Exploiting support information and Lamé curves in 2D inverse scattering problems

Inverse scattering problems are usually cast as optimization ones, in which the global minimum of a cost functional denes the solution. Such a minimization can be tackled by either adopting global or local methods and many dierent approaches have been proposed in the literature. However, while applicability of global schemes (see [1] and references therein) is actually limited by the computational cost that grows exponentially fast with the number of unknowns, local schemes (adopted f.i. in [2-4]) may conversely lead to \false solutions", so that their reliability actually depends on the starting guess. Therefore, electrically and geometrically poor images are often obtained from the inversion process, unless a priori information about targets (positivity constraints, lossless nature, etc..) are enforced. Comprehension of the factors aecting the diculty of the problem may give hints to devise new and more eective imaging strategies. For instance, the analysis of the \degree of non-linearity" of the inverse scattering problem [2] allows to understand how the (approximate) knowledge of position, shape and aver- age permittivity of the unknown targets can strongly improve the eectiveness of the inversion procedures [2]. By exploiting these results, an innovative two-step strategy has been recently proposed and tested on experimental data [5]. In the rst step, the Linear Sampling Method (LSM) [6] is adopted to eectively retrieve the geometrical features of the targets. Then, this information is exploited in the second step, devoted to the electromagnetic characterization of targets. This step is based on local optimization scheme and takes decisive advantage from a proper optimization of the Contrast Source - Extended Born (CS-EB) inversion method [3,5]. Along the same path, in this contribution, we propose a new two-step inversion strategy, wherein, unlike [5], a global optimization method is exploited in the electromagnetic characterization step. As recalled, the crucial point in this case is the \curse of dimensionality", which we eectively tackle by lowering the number of unknown parameters using a representation of the unknown contrast based on Lame curves. As a matter of fact, these curves allow to map a large class of dierent shapes by means of a reduced number of parameters, so that they can be of interest in several applications, ranging from biomedical diagnostics to subsurface sensing. A key role in the success of the overall method is also played by the result achieved in the rst step using LSM [6]. As a matter of fact, the preliminary, possibly rough, shape estimation allows to x the number of targets to be retrieved and their (approximate) locations in the test domain, thus providing not only a reliable starting guess for the following step, but also reducing the search-space in the global minimization scheme, thus remarkably reducing the overall computational cost. Numerical examples conrming the eectiveness of the proposed strategy will be presented at the Conference. [1] M. Pastorino, IEEE Trans. Antennas Propagat., 55, pp. 538 - 548, 2007. [2] O. M. Bucci et al., J. Opt. Soc. Am. A., 18, pp. 1832-1845, 2001. [3] T. Isernia et al., IEEE Geosc. Remote Sens. Letters, 1, pp. 331-337, 2004. [4] P. M. van den Berg et al. , Inv. Probl., 15, pp. 1325-1344, 1999. [5] I. Catapano et al., IEEE Trans. Antennas Propagat., 55, pp.1895-1899, 2007. [6] D. Colton et al., Inv. Probl., 19, pp. S105-S137, 2003.