RECONSTRUCTION OF A 2-DIMENSIONAL BINARY OBSTACLE BY CONTROLLED EVOLUTION OF A LEVEL-SET

We are concerned with the retrieval of the unknown cross section of a homogeneous cylindrical obstacle embedded in a homogeneous medium and illuminated by time-harmonic electromagnetic line sources. The dielect ric parameters of the obstacle and embedding materials are known and p iecewise constant. That is, the shape (here, the contour) of the obsta cle is sufficient for its full characterization. The inverse scatterin g problem is then to determine the contour from the knowledge of the s cattered field measured for several locations of the sources and/or fr equencies. An iterative process is implemented: given an initial conto ur, this contour is progressively evolved such as to minimize the resi dual in the data fit. This algorithm presents two main important point s. The first concerns the choice of the transformation enforced on the contour. We will show that this involves the design of a velocity fie ld whose expression only requires the resolution of an adjoint problem at each step. The second concerns the use of a level-set function in order to represent the obstacle. This level-set function will be of gr eat use to handle in a natural way splitting or merging of obstacles a long the iterative process. The evolution of this level-set is control led by a Hamilton-Jacobi-type equation which will be solved by using a n appropriate finite-difference scheme. Numerical results of inversion obtained from both noiseless and noisy synthetic data illustrate the behaviour of the algorithm for a variety of obstacles.