An asymptotic representation formula for scattering by thin tubular structures and an application in inverse scattering

TL;DR

This paper derives an asymptotic formula for electromagnetic scattering by thin tubular objects, linking the scattered wave to the shape and material properties, and applies it to improve inverse scattering reconstructions.

Contribution

It introduces a new asymptotic representation formula for scattering by thin tubes, incorporating shape and material tensors, and demonstrates its use in inverse problem algorithms.

Findings

Explicit polarization tensors characterized in terms of shape and cross-section

Efficient shape reconstruction demonstrated through numerical results

Asymptotic formula accurately approximates scattered waves for thin structures

Abstract

We consider the scattering of time-harmonic electromagnetic waves by a penetrable thin tubular scattering object in three-dimensional free space. We establish an asymptotic representation formula for the scattered wave away from the thin tubular scatterer as the radius of its cross-section tends to zero. The shape, the relative electric permeability and the relative magnetic permittivity of the scattering object enter this asymptotic representation formula by means of the center curve of the thin tubular scatterer and two electric and magnetic polarization tensors. We give an explicit characterization of these two three-dimensional polarization tensors in terms of the center curve and of the two two-dimensional polarization tensor for the cross-section of the scattering object. As an application we demonstrate how this formula may be used to evaluate the residual and the shape derivative in an efficient iterative reconstruction algorithm for an inverse scattering problem with thin tubular scattering objects. We present numerical results to illustrate our theoretical findings. Mathematics subject classifications (MSC2010): 35C20, (65N21, 78A46)