过去的二十年中，这些和其他应用的重要性引发了各种工具的发展，如样条和小波基适用于单位球$\mathbb{S}^{2}$，$\>\>\mathbb{S}^{3}$和旋转群$SO(3)$。本文总结了作者及其合作者关于紧致黎曼流形上的广义（平均）变分样条和局部化框架（小波）的一些结果。结果示出的应用程序的Radon型变换的$\mathbb{S}^{d}$和$SO(3)$。
英文摘要：
Analysis on the unit sphere $\mathbb{S}^{2}$ found many applications in seismology, weather prediction, astrophysics, signal analysis, crystallography, computer vision, computerized tomography, neuroscience, and statistics. In the last two decades, the importance of these and other applications triggered the development of various tools such as splines and wavelet bases suitable for the unit spheres $\mathbb{S}^{2}$, $\>\>\mathbb{S}^{3}$ and the rotation group $SO(3)$. Present paper is a summary of some of results of the author and his collaborators on generalized (average) variational splines and localized frames (wavelets) on compact Riemannian manifolds. The results are illustrated by applications to Radon-type transforms on $\mathbb{S}^{d}$ and $SO(3)$.
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