Detection of small inhomogeneities via direct sampling method in transverse electric polarization

TL;DR

This paper investigates the limitations of the direct sampling method in detecting small inhomogeneities with different permeability in transverse electric polarization, providing theoretical insights and numerical validation.

Contribution

It offers a theoretical explanation for why the direct sampling method fails to locate inhomogeneities with differing permeability, linking the indicator function to Bessel functions.

Findings

Indicator function expressed via Bessel function of order one

Theoretical explanation for detection failure with permeability contrast

Numerical results support the theoretical analysis

Abstract

Various studies have confirmed the possibility of identifying the location of a set of small inhomogeneities via a direct sampling method; however, when their permeability differs from that of the background, their location cannot be satisfactorily identified. However, no theoretical explanation for this phenomenon has been verified. In this study, we demonstrate that the indicator function of the direct sampling method can be expressed by the Bessel function of order one of the first kind and explain why the exact locations of inhomogeneities cannot be identified. Numerical results with noisy data are exhibited to support our examination.