Shape reconstruction of buried obstacles by controlled evolution of a level set : from a MinMax formulation to numerical experimentation
Résumé
The nonlinearized reconstruction of the cross-sectional contour of a homogeneous, possibly multiply connected obstacle buried in a half-space from time-harmonic wave field data collected above this half-space in both transverse magnetic (TM) and transverse electric (TE) polarization cases is investigated. The reconstruction is performed via controlled evolution of a level set that was pioneered by Litman et al (Litman A, Lesselier D and Santosa F 1998 Inverse Problems 14 685-706) but at this time restricted to free space for TM data collected all around the sought obstacle. The main novelty of the investigation lies in the following points: from the rigorous contrast-source domain integral formulation (TM) and integral-differential formulation (TE) of the direct scattering problems in the buried obstacle configuration, and from appropriately cast adjoint scattering problems, we demonstrate, by processing min-max formulations of an objective functional J made of the data error to be minimized, that its derivatives with respect to the evolution time t are given in closed form. They are contour integrals involving the normal component of the velocity field of evolution times the product of direct and adjoint fields (TM), or of the scalar product of gradients of such fields (TE) at t. This approach only calls for the analysis of the well posed direct and adjoint scattering problems formulated from the TM and TE Green systems of the unperturbed layered environment and, unusually, it avoids the differentiation of state fields. Other contributions of the paper come from exhibiting and analysing via comprehensive numerical experimentation how and under which conditions evolutions of level sets involving velocities opposite to shape gradients perform in demanding configurations including two disjoint obstacles, constitutive materials strongly less or more refractive than the embedding material, aspect-limited and noisy monochromatic data, in the severe TE case as well as in the more menial TM case. A comparison with a binary-specialized modified-gradient solution method is also led for several, more and more lossy embedding half-spaces. Rules of thumb for effectiveness of the inversions and pending theoretical and computational questions are outlined in conclusion.