Reconstruction of functions on the sphere from their integrals over hyperplane sections

TL;DR

This paper introduces new inversion formulas for Funk-type transforms on the sphere, relating integrals over hyperplane sections to classical Radon transforms, using advanced integral transforms and stereographic projections.

Contribution

It provides novel inversion formulas for two types of Funk transforms associated with hyperplane sections passing through a point inside or on the sphere.

Findings

Derived inversion formulas for Funk-type transforms.

Reduced transforms to classical Radon transforms.

Utilized cosine transforms, Semyanisyi's integrals, and stereographic projection.

Abstract

We obtain new inversion formulas for the Funk type transforms of two kinds associated to spherical sections by hyperplanes passing through a common point $A$ which lies inside the n-dimensional unit sphere or on the sphere itself. Transforms of the first kind are defined by integration over complete subspheres and can be reduced to the classical Funk transform. Transforms of the second kind perform integration over truncated subspheres, like spherical caps or bowls, and can be reduced to the hyperplane Radon transform. The main tools are analytic families of cosine transforms, Semyanisyi's integrals, and modified stereorgraphic projection with the pole at $A$. Assumptions for functions are close to minimal.