# The Imaging of Small Perturbations in an Anisotropic Media

## TL;DR

This paper introduces an asymptotic analysis-based method for imaging small defects in anisotropic media using far field data, combining the MUSIC algorithm and transmission eigenvalue analysis.

## Contribution

It develops a novel approach integrating asymptotic expansion, MUSIC algorithm, and eigenvalue analysis for detecting and characterizing small perturbations in anisotropic materials.

## Key findings

- Successful reconstruction of defect locations using MUSIC algorithm.
- Analysis of how defects influence transmission eigenvalues.
- Numerical validation demonstrating method effectiveness.

## Abstract

In this paper, we employ asymptotic analysis to determine information about small volume defects in a known anisotropic scattering medium from far field scattering data. The location of the defects is reconstructed via the MUSIC algorithm from the range of the multi-static response matrix derived from the asymptotic expansion of the far field pattern in the presence of small defects. Since the same data determines the transmission eigenvalues corre- sponding to the perturbed media, we investigate how the presence of the defects changes the transmission eigenvalues and use this information to recover the strength of the small defects. We provide convergence results on transmission eigenvalues as the size of the defects tends to zero as well as derive the first correction term in the asymptotic expansion of the simple transmission eigen- values. Numerical examples are presented to show the viability of our imaging method.