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 变换过程指派了单位球范围内界定的函数, 其组成部分在球体的所有大圆圈中。 在本文中, 我们考虑了Funk- Radon变换的框架分解, 这是单值分解的灵活替代物 。 特别是, 我们根据三角多义构建了一个新的框架分解过程, 并展示了它用于Funk- Radon变换的应用程序 。 我们的理论结论通过数字实验得到验证, 同时也包含一个正规化计划 。</s>