Stability and Resolution Analysis of Topological Derivative Based Localization of Small Electromagnetic Inclusions

TL;DR

This paper rigorously analyzes a topological derivative imaging method for locating small electromagnetic inclusions, demonstrating its resolution limit and stability under noise, with implications for improved electromagnetic imaging techniques.

Contribution

The paper provides a detailed theoretical analysis of the resolution and stability of a topological derivative based imaging framework for electromagnetic inclusion detection.

Findings

Achieves Rayleigh resolution limit

Provides stability analysis with respect to noise

Evaluates signal-to-noise ratio in the imaging functional

Abstract

The aim of this article is to elaborate and rigorously analyze a topological derivative based imaging framework for locating an electromagnetic inclusion of diminishing size from boundary measurements of the tangential component of scattered magnetic field at a fixed frequency. The inverse problem of inclusion detection is formulated as an optimization problem in terms of a filtered discrepancy functional and the topological derivative based imaging functional obtained therefrom. The sensitivity and resolution analysis of the imaging functional is rigorously performed. It is substantiated that the Rayleigh resolution limit is achieved. Further, the stability of the reconstruction with respect to measurement and medium noises is investigated and the signal-to-noise ratio is evaluated in terms of the imaginary part of free space fundamental magnetic solution.