On reconstruction of complex-valued once differentiable conductivities

Abstract

The classical ∂-method has been generalized recently [13], [14] to be used in the presence of exceptional points. We apply this generalization to solve Dirac inverse scattering problem with weak assumptions on smoothness of potentials. As a consequence, this provides an effective method of reconstruction of complex-valued one time differentiable conductivities in the inverse impedance tomography problem.

CGO-Faddeev approach for complex conductivities with regular jumps in two dimensions

“…Hereby, we want to point out that the Dirichlet-to-Neumann map determines the scattering data uniquely can be proven similarly to [23] (see Section 4). Now, Theorem 3.5 will even provide a reconstruction formula for the potential q in so-called proper admissible points.…”

“…Proof. We divide the integral (40) 1 |λ| Γ + e ln |λ|λs(z−w) 2 µ 2 (z)dz, into two pieces, according to the decomposition of µ 2 given by formula (23), that is…”

“…If the frequency ω is sufficiently small, then one can approximate γ by a real-valued function. Previous approaches for EIT with isotropic admittivity, which were in active use for the last three decades, can be divided in two groups: closed-form solution or sample methods where we refer to reviews [9] and [29] for further details as well as latest articles [5,3,6,8,11,12,16,17,18,19,21,22,23,24,32]. These approaches do not fully coincide, for example the Linear Sampling Method (LSM) allows the reconstruction of the parameter's jump location, but it assumes the medium is known outside of the jump.…”

“…[12], [16]. In turn, E. Lakshtanov, R. Novikov and B. Vainberg also got some closed-form reconstruction/uniqueness results [22], [23]. Furthermore, E. Lakshtanov, and B.Vainberg got a feeling that LSM and CGO methods can be applied simultaneously to reconstruct the shape of the jump even if the potential is unknown.…”

Uniqueness in the Inverse Conductivity Problem for Complex-Valued Lipschitz Conductivities in the Plane

“…On the other hand, the work of Francini [10], where the ideas of [7] were extended to deal with complex conductivities with small imaginary part, are not applicable to general complex conductivities due to possible existence of the so called exceptional points. In [13], Lakstanov and Vainberg extended the ideas of [12] to apply the ∂-method in the presence of exceptional points and reconstructed generic conductivities under the assumption that γ − 1 ∈ W 1,p comp (R 2 ), p > 4, and F (∇γ) ∈ L 2−ε (R 2 ) (here F is the Fourier transform). In this paper, we will prove that complex-valued Lipschitz conductivities are uniquely determined by information on the boundary.…”