Direct sampling methods for isotropic and anisotropic scatterers with point source measurements

TL;DR

This paper introduces two novel imaging functionals for inverse scattering problems involving point source measurements, providing explicit decay rates and analysis for both isotropic and anisotropic scatterers, supported by numerical experiments.

Contribution

The paper proposes two new imaging functionals for inverse scattering, utilizing far-field transforms and Cauchy data, with explicit decay analysis and numerical validation.

Findings

Explicit decay rate derived for the first imaging functional.

Behavior of the second functional analyzed using Green's identities.

Numerical experiments demonstrate effectiveness for 2D isotropic and anisotropic scatterers.

Abstract

In this paper, we consider the inverse scattering problem for recovering either an isotropic or anisotropic scatterer from the measured scattered field initiated by a point source. We propose two new imaging functionals for solving the inverse problem. The first one employs a far-field transform to the data which we then use to derive and provide an explicit decay rate for the imaging functional. In order to analyze the behavior of this imaging functional we use the factorization of the near field operator as well as the Funk-Hecke integral identity. For the second imaging functional the Cauchy data is used to define the functional and its behavior is analyzed using the Green's identities. Numerical experiments are given in two dimensions for both isotropic and anisotropic scatterers.