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

TL;DR

This paper presents a method for uniquely reconstructing complex anisotropic tensors in 2D Maxwell's equations from magnetic field data, with explicit inversion procedures and numerical validation, relevant for medical imaging techniques.

Contribution

It introduces a new approach for local reconstruction of complex tensors in Maxwell's equations with explicit inversion and stability analysis.

Findings

Unique reconstruction of tensor b3 0 with two-derivative loss.

Minimum of five functionals needed for local reconstruction.

Numerical simulations demonstrate boundary condition effects on stability.

Abstract

This paper concerns the imaging of a complex-valued anisotropic tensor {\gamma} = {\sigma}+{\iota}{\omega}{\epsilon} from knowledge of several inter magnetic fields H where H satisfies the anisotropic Maxwell system on a bounded domain with prescribed boundary conditions. We show that {\gamma} 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 {\gamma} in dimension two. The explicit inversion procedure is presented in several numerical simulations, which demonstrate the influence of the choice 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.