Imaging of anisotropic conductivities from current densities in two dimensions

TL;DR

This paper presents a method to uniquely reconstruct anisotropic conductivity tensors in two dimensions from internal current density measurements, demonstrating stability and practical reconstruction techniques with applications in medical imaging.

Contribution

It introduces a local inversion method for anisotropic conductivities using four specific internal functionals and establishes conditions for stable reconstruction via complex geometric optics solutions.

Findings

Unique reconstruction from four functionals

Stability depends on boundary conditions

Numerical simulations validate the approach

Abstract

We consider the imaging of anisotropic conductivity tensors $\gamma=(\gamma_{ij})_{1\leq i,j\leq 2}$ from knowledge of several internal current densities $\mathcal{J}=\gamma\nabla u$ where $u$ satisfies a second order elliptic equation $\nabla\cdot(\gamma\nabla u)=0$ on a bounded domain $X\subset\mathbb{R}^2$ with prescribed boundary conditions on $\partial X$. We show that $\gamma$ can be uniquely reconstructed from four {\em well-chosen} functionals $\mathcal{J}$ and that noise in the data is differentiated once during the reconstruction. The inversion procedure is local in the sense that (most of) the tensor $\gamma(x)$ can be reconstructed from knowledge of the functionals $\mathcal{J}$ in the vicinity of $x$. We obtain the existence of an open set of boundary conditions on $\partial X$ that guaranty stable reconstructions by using the technique of complex geometric optics (CGO) solutions. 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 modality called Current Density Imaging or Magnetic Resonance Electrical Impedance Tomography.