Structure and properties of linear sampling method for perfectly conducting, arc-like cracks

TL;DR

This paper analyzes the linear sampling method for imaging arc-like perfectly conducting cracks, revealing its mathematical structure, explaining its effectiveness, and proposing a multi-frequency enhancement supported by numerical experiments.

Contribution

It uncovers the eigenvector structure of the MSR matrix and links the imaging functional to Bessel functions, providing theoretical insight and an improved multi-frequency approach.

Findings

Linear sampling method's effectiveness is explained via Bessel functions.

The proposed multi-frequency imaging functional improves imaging quality.

Numerical experiments validate the theoretical analysis.

Abstract

We consider the imaging of arbitrary shaped, arc-like perfectly conducting cracks located in the two-dimensional homogeneous space via linear sampling method. Based on the structure of eigenvectors of so-called Multi Static Response (MSR) matrix, we discover the relationship between imaging functional adopted in the linear sampling method and Bessel function of integer order of the first kind. This relationship tells us that why linear sampling method works for imaging of perfectly conducting cracks in either Transverse Magnetic (Dirichlet boundary condition) and Transverse Electric (Neumann boundary condition), and explains its certain properties. Furthermore, we suggest multi-frequency imaging functional, which improves traditional linear sampling method. Various numerical experiments are performed for supporting our explores.