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  • Author: Guillaume Bal x
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Abstract

This paper concerns the imaging of a complex-valued anisotropic tensor γ=σ+𝜾ωε from knowledge of several inter magnetic fields H, where H satisfies the anisotropic Maxwell system on a bounded domain X2 with prescribed boundary conditions on X. We show that γ can be uniquely reconstructed with a loss of two derivatives from errors in the acquisition H. A minimum number of five well-chosen functionals guaranties a local reconstruction of γ in dimension two. The explicit inversion procedure is presented in several numerical simulations, which demonstrate the influence of the choice of boundary conditions on the stability of the reconstruction. This problem finds applications in the medical imaging modalities Current Density Imaging and Magnetic Resonance Electrical Impedance Tomography.

Abstract.

In this paper we develop a convergence analysis in an infinite dimensional setting of the Levenberg–Marquardt iteration for the solution of a hybrid conductivity imaging problem. The problem consists in determining the spatially varying conductivity σ from a series of measurements of power densities for various voltage inductions. Although this problem has been very well studied in the literature, convergence and regularizing properties of iterative algorithms in an infinite dimensional setting are still rudimentary. We provide a partial result under the assumptions that the derivative of the operator, mapping conductivities to power densities, is injective and the data is noise-free. Moreover, we implemented the Levenberg–Marquardt algorithm and tested it on simulated data.