We give an overview of recent techniques which use a level set representation of shapes for solving inverse scattering problems. The main focus is on electromagnetic scattering using dierent popular models, as for example Maxwell's equations, TM-polarized and TE-polarized waves, impedance tomography, a transport equation or its diusion approximation. These models also are representative for a broader class of inverse problems. Starting out from the original binary approach of Santosa for solving the corresponding shape reconstruction problem, we successively develop more recent generalizations, as for example using colour or vector level sets. Shape sensitivity analysis and topological derivatives are discussed as well in this framework. Moreover, various techniques for incorporating regularization into the shape inverse problem using level sets are demonstrated, which also include the choice of subclasses of simple shapes, such as ellipsoids, for the inversion. Finally, we present various numerical examples in 2D and in 3D for demonstrating the performance of level set techniques in realistic applications.

In this paper a MUSIC (standing for multiple signal classification) algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency is developed. The underlying application area which motivated this work is the imaging of antipersonnel mines from electromagnetic data, formulated as an inverse scattering problem for the Helmholtz equation. The algorithm makes use of an asymptotic expansion of the scattering amplitude. A derivation of the leading-order term in this asymptotic expansion and its application for designing a MUSIC type of algorithm are presented. The viability of this algorithm is documented by a variety of numerical results.

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 dielectric parameters of the obstacle and embedding materials are known and piecewise constant. That is, the shape (here, the contour) of the obstacle is sufficient for its full characterization. The inverse scattering problem is then to determine the contour from the knowledge of the scattered field measured for several locations of the sources and/or frequencies. An iterative process is implemented: given an initial contour, this contour is progressively evolved such as to minimize the residual in the data fit. This algorithm presents two main important points. The first concerns the choice of the transformation enforced on the contour. We will show that this involves the design of a velocity field 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 great use to handle in a natural way splitting or merging of obstacles along the iterative process. The evolution of this level-set is controlled by a Hamilton-Jacobi-type equation which will be solved by using an 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.

In this paper we consider the localization of a collection of small, three‐dimensional bounded homogeneous inclusions via time‐harmonic electromagnetic means, typically using arrays of electric or magnetic dipole transmitters and receivers with given polarization(s) at some distance from the collection, possibly also lying in the far field. The inclusions, somewhat apart or closely spaced, are buried within a homogeneous medium, and are of arbitrary contrast of permittivity, conductivity, and permeability vis‐à‐vis this embedding medium. The problem is formulated as an inverse scattering problem for the full Maxwell equations and it involves a robust asymptotic modeling of the multistatic response matrix. No specific application is studied at this stage, but characterization of obstacles in subsoils, nondestructive evaluation of man‐made structures, and medical imaging are primary fields of application envisaged. The proposed approach uses a MUSIC (multiple signal classification)‐type algorithm, and it yields fast numbering, accurate localization, and estimates of the electromagnetic and geometric parameters (polarization tensors) of the inclusions. The mathematical machinery is detailed first, some specific attention being given to triaxial ellipsoidal inclusions and degenerate spherical shapes (for the latter known results are retrieved). Then, the viability of this algorithm—which would be easily extended to planarly layered environments by introduction of their Green’s functions—is documented by a variety of numerical results from synthetic, noiseless, and severely noisy field data.

A family of new integral equations (NIE) is proposed in this paper, which are transformed from the original Lippmann-Schwinger integral equation. It can be shown that the NIE can effectively reduce the nonlinearity of inverse scattering problems by reducing the global nonlinear effects, introduced by the multiple scattering behaviors, in estimating the contrast. Equipping the previously proposed twofold subspace-based optimization method with such NIE, the new inversion method is able to solve inverse scattering problems with strong scatterers, like with high contrast and/or large dimensions (in terms of wavelength) ones. Furthermore, such a family of NIE could provide a convenient tool to appraise reconstructed results. Several representative numerical tests are carried out, using both synthetic and experimental data, to verify the efficacy of the new inversion method.