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 different popular models, as for example Maxwell's equations, TM-polarized and TE-polarized waves, impedance tomography, a transport equation or its diffusion 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.

A t wo-step shape reconstruction method for electromagnetic (EM) tomography is presented which uses adjoint elds and level sets. The inhomogeneous background permittivity distribution and the values of the permittivities in some penetrable obstacles are assumed to be known, and the number, sizes, shapes, and locations of these obstacles have to be reconstructed given noisy limited-view EM data. The main application we address in the paper is the imaging and monitoring of pollutant plumes in environmental cleanup sites based on cross-borehole EM data. The rst step of the reconstruction scheme makes use of an inverse scattering solver which rst recovers equivalent scattering sources for a number of experiments, and then calculates from these an approximation for the permittivity distribution in the medium. The second step uses this result as an initial guess for solving the shape reconstruction problem. A key point in this second step is the fusion of the 'level set technique' for representing the shapes of the reconstructed obstacles, and an 'adjoint eld technique' for solving the nonlinear inverse problem. In each step, a forward and an adjoint Helmholtz problem are solved based on the permittivity d i s t r ibution which corresponds to the latest best guess for the representing level set function. A correction for this level set function is then calculated directly by combining the results of these two r u n s. Numerical experiments are presented which s h o w that the derived method is able to recover one or more objects with nontrivial shapes given noisy cross-borehole EM data.

Optical tomography is modelled by the inverse problem of the time-dependent linear transport equation in n spatial dimensions (n = 2, 3). Based on the measurements which consist of some functionals of the outgoing density at the boundary ∂ for different sources q j , j = 1, . . . , p, two coefficients of the equation, the absorption coefficient σ a (x) and the scattering coefficient b(x), are reconstructed simultaneously inside . Starting out from some initial guess (σ a , b) T for these coefficients, the transport-backtransport (TBT) algorithm calculates the difference between the computed and the physically given measurements for a fixed source q j by solving a 'direct' transport problem, and then transports these residuals back into the medium by solving a corresponding adjoint transport problem. The correction (h, k) T j to the guess (σ a , b) T is calculated from the densities of the direct and the adjoint problem inside the medium. Doing this for all source positions q j , j = 1, . . . , p, one after the other yields one sweep of the algorithm. Numerical experiments are presented for the case when n = 2. They show that the TBT-method is able to reconstruct and to distinguish between scattering and absorbing objects in the case of large mean free path (which corresponds to x-ray tomography with scattering). In the case of very small mean free path (which corresponds to optical tomography), scattering and absorbing objects are located during the early sweeps, but phantoms are built up in the reconstructed scattering coefficient at positions where an absorber is situated and vice versa.

Electromagnetic imaging is modelled as an inverse problem for the 3D system of Maxwell's equations of which the isotropic conductivity distribution in the domain of interest has to be reconstructed. The main application we have in mind is the monitoring of conducting contaminant plumes out of surface and borehole electromagnetic imaging data. The essential feature of the method developed here is the use of adjoint fields for the reconstruction task, combined with a splitting of the data into smaller groups which define subproblems of the inversion problem. The method works iteratively, and can be considered as a nonlinear generalization of the algebraic reconstruction technique in x-ray tomography. Starting out from some initial guess for the conductivity distribution, an update for this guess is computed by solving one forward and one adjoint problem of the 3D Maxwell system at a time. Numerical experiments are performed for a layered background medium in which one or two localized (3D) inclusions are immersed. These have to be monitored out of surface to borehole and cross-borehole electromagnetic data. We show that the algorithm is able to recover a single inclusion in the earth which has high contrast to the background, and to distinguish between two separated inclusions in the earth given certain borehole geometries.

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 narrowband 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.