On the Determination of a Function from an Elliptical Radon Transform

TL;DR

This paper investigates the mathematical properties and inversion formulas of an elliptical Radon transform integrating functions over solid ellipsoids of rotation, with applications in tomography and imaging.

Contribution

It introduces new inversion formulas, stability estimates, and range descriptions for the elliptical Radon transform with fixed eccentricity and hyperplane foci.

Findings

Derived explicit inversion formulas for the transform.

Established stability estimates and range conditions.

Proved local uniqueness results for the inversion.

Abstract

In recent years, Radon type transforms that integrate functions over various sets of ellipses/ellipsoids have been considered in SAR, ultrasound reflection tomography, and radio tomography. In this paper, we consider the transform that integrates a given function in $\RR^n$ over a set of solid ellipsoids of rotation with a fixed eccentricity and foci restricted to a hyperplane. Inversion formulas are obtained for appropriate classes of functions that are even with respect to the hyperplane. Stability estimates, range description, and local uniqueness results are also provided.