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

Title: 球面上的斯莱皮安空间-谱集中问题：有限余纬-经度空间区域的解析表述

Abstract: In this paper, 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 that 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 that 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 a spatially concentrated signal is indeed sparse in the Slepian basis. Show Chinese abstract

Abstract: 本文中，我们为球面上的某一有限余纬-经度空间区域（定义为正余纬和经度范围的笛卡尔积）上的 Slepian 空间-谱集中问题建立了分析公式。 Slepian 问题的解是一组在空间或谱域内最优集中且正交的函数。 这些性质使它们在测量局限于球面某空间区域和/或信号仅需在球面某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化谱/空域分析以及数据的稀疏表示，我们利用球谐函数在复指数基下的展开，为有限余纬-经度空间区域的 Slepian 集中问题建立了分析公式。 我们还扩展了适用于由旋转后的有限余纬-经度子区域组成的空间区域的分析公式。 通过利用所提出公式的各种对称性，我们设计了一个计算高效的算法来实现该分析公式。 通过数值实验验证了这种计算时间的减少。 我们借助数值实例展示了结果，并表明在 Slepian 基中空间集中的信号确实具有稀疏表示。