On Radon transforms between lines and hyperplanes

TL;DR

This paper develops new inversion formulas for the Radon transform and its dual between lines and hyperplanes in n-dimensional space, focusing on quasi-radial functions and employing advanced integral transforms.

Contribution

It introduces explicit inversion formulas for the Radon transform and its dual in this setting, including the symmetric and general cases, using sophisticated mathematical tools.

Findings

New inversion formulas for Radon transform between lines and hyperplanes.

Explicit formulas for the dual transform in symmetric and general cases.

Application of Funk, Radon-John, Kelvin, and Erdélyi-Kober transforms.

Abstract

We obtain new inversion formulas for the Radon transform and its dual between lines and hyperplanes in $\rn$. The Radon transform in this setting is non-injective and the consideration is restricted to the so-called quasi-radial functions that are constant on symmetric clusters of lines. For the corresponding dual transform, which is injective, explicit inversion formulas are obtained both in the symmetric case and in full generality. The main tools are the Funk transform on the sphere, the Radon-John $d$-plane transform in $\rn$, the Grassmannian modification of the Kelvin transform, and the Erd\'elyi-Kober fractional integrals.