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

TL;DR

This paper explores the invertibility of the Funk transform on the sphere, extending its applicability to non-even functions by introducing additional conditions and iterative inversion methods, with implications for geometry, tomography, and MRI.

Contribution

It introduces conditions for reconstructing non-even functions from the Funk transform and presents an iterative inversion formula, expanding the transform's utility beyond symmetric functions.

Findings

Established conditions for reconstructing odd functions.

Developed an iterative inversion formula.

Analyzed injectivity of the two data Funk transform.

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 S^2. In this article, for the reconstruction of f from C(S^2) (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.