Structure analysis of direct sampling method in 3D electromagnetic inverse problem: near- and far-field configuration

Abstract

In this paper, we study the structure analysis of the Direct Sampling Method (DSM) in the threedimensional inverse electromagnetic scattering problem. Even though the DSM is well-known to the robust, fast, and efficient non-iterative type algorithm to estimate the support and shape of unknown inhomogeneities from scattered data, the inhomogeneities might not be detected in the DSM with improper test polarization. To explain the reason for the phenomenon, we carefully analyze the indicator function of DSM using the asymptotic formula of the scattered field under a small volume hypothesis of well-separated inhomogeneities. The analytic representation formula of the indicator function of DSM is presented by establishing the relationship of spherical Bessel functions, the polarization of the incident wave and the test dipole, and the polarization tensor of the inhomogeneities, which depends on their sizes, permittivity, etc. Through such theoretical results, we also show the validity of the guideline to choose the test polarization given by other work. Furthermore, we propose our method to select the polarization of the test dipole for better efficiency. With a similar path of derivation, we also introduce and analyze the indicator function of DSM in far-field configurations. Various numerical simulations with synthetic and experimental data validate our theoretical results and proposals.

Theoretical Study on Non-Improvement of the Multi-Frequency Direct Sampling Method in Inverse Scattering Problems

“…The direct sampling method (DSM) is a sampling-type, non-iterative technique for retrieving unknown scatterers' location and/or shape. It was first introduced for 2D inverse medium scattering for fixed plane incident waves [1], and it was subsequently applied and extended to various inverse problems, including identifying 2D and 3D electromagnetic inhomogeneities [2][3][4][5], electrical impedance tomography [6], diffusive optical tomography [7], inhomogeneity detection in mono-static [8] measurement configurations, source detection in stratified ocean waveguide [9,10], localizing short, linear, and perfectly conducting cracks [11], retrieving magnetic inhomogeneity locations in transverse electric (TE) polarization [12], anomaly detection in microwave imaging [13], phaseless inverse source scattering [14], and real-world microwave imaging [15]. We reference various studies [16,17] related to the direct sampling method.…”

Imaging of 3D objects with experimental data using orthogonality sampling methods

“…The second goal is to apply the modified OSM, the OSM in [15], and the FM to inversion of 3D experimental data from the Fresnel Institute [12]. It is worth mentioning that there have been results on experimental data verification for sampling methods, see, e.g., [2,8,18,23,29]. Most of these results are for 2D data.…”

“…Most of these results are for 2D data. The 3D data for two spheres from the Fresnel Institute was studied in [18] with a direct sampling method under a small volume hypothesis of well-separated inhomogeneities. Note that the modified OSM studied in this paper does not require this hypothesis.…”