Reconstruction of shapes and refractive indices from backscattering experimental data using the adaptivity

TL;DR

This paper presents a two-stage adaptive method for reconstructing the dielectric constant, shape, and refractive index of targets from backscattering radar data, combining global and local convergence techniques for improved accuracy.

Contribution

The paper introduces a novel two-stage reconstruction approach that combines a globally convergent method with an adaptive finite element method for enhanced imaging accuracy.

Findings

Accurate imaging of refractive indices, shapes, and locations of targets.

Effective combination of global and local convergence techniques.

Successful application to experimental backscattering data.

Abstract

We consider the inverse problem of the reconstruction of the spatially distributed dielectric constant $\varepsilon_{r}\left(\mathbf{x}\right), \ \mathbf{x}\in \mathbb{R}^{3}$, which is an unknown coefficient in the Maxwell's equations, from time-dependent backscattering experimental radar data associated with a single source of electric pulses. The refractive index is $n\left(\mathbf{x}\right) =\sqrt{\varepsilon_{r}\left(\mathbf{x}\right)}.$ The coefficient $\varepsilon_{r}\left(\mathbf{x}\right) $ is reconstructed using a two-stage reconstruction procedure. In the first stage an approximately globally convergent method proposed is applied to get a good first approximation of the exact solution. In the second stage a locally convergent adaptive finite element method is applied, taking the solution of the first stage as the starting point of the minimization of the Tikhonov functional. This functional is minimized on a sequence of locally refined meshes. It is shown here that all three components of interest of targets can be simultaneously accurately imaged: refractive indices, shapes and locations.