Theoretical and computational aspects of 2-D inverse profiling

Iterative Solution of Field Problems with a Varying Physical Parameter

“…Until now, we have only dealt with the inverse-profiling problem for a twodimensionally inhomogeneous, lossy dielectric cylinder discussed in [16]. As mentioned above, for this configuration the full scattering operator can be determined at the cost of a few individual field computations.…”

“…This configuration was inspired by a scanner that is presently under construction at CNRS/Supélec, Gif-sur-Yvette, Paris, France [25]. The aim of our research is to generalize the linear and nonlinear inversion schemes for a dielectric cylinder in a homogeneous medium as outlined in [16] to the geometry shown in Figure 13. Conducting Cylinder Figure 13.…”

3d Microwave Tomography With Huber Regularization Applied to Realistic Numerical Breast Phantoms

“…In this case a regularization is applied to the permittivity update in each iteration of the optimization: Tikhonov regularization -similar to [11,13] -in [14,34,37], a combination of L1 and L2 norm sparsity promoting regularization in [38] and other compressive sensing schemes in, e.g., [44] and a conjugate gradient for least squares (CGLS) algorithm in [3,33,39,40]. In the present paper, as in [26,42,43,45] for microwave breast imaging and as in, e.g., [19,30,[46][47][48][49][50][51][52] for microwave imaging, we adopt a regularized cost function consisting of the data fit term and a regularization term. The regularization term allows for easy incorporation of a priori information on the complex permittivity profile.…”

“…Quadratic (L2-norm) smoothing regularization [19,46], or a Tikhonov potential function [53,55], penalizes small differences between neighboring permittivity values, but also smooths out true discontinuities; it is applied to 3D simulated breast imaging in an additive multiplicative fashion in [43]. For (quasi) piecewise-constant profiles strong edge-preserving capabilities are shown by value picking (VP) regularization [48,56], which is a nonspatially structured approach, and by piecewise-smoothed VP regularization (a combined nonspatially-spatially structured approach) [49], when applied to measured data from the 3D and 2D Fresnel databases, respectively, but these techniques perform best for a limited number of different permittivity values.…”

The Contrast Source-Extended Born Model for 2d Subsurface Scattering Problems

“…Hence, approximate solutions, such as the Extended Born (EB) approximation [4] and its heuristic extensions [5] or algebraic preconditioners [6,7] are worth studying as means to improve the convergence rate. The inverse scattering problem is non-linear and ill-posed [8] and it is usually cast as the global optimization of a suitable cost functional [9][10][11][12][13][14][15].…”