The cone-beam transform consists of integrating a function defined on the three-dimensional space along every ray that starts on a certain scanning set. Based on Grangeat's formula, Louis [2016, Inverse Problems 32 115005] states reconstruction formulas based on a new generalized Funk-Radon transform on the sphere. In this article, we give a singular value decomposition of this generalized Funk-Radon transform. We use this result to derive a singular value decomposition of the cone-beam transform with sources on the sphere thus generalizing a result of Kazantsev [2015, J. Inverse Ill-Posed Probl. 23(2):173-185].
翻译：共波波束变换包括将三维空间上定义的功能整合到从特定扫描器组开始的每个射线上。根据Grangeat的公式,Louis [2016] 反问题32 115005 表示重建公式,以新的通用Funk-Radon的球体变换为基础。在本条中,我们给出了这种广义的Funk-Radon变换的单一价值分解。我们利用这一结果得出了共波变换的单值分解,其来源在球体上,从而将Kazantsev[2015,J. Inverse Ill-Posed Probl. 23(2):173-185]的结果普遍化。(2015,J. Invers Ill-Posed Probl. 23(2):173-185) 。
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