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.

Atwo-step shape reconstruction method for electromagnetic (EM) tomographyis presented which uses adjoint fields and level sets. The inhomogeneousbackground permittivity distribution and the values of the permittivities insome penetrable obstacles are assumed to be known, and the number, sizes,shapes, and locations of these obstacles have to be reconstructed given noisylimited-view EM data. The main application we address in the paper is theimaging and monitoring of pollutant plumes...

A shape reconstruction method for electrical resistance and capacitance tomography is presented using a level set formulation. In this shape reconstruction approach, the conductivity (or permittivity) values of the inhomogeneous background and the obstacles are assumed to be (approximately) known, but the number, sizes, shapes, and locations of these obstacles have to be recovered from the data. A key point in this shape identification technique is to represent geometrical boundaries of the obstacles by using a level set function. This representation of the shapes has the advantage that the level set function automatically handles the splitting or merging of the objects during the reconstruction. Another key point of the algorithm is to solve the inverse problem of finding the interfaces between two materials using a narrow-band method, which not only decreases the number of unknowns and therefore the computational cost of the inversion, but also tends to improve the condition number of the discrete inverse problem compared to pixel (voxel)-based image reconstruction. Level set shape reconstruction results shown in this article are some of the first ones using experimental electrical tomography data. The experimental results also show some improvements in image quality compared with the pixel-based image reconstruction. The proposed technique is applied to 2D resistance and capacitance tomography for both simulated and experimental data. In addition, a full 3D inversion is performed on simulated 3D resistance tomography data.