Reconstruction of small and extended regions in EIT with a Robin transmission condition

Abstract

We consider an inverse shape problem coming from electrical impedance tomography with a Robin transmission condition. In general, a boundary condition of Robin type models corrosion. In this paper, we study two methods for recovering an interior corroded region from electrostatic data. We consider the case where we have small volume and extended regions. For the case where the region has small volume, we will derive an asymptotic expansion of the current gap operator and prove that a MUSICtype algorithm can be used to recover the region. In the case where one has an extended region, we will show that the regularized factorization method can be used to recover said region. Numerical examples will be presented for both cases in two dimensions in the unit circle.

Regularization of the factorization method with applications to inverse scattering

“…We are motivate by the work in [24,27] where regularized variants of the factorization method were developed. This method has been applied to diffuse optical tomography [24], electrical impedance tomography [22] and inverse scattering [25]. In the aforementioned papers, the results hold when one has the unperturbed data operator whereas we wish to extend the results when one only has access to the perturbed data operator.…”

“…For an application of this method, we have applied the regularized factorization method to an inverse scattering problem for recovering a scatterer from the far-field data. As in [22,24] we know that this method can be applied to other imaging modalities such as electrical and diffuse optical tomography.…”

“…Theorem 4.2. Assume that F δ : L 2 (S d−1 ) −→ L 2 (S d−1 ) is a perturbation of the far-field operator F given by (22) with F δ being a positive compact operator. Provided that k is not a transmission eigenvalue with (n − 1) and (η) both uniformly positive (or negative) definite, then z ∈ D ⇐⇒ lim inf…”

“…Over the years the factorization method has become a useful analytical and computational tool for shape reconstruction. See the papers [9,11,13,14,16,22,23,27,37,40] and the references therein for applications of the factorization method for solving inverse shape problems for elliptic and hyperbolic PDEs.…”

Regularized factorization method for a perturbed positive compact operator applied to inverse scattering

“…Over the years the factorization method has become a useful analytical and computational tool for shape reconstruction. See the papers [8,9,11,12,14,20,21,25,35,38] and the references therein for applications of the factorization method for solving inverse shape problems for elliptic and hyperbolic partial differential equations(PDEs).…”