Exact inversion of Funk-Radon transforms with non-algebraic geometries

TL;DR

This paper presents explicit inversion formulas for Funk-Radon transforms on spheres and Riemannian hypersurfaces, enabling exact reconstruction of functions from their integrals over specific geometric subsets, extending classical results to non-algebraic geometries.

Contribution

It introduces explicit inversion formulas for non-geodesic Funk transforms on spheres of arbitrary dimension and generalizes these methods to Riemannian hypersurfaces in affine spaces.

Findings

Explicit inversion formulas for non-geodesic Funk transforms.

Generalization to Riemannian hypersurfaces.

Exact reconstruction of functions from integral data.

Abstract

Any even function defined on 2-sphere is reconstructed from its integrals over big circles by means of the classical Funk formula. For the non-geodesic Funk transform on the sphere of arbitrary dimension, there is the explicit inversion formula similar to that for the geodesic transform. A function defined on the sphere of radius one is integrated over traces of hyperplanes tangent to a sphere contained in the unit ball. This reconstruction is generalized in the paper for Riemannian hypersurfaces in an affine space.