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

Abstract

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.

On novel developments of controlled evolution of level sets in the field of inverse shape problems

“…[35] The level set representation of shapes, combined with the speed method, enables many kind of evolutions, given the velocity field, such as deformations, splitting or merging, and performs a worthwhile role in the field of inverse scattering problems as already underlined in previous publications [Ramananjaona et al, 2001a[Ramananjaona et al, , 2001b.…”

“…[13] The method has been pioneered by Litman et al [1998] and investigated since then by the authors [e.g., Ramananjaona et al, 2001b], noteworthy investigations being led elsewhere [e.g., Dorn et al, 2000;Ito et al, 2001]. As already said, it combines a level set representation of shapes [Sethian, 1999] the first use of which in an inversion procedure goes back to Santosa [1996] and the Speed Method which has been much studied in design and sensitivity problems [Sokolowski and Zolésio, 1992].…”

“…Its derivative (as demonstrated by Ramananjaona et al [2001b] by a Min Max analysis developed from the Green-based integral formulations of the field) follows as…”

“…This time step is indeed one of the few key parameters at the implementation stage of the solution algorithm-as expected, too small a time step leads to a slow decrease of the objective functional, and too high a time step leads to divergence of the algorithm. Up until now [Ramananjaona et al, 2001a[Ramananjaona et al, , 2001b, Dt is chosen by pure numerical experimentation. Here, two adaptative schemes are proposed: one is based on a CFL criterion [Sethian, 1999], where Dt is a function of the spatial discretization and of the maximum amplitude of the speed of motion of the level set; the other assumes a maximum time step chosen so as a properly introduced L 1 norm of the level set is decreased from one iteration to the next.…”

“…

[1] Novel developments of the so-called controlled evolution of level sets [Ramananjaona et al, 2001b], which is devoted to shape identification of homogeneous scattering obstacles buried in a known space from time-harmonic wave field data, are considered herein. The emphasis is twofold: regularization of the geometry of the sought shapeenforced via a speed of motion of the level set in ( pseudo)time and space resulting from the minimization of a properly penalized objective functional; and improvement of the convergence of the scheme itself by a choice of a time step adapted to the evolution of the level set and to the sought decrease of the objective functional.

…”