Reconstruction Analysis for Inverse Scattering Using the Topological Derivative and the Boundary Element Method

A.P. CISILINO; S.BECK; S.LANGER

Tandil, Argentina

Congreso; XVIII Congreso sobre Métodos Numéricos y sus Aplicaciones ENIEF2009; 2009

Asocición Argentina de Mecánica Computcional

It is presented in this work a direct (non-iterative) method for solving the inverse scattering problem using the framework provided by the topological derivative and the boundary element method. The application is based on the topological derivative for scattering problems introduced by Feijoó (Feijoó G.R., “A new method in inverse scattering based on the topological derivative”, Inverse Problems 20 (2004) 1819-1840). The method allows imaging the boundary of an impenetrable object immersed in a homogeneous medium by using a set of measurements of the radiated scattering pattern resulting from illuminating the object from different directions. This leads to an optimization problem consisting in the minimization of the mismatch between the measured scattering pattern and the scattering pattern resulting from an impenetrable inclusion placed at a point in the medium. The rate of change of this mismatch with respect to the size of the inclusion is the topological derivative field. Based on the heuristics that the boundary of the object can be assimilated to a group of inclusions, the boundary of the object is identified as the locus defined by the positions of the inclusions resulting in the highest values of the topological derivative. The computation of the topological derivate requires the pressure solutions of the adjoint and forward problems. The solution of the forward problem is that of the incident wave for a medium without obstacles. On the other hand, the adjoint problem accounts for the difference between the forward solution and the scatter measures in the field around the object. Both, the forward and the adjoint problems can be solved analytically. Reconstructions are done in this work by using synthetic data produced by means of boundary element models for an infinite medium which requires of the discretization of the object boundary only. The scatter solutions for a number of measuring points placed circularly around the inclusion are used as input data to compute the adjoint solution referenced in the previous paragraph. The efficacy of the method is demonstrated for a number of examples. The effects of the wave length and the number and spatial distribution of the measuring points is assessed. This work can be seen as a first step towards the implementation of an optimization procedure for optimizing the shape of the objects in order to minimize their scattering behavior.