A new technique for finding the convex hull of a scattering system: performance analysis and application to inhomogeneous backgrounds (2004)

This paper presents a systematic numerical analysis of the performances of a recent approach to the estimation of the convex hull of a scattering systems from electromagnetic field data also. The algorithm has been applied to the dielectric scattering case never considered before and its behaviour has been investigated, for the first time, for many complex configurations of practical interest. In particular, the simultaneous presence of dielectric and metallic scatterers as well as scattering systems embedded in an inhomogeneous medium have been considered. By introducing a numerical figure that synthetically describes the quality of the convex hull estimates, the analysis is systematically and compactly presented. Inverse Electromagnetic Scattering (ISC) problems amounts to find the location, the excitation and the electromagnetic properties of a system of scatterers from field data on an observation domain (1-3). However, the solution of the full inverse problem is significantly affected by the ill-positioning, the ill-conditioning, a high computation burden and the occurrence of false solutions for local optimisation based algorithms. Any a priori information on the observed system can be exploited to improve the performances of full inverse scattering algorithms. In particular, a good estimate of the region containing the scatterers it is of primary interest. In fact, as long as the dimension of the searching region is assumed larger than the bare necessity, the computational complexity grows and the reliability of the solution degrades. Accordingly, a simple and effective technique which is able to evaluate the convex hull of a scattering system without facing the complete inversion of the observed data can be very helpful in solving ISC problems. Recently, a simple method to estimate the convex hull of a scattering system from field data has been introduced and tested for few configurations of scatterers embedded in the free space (2-3). The aim of this communication is to present a survey of the performances of the approach for a 2D scalar geometry in a much wider number of cases. First, the case of metallic and/or dielectric scatterers immersed in the free space is considered (case A) and the influence of the dielectric contrast is investigated. Then, to study the effectiveness of the approach in realistic applications, the case of metallic and/or dielectric scatterers included in a dielectric circular domain immersed in an unbounded homogeneous medium (void) is considered (case B). In all cases, the scattered field is observed on a circle enclosing the scattering system. In case B, since the available technique can be applied in the inhomogeneous case only for observation curves very close to the discontinuity, the observation circle has been assumed equal to the discontinuity dielectric/void contour. It is worth noting that, to simplify the analysis, the coupling probe-discontinuity has been dismissed in case B. 2. Convex envelope finding technique and performance analysis Let us now give a concise description of the convex envelope finding technique. For a complete discussion the reader can refer to (3). The geometry of the problem is presented in Fig. 1: S represent the scattering systems, / the observation line, P(s) the observation point and s the natural parameter on / . For each P(s), the minimum angular sector 6(s) containing S and with vertex in P(s) is estimated by exploiting two local properties of the scattered field, the Point Source Spectral Content and the Local Spectral Content (2,4). An estimate He of the convex hull H of S is obtained by intersecting all 6(s). To synthetically present the results of a systematic analysis including many test-cases, we introduce the parameterU characterising the quality of He: