Shape reconstruction of buried obstacles by controlled evolution of a level set: from a min-max formulation to numerical experimentation

The nonlinearized reconstruction of the cross-sectional contour of a homoge neous, possibly multiply connected obstacle buried in a half-space from tim e-harmonic wave field data collected above this half-space in both transver se magnetic (TM) and transverse electric (TE) polarization cases is investi gated. 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 spa ce for TM data collected all around the sought obstacle. The main novelty o f the investigation lies in the following points: from the rigorous contras t-source domain integral formulation (TM) and integral-differential formula tion (TE) of the direct scattering problems in the buried obstacle configur ation, and from appropriately cast adjoint scattering problems, we demonstr ate, by processing min-max formulations of an objective functional J made o f 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 invol ving the normal component of the velocity field of evolution times the prod uct of direct and adjoint fields (TM), or of the scalar product of gradient s of such fields (TE) at t. This approach only calls for the analysis of th e 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 experi mentation 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 monoc hromatic 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 i s also led for several, more and more lossy embedding half-spaces. Rules of thumb for effectiveness of the inversions and pending theoretical and comp utational questions are outlined in conclusion.