The Spherical Mean Transform with Data on a Parabola in the Plane

TL;DR

This paper presents a method for reconstructing functions from their spherical mean transform with centers on a parabola, enabling recovery outside the parabola using known inversion formulas.

Contribution

It introduces a new approach to recover functions from spherical mean data on a parabola, extending existing inversion techniques to exterior regions.

Findings

Reconstruction of functions outside the parabola from data on the parabola.

Derivation of a formula to extract exterior values from interior data.

Application of known inversion formulas for complete reconstruction.

Abstract

In this paper we deal with the problem of recovering functions from their spherical mean transform $\mathcal{R}$, which integrates functions on circles in the plane, in case where the centers of the circles of integration are located on a parabola $\mathcal{P}$ while their radii can be chosen arbitrarily. Using our data, on the values of $\mathcal{R}$ on $\mathcal{P}$, we show how to extract its values in the exterior of $\mathcal{P}$ in case where the functions in question have compact support inside $\mathcal{P}$. Hence, one can use known inversion formulas for $\mathcal{R}$ in the exterior of $\mathcal{P}$ in order to obtain a reconstruction formula.