A direct three dimensional EIT reconstruction algorithm based on complex geometrical optics solutions and a nonlinear scattering transform is presented and implemented for spherically symmetric conductivity distributions. The scattering transform is computed both with a Born approximation and from the forward problem for purposes of comparison. Reconstructions are computed for several test problems. A connection to Calderón's linear reconstruction algorithm is established, and reconstructions using both methods are compared.
Reconstruction from boundary measurements for less regular conductivities
本文遵循Nachman的思想以及Haberman和Tataru的思想，从Dirichlet-to-Neumann映射$Λ_γ$出发，在Lipschitz域$Ω$中重建$C^1$电导率$γ$或具有足够小的$|\nabla logγ|$值的Lipchitz电导率$γ$.在附录中，作者和R. M. Brown从Dirichlet-Neumann映射$Λ_γ$恢复了Lipschitz域边界处$C^1$ -电导率的梯度。
In this paper we show that following Nachman's method we can still reconstruct complex conductivities in $C^{1,1}$ from its Dirichlet-to-Neumann map in three and higher dimensions. For such, we analyze all of the results in Nachman and pinpoint what really needs to be shown for complex conductivities. Moreover, we also obtain low frequency estimates for $C^{1,1}$-boundaries following the approach established by Cornean, Knudsen and Siltanen. As far as we aware, this is the first reconstruction procedure for complex conductivities, even though the proof follows trivially by extending some of Na…
Two reconstruction methods of Electrical Impedance Tomography (EIT) are numerically compared for nonsmooth conductivities in the plane based on the use of complex geometrical optics (CGO) solutions to D-bar equations involving the global uniqueness proofs for Calderón problem exposed in [Nachman; Annals of Mathematics 143, 1996] and [Astala and Päivärinta; Annals of Mathematics 163, 2006]: the Astala-Päivärinta theory-based "low-pass transport matrix method" implemented in [Astala et al.; Inverse Problems and Imaging 5, 2011] and the "shortcut method" which considers ingredients of both theori…
A direct reconstruction algorithm based on Calderón's linearization method for the reconstruction of isotropic conductivities is proposed for anisotropic conductivities in two-dimensions. To overcome the non-uniqueness of the anisotropic inverse conductivity problem, the entries of the unperturbed anisotropic tensors are assumed known \emph{a priori}, and it remains to reconstruct the multiplicative scalar field. The quasi-conformal map in the plane facilitates the Calderón-based approach for anisotropic conductivities. The method is demonstrated on discontinuous radially symmetric conductivit…
In this paper, we propose and analyze a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. It is based on a regularized output least-squares formulation with the standard $L^2(Ω)^{d,d}$ penalty, which is then discretized by the standard Galerkin finite element method. We establish the continuity and differentiability of the forward map with respect to the conductivity tensor in the $L^p(Ω)^{d,d}$-norms, the existence of minimizers and optimality systems of the regula…
Electrical Impedance Tomography gives rise to the severely ill-posed Calderón problem of determining the electrical conductivity distribution in a bounded domain from knowledge of the associated Dirichlet-to-Neumann map for the governing equation. The uniqueness and stability questions for the three-dimensional problem were largely answered in the affirmative in the 1980's using complex geometrical optics solutions, and this led further to a direct reconstruction method relying on a non-physical scattering transform. In this paper, the reconstruction problem is taken one step further towards p…
