To support localized signal analysis on the sphere for applications in geophysics, cosmology, acoustics and beyond, we develop a framework for the analytic evaluation of the integral of spherical signals and for the analytic solution of the Slepian spatialspectral concentration problem over simple spherical polygons. We propose a polygon right angle triangulation method for the division of a simple spherical polygon into spherical right angle triangles, which allows us to decompose the problem of integrating signals or solving the spatial-spectral concentration problem over the polygon region into sub-problems that require the evaluation of the integral of complex exponential functions over spherical right angle triangles of arbitrary orientation and position. We derive closed-form expressions for the evaluation of such integrals by appropriately choosing the rotations and using Wigner-D functions. The proposed framework enables us to solve the Slepian spatial-spectral concentration problem for simple spherical polygons, resulting in bandlimited, spatially optimally concentrated basis functions for signal representation and reconstruction, localized analysis and signal modeling on the sphere. We also present convergence criterion for the infinite series expansions involved in the evaluation of the integral of complex exponential functions, and establish the validity of the proposed developments by evaluating the integral and computing the Slepian basis functions over the geographical region of Australia and the volcanic plateau of Tharsis, using the Earth and Mars topography maps respectively.

中文翻译：

---

多边形区域上球体信号的局部分析


为了支持球体上的局部信号分析，以应用于地球物理学、宇宙学、声学等领域，我们开发了一个框架，用于分析评估球体信号的积分以及解析简单球面多边形上的 Slepian 空间谱浓度问题。我们提出了一种多边形直角三角剖分方法，用于将简单的球面多边形划分为球面直角三角形，该方法使我们能够将积分信号问题或解决多边形区域上的空间光谱集中问题分解为需要的子问题任意方向和位置的球面直角三角形上的复指数函数积分的评估。我们通过适当选择旋转并使用 Wigner-D 函数导出用于评估此类积分的闭合形式表达式。所提出的框架使我们能够解决简单球形多边形的斯莱普空间谱集中问题，从而产生带限的、空间最优集中的基函数，用于球体上的信号表示和重建、局部分析和信号建模。我们还提出了复杂指数函数积分评估中涉及的无限级数展开的收敛准则，并通过评估积分和计算澳大利亚和火山高原地理区域的斯莱皮基函数来确定所提出的发展的有效性塔尔西斯，分别使用地球和火星地形图。