Reconstruction of less regular conductivities in the plane

We study the inverse conductivity problem of how to reconstruct an isotropic electrical conductivity distribution $\gamma$ in an object from static electrical measurements on the boundary of the object. We give an exact reconstruction algorithm for the conductivity \gamma\in C^{1+\epsilon}(\ol \Om) in the plane domain $\Omega$ from the associated Dirichlet to Neumann map on \partial \Om. Hence we improve earlier reconstruction results. The method used relies on a well-known reduction to a first order system, for which the \ol\partial-method of inverse scattering theory can be applied

Stability in Conductivity Imaging from Partial Measurements of One Interior Current

We prove a stability result in the hybrid inverse problem of recovering the electrical conductivity from partial knowledge of one current density field generated inside a body by an imposed boundary voltage. The region where interior data stably reconstructs the conductivity is well defined by a combination of the exact and perturbed data

A weighted minimum gradient problem with complete electrode model boundary conditions for conductivity imaging

We consider the inverse problem of recovering an isotropic electrical conductivity from interior knowledge of the magnitude of one current density field generated by applying current on a set of electrodes. The required interior data can be obtained by means of MRI measurements. On the boundary we only require knowledge of the electrodes, their impedances, and the corresponding average input currents. From the mathematical point of view, this practical question leads us to consider a new weighted minimum gradient problem for functions satisfying the boundary conditions coming from the Complete Electrode Model of Somersalo, Cheney and Isaacson. This variational problem has non-unique solutions. The surprising discovery is that the physical data is still sufficient to determine the geometry of the level sets of the minimizers. In particular, we obtain an interesting phase retrieval result: knowledge of the input current at the boundary allows determination of the full current vector field from its magnitude. We characterize the non-uniqueness in the variational problem. We also show that additional measurements of the voltage potential along one curve joining the electrodes yield unique determination of the conductivity. A nonlinear algorithm is proposed and implemented to illustrate the theoretical results.Comment: 20 pages, 5 figure

Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions

We consider the problem of recovering an isotropic conductivity outside some perfectly conducting or insulating inclusions from the interior measurement of the magnitude of one current density field $|J|$. We prove that the conductivity outside the inclusions, and the shape and position of the perfectly conducting and insulating inclusions are uniquely determined (except in an exceptional case) by the magnitude of the current generated by imposing a given boundary voltage. We have found an extension of the notion of admissibility to the case of possible presence of perfectly conducting and insulating inclusions. This also makes it possible to extend the results on uniqueness of the minimizers of the least gradient problem $F(u)=\int_{\Omega}a|\nabla u|$ with $u|_{\partial \Omega}=f$ to cases where $u$ has flat regions (is constant on open sets)

On the XX-ray transform of planar symmetric 2-tensors

In this paper we study the attenuated $X$-ray transform of 2-tensors supported in strictly convex bounded subsets in the Euclidean plane. We characterize its range and reconstruct all possible 2-tensors yielding identical $X$-ray data. The characterization is in terms of a Hilbert-transform associated with $A$-analytic maps in the sense of Bukhgeim.Comment: 1 figure. arXiv admin note: text overlap with arXiv:1411.492