Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields

TL;DR

This paper demonstrates the unique reconstruction of complex anisotropic tensors in the Maxwell system from internal magnetic field data, with implications for medical imaging techniques like MRI and electrical impedance tomography.

Contribution

It introduces a method for reconstructing complex-valued anisotropic tensors from internal magnetic field measurements, including a local reconstruction approach and partial results for symmetric tensors.

Findings

Unique reconstruction of $oldsymbol{ ext{γ}}$ with two derivatives loss.

Minimum of 6 internal magnetic field measurements needed.

Application to medical imaging modalities like MRI and EIT.

Abstract

This paper concerns the reconstruction of a complex-valued anisotropic tensor $\gamma=\sigma+\i\omega\varepsilon$ from knowledge of several internal 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 of $H$. A minimum number of 6 such functionals is sufficient to obtain a local reconstruction of $\gamma$. In the special case where $\gamma$ is close to a scalar tensor, boundary conditions are chosen by means of complex geometric optics (CGO) solutions. For arbitrary symmetric tensors $\gamma$, a Runge approximation property is used to obtain partial results. This problem finds applications in the medical imaging modalities Current Density Imaging and Magnetic Resonance Electrical Impedance Tomography.