Topological Derivative for Imaging of Thin Electromagnetic Inhomogeneity: Least Condition of Incident Directions

Won-Kwang Park

doi:10.1155/2018/2096058

Abstract

It is well-known that using topological derivative is an effective noniterative technique for imaging of crack-like electromagnetic inhomogeneity with small thickness when small number of incident directions are applied. However, there is no theoretical investigation about the configuration of the range of incident directions. In this paper, we carefully explore the mathematical structure of topological derivative imaging functional by establishing a relationship with an infinite series of Bessel functions of integer order of the first kind. Based on this, we identify the condition of the range of incident directions and it is highly depending on the shape of unknown defect. Results of numerical simulations with noisy data support our identification.

Highlights

In this paper, we consider topological derivative [1] based imaging technique for thin, curve-like penetrable electromagnetic inhomogeneity with small thickness
We carefully explore the mathematical structure of topological derivative imaging functional by establishing a relationship with an infinite series of Bessel functions of integer order of the first kind
We identify a least condition of the range of incident directions for a successful application in limited-aperture inverse scattering problem and confirm this condition is highly depending on the unknown shape of thin inhomogeneity

Summary

Introduction

We consider topological derivative [1] based imaging technique for thin, curve-like penetrable electromagnetic inhomogeneity with small thickness This has been considered for shape optimization problems [2,3,4,5,6,7] and was successfully combined with the level-set method (see [8,9,10,11,12,13]) for various inverse scattering problems. Throughout results of numerical simulations, it has been confirmed that topological derivatives can be applied in limited-aperture problems, and an analysis in limitedaperture problems has been performed in [22] In this interesting research, a relationship between topological derivative imaging function and an infinite series of Bessel functions of first kind has been established and, correspondingly, a sufficient condition of the range of incident directions for application has been identified theoretically.

Introduction to Direct Scattering Problem and Topological Derivative

Least Condition of Incident Directions

Simulations Results and Discussions

Conclusion