Imaging of complex-valued tensors for two-dimensional Maxwell’s equations

Abstract

Abstract This paper concerns the imaging of a complex-valued anisotropic tensor γ = σ + 𝜾 ⁢ ω ⁢ ε ${\gamma=\sigma+\boldsymbol{\iota}\omega\varepsilon}$ from knowledge of several inter magnetic fields H, where H satisfies the anisotropic Maxwell system on a bounded domain X ⊂ ℝ 2 ${X\subset\mathbb{R}^{2}}$ with prescribed boundary conditions on ∂ ⁡ X ${\partial X}$ . 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.