The Funk-Radon transform assigns to a function defined on the unit sphere its integrals along all great circles of the sphere. In this paper, we consider a frame decomposition of the Funk-Radon transform, which is a flexible alternative to the singular value decomposition. In particular, we construct a novel frame decomposition based on trigonometric polynomials and show its application for the inversion of the Funk-Radon transform. Our theoretical findings are verified by numerical experiments, which also incorporate a regularization scheme.

翻译：Funk-Radon变换将定义在单位球面上的函数映射为该函数沿所有球面大圆的积分。本文研究Funk-Radon变换的框架分解，该分解为奇异值分解提供了一种灵活的替代方案。特别地，我们基于三角多项式构建了一种新型框架分解，并展示了其在Funk-Radon变换反演中的应用。数值实验验证了理论结果，其中还包含正则化方案。