Reconstruction of a function defined on a sphere using the Funk transform

2024/11/08 by Rafik Aramyan, Aramyan, Rafik
Engineering · #44A12 #45Q05 #65R32 #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2411.05365

openalex publication_date 2024/11/08 · openalex created_date 2024/11/15 · openalex updated_date 2026/07/28

Abstract

It is known that the Funk transform (the Funk-Radon transform) is invertible in the class of even (symmetric) continuous functions defined on the unit 2-sphere S². In this article, for the reconstruction of f from C(S²) (can be non-even), an additional condition (to reconstruct an odd function) is found, and the injectivity of the so-called two data Funk transform is considered. An iterative inversion formula of the transform is presented. Such inversions have theoretical significance in convexity theory, integral geometry and spherical tomography. Also, the Funk-Radon transform is used in Diffusion-weighted magnetic resonance imaging.

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