Title: Slepian Spatial-Spectral Concentration Problem on the Sphere: Analytical Formulation for Limited Colatitude-Longitude Spatial Region Show Chinese title

Title: 球面上的Slepian时空浓度问题：有限纬度-经度空间区域的解析公式

Abstract: In this work, we develop an analytical formulation for the Slepian spatial-spectral concentration problem on the sphere for a limited colatitude-longitude spatial region on the sphere, defined as the Cartesian product of a range of positive colatitudes and longitudes. The solution of the Slepian problem is a set of functions which are optimally concentrated and orthogonal within a spatial or spectral region. These properties make them useful for applications where measurements are taken within a spatially limited region of the sphere and/or a signal is only to be analyzed within a region of the sphere. To support localized spectral/spatial analysis, and estimation and sparse representation of localized data in these applications, we exploit the expansion of spherical harmonics in the complex exponential basis to develop an analytical formulation for the Slepian concentration problem for a limited colatitude-longitude spatial region. We also extend the analytical formulation for spatial regions which are comprised of a union of rotated limited colatitude-longitude subregions. By exploiting various symmetries of the proposed formulation, we design a computationally efficient algorithm for the implementation of the proposed analytical formulation. Such a reduction in computation time is demonstrated through numerical experiments. We present illustrations of our results with the help of numerical examples and show that the representation of spatially concentrated signal is indeed sparse in the Slepian basis. Show Chinese abstract

Abstract: 在本工作中，我们为球面上有限余纬度-经度空间区域上的Slepian时空浓度问题开发了一个解析公式，该区域定义为一系列正余纬度和经度的笛卡尔积。 Slepian问题的解是一组在空间或频谱区域内最优集中且正交的函数。 这些特性使它们在测量仅在球面的有限空间区域内进行和/或信号仅需在球面的一个区域内进行分析的应用中非常有用。 为了支持这些应用中的局部频谱/空间分析，以及局部数据的估计和稀疏表示，我们利用球面谐波在复指数基下的展开，为有限余纬度-经度空间区域上的Slepian浓度问题开发了一个解析公式。 我们还扩展了该解析公式，以适用于由旋转后的有限余纬度-经度子区域的并集构成的空间区域。 通过利用所提出公式的各种对称性，我们设计了一个计算效率高的算法来实现所提出的解析公式。 通过数值实验展示了计算时间的减少。 我们借助数值例子展示了我们的结果，并表明空间集中信号在Slepian基中确实具有稀疏性。