## 1. Introduction

[2] Level set methods are increasingly used to compute and analyze the motion of a contour. Investigated in-depth by *Osher and Senthian* [1988] for the modeling of the propagation of fronts, they are nowadays considered for various applications as is illustrated, e.g., by *Osher and Fedkiw* [2001] and references therein. One of the major advantages of these methods appears to be their capability of merging or splitting obstacles without topological constraints.

[3] The so-called *controlled evolution of level sets* whose novel developments are considered herein is based on a combination of two methods:

- The level set representation of shapes [e.g.,
*Osher and Senthian*, 1988], which was first proposed by*Santosa*[1996] for an inversion procedure, enables us to seek the closed contours of a scatterer as the zero-level of a level set. This level set is evolved in space and (pseudo)time in a prescribed search domain so as a suitably chosen objective function is minimized. The evolution is governed by a Hamilton–Jacobi equation which links, through a velocity motion, its partial derivative with respect to time to the norm of its gradient in space. - The Speed or Velocity Method, whose theoretical features are now well established for the solution of design and sensitivity problems [
*Sokolowski and Zolésio*, 1992], enables us to choose a velocity such as the time derivative of the objective functional is negative or null, and consequently to decrease this functional with time (pending limitations that might arise at the discretization stage).

[4] The present paper is focused on an electromagnetic inverse scattering problem where one has to retrieve a homogeneous, closed obstacle (a possibly multiply connected lossy dielectric cylinder with same *z*-orientation and unknown cross-sectional shape in the orthogonal (*x*-*y*) plane) which is buried in a known lossy dielectric half-space. This must be done from scattered fields that are observed above this half-space when the said obstacles are probed by prescribed wave fields. More specifically, the polarization of the wave field is transverse electric (TE), i.e., the only nonnull component of the magnetic (*H*) field (incident as well as scattered) is parallel to the *z* axis. Pioneered by *Litman et al.* [1998] for the reconstruction of penetrable obstacles in free space in the dual Transverse Magnetic polarization (the only nonnull component of the electric (*E*) field is parallel to the cylinders' axes), the controlled evolution of level sets has been extended by *Ramananjaona et al.* [2001a] to the retrieval of buried obstacles in both TM and TE polarizations and it has been applied to the reconstruction of obstacles in free space from experimental microwave data [*Ramananjaona et al.*, 2001a]. Here, the purpose of our investigation is twofold.

[5] First, regularization procedures that are involving either the length of the cross-sectional contour or the area of the cross-section itself are proposed in order to avoid unrealistically jagged boundary curves while still preserving the ability to produce multiply connected objects. Such a regularization is enforced via the objective functional and consequently is shown to only entail changes in the velocity motion, with the occurrence of either curvature-driven or advection terms—in effect as it was suggested earlier by *Santosa* [1996] from inspection of existing material on front propagation.

[6] Second, a numerical scheme for the choice of the time step Δ*t* (needed at the discretization stage of the Hamilton–Jacobi equation involved in the evolution of the level set) is put forth by combining theoretical analysis and numerical experimentation. 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, 2001b], Δ*t* is chosen by pure numerical experimentation. Here, two adaptative schemes are proposed: one is based on a CFL criterion [*Sethian*, 1999], where Δ*t* 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*^{∞} norm of the level set is decreased from one iteration to the next.

[7] The plan of the paper is as follows. In section 2 geometry and the TE formulation of the wave field are sketched. In section 3 key features of the inversion method are reminded. In section 4 regularization of the inversion via geometrical constraints is discussed. In section 5 adaptative time steps are introduced. In section 6 a number of illustrative numerical examples are given. A short outline of forthcoming studies follows in section 7.