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

Abstract

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 Feijóo (2004a). 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 a threedimensional boundary element analysis which requires 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 BEM calculations are straight forward since the measuring points can be associated to internal points in the BEM model. Obtained results allow concluding that the BEM implementation of the method has the potential to further develop and implement algorithms which can improve the quality of the reconstructions. To this end an extended version of the present method can be coupled with the algorithms introduced in previous works for the topological optimization of potential ( Cisilino, 2006) and elasticity (Carretero et al, 2008) problems using the topological derivative and BEM.