Reconstruction procedures for two inverse scattering problems without the phase information

TL;DR

This paper introduces new reconstruction methods for 3D phaseless inverse scattering problems without relying on the Born approximation, transforming the problem into an inverse kinematic problem and providing explicit formulas.

Contribution

The paper develops two novel reconstruction procedures for 3D phaseless inverse scattering problems that do not depend on the Born approximation, including an explicit formula via the inverse Radon transform.

Findings

Established the connection to the inverse kinematic problem.

Derived an explicit reconstruction formula using the inverse Radon transform.

Proposed an alternative method based on integral equations of Abel type.

Abstract

This is a continuation of two recent publications of the authors about reconstruction procedures for 3-d phaseless inverse scattering problems. The main novelty of this paper is that the Born approximation for the case of the wave-like equation is not considered. It is shown here that the phaseless inverse scattering problem for the 3-d wave-like equation in the frequency domain leads to the well known Inverse Kinematic Problem. Uniqueness theorem follows. Still, since the Inverse Kinematic Problem is very hard to solve, a linearization is applied. More precisely, geodesic lines are replaced with straight lines. As a result, an approximate explicit reconstruction formula is obtained via the inverse Radon transform. The second reconstruction method is via solving a problem of the integral geometry using integral equations of the Abel type.