Inversion of the elliptical Radon transform arising in migration imaging using the regular Radon transform

TL;DR

This paper establishes a relation between the elliptical Radon transform and the regular Radon transform, providing an inversion formula and demonstrating its effectiveness through numerical simulations in two dimensions.

Contribution

It introduces a method to invert the elliptical Radon transform by relating it to the well-known regular Radon transform, enabling new applications in imaging.

Findings

Derived an inversion formula for the elliptical Radon transform.

Validated the approach with numerical simulations in 2D.

Showed the practical applicability in migration imaging.

Abstract

In recent years, many types of elliptical Radon transforms that integrate functions over various sets of ellipses/ellipsoids have been considered, relating to studies in bistatic synthetic aperture radar, ultrasound reflection tomography, radio tomography, and migration imaging. In this article, we consider the transform that integrates a given function in $\mathbf R^n$ over a set of ellipses (when $n=2$) or ellipsoids of rotation (when $n\geq 3$) with foci restricted to a hyperplane. We show a relation between this elliptical Radon transform and the regular Radon transform, and provide the inversion formula for the elliptical Radon transform using this relation. Numerical simulations are performed to demonstrate the suggested algorithms in two dimensions, and these simulations are also provided in this article.