# Linearized 3-D Electromagnetic Contrast Source Inversion and Its Applications to Half-space Configurations

## TL;DR

This paper introduces a linearized 3-D electromagnetic inversion method that reduces computational costs by calculating total fields only once, effectively reconstructing contrasts in ground-penetrating and through-the-wall imaging applications.

## Contribution

It presents a novel linearized inversion approach using a cascade of linear functionals and a sum-of-$\\ell_1$-norm scheme to handle nonuniqueness, improving efficiency in 3-D electromagnetic imaging.

## Key findings

- Efficient reconstruction with only one total field computation per iteration.
- Successful application to ground-penetrating radar and through-the-wall imaging.
- Maintains high accuracy and quality of reconstructions.

## Abstract

One of the main computational drawbacks in the application of 3-D iterative inversion techniques is the requirement of solving the field quantities for the updated contrast in every iteration. In this paper, the 3-D electromagnetic inverse scattering problem is put into a discretized finite-difference frequency-domain scheme and linearized into a cascade of two linear functionals. To deal with the nonuniqueness effectively, the joint structure of the contrast sources is exploited using a sum-of-$\ell_1$-norm optimization scheme. A cross-validation technique is used to check whether the optimization process is accurate enough. The total fields are, then, calculated and used to reconstruct the contrast by minimizing a cost functional defined as the sum of the data error and state error. In this procedure, the total fields in the inversion domain are computed only once, while the quality and accuracy of the obtained reconstructions are maintained. The novel method is applied to ground-penetrating radar imaging and through-the-wall imaging, in which the validity and efficiency of the method is demonstrated.