Local recovery of a piecewise constant anisotropic conductivity in EIT on domains with exposed corners

TL;DR

This paper demonstrates that local reconstruction of a piecewise constant anisotropic conductivity in EIT is almost surely possible near a known background, using boundary measurements on domains with corners and known polygonal decompositions.

Contribution

It proves the almost sure local recoverability of anisotropic conductivities in EIT on polygonal domains with corners, extending previous results to more complex geometries.

Findings

Injectivity of the Fréchet derivative is almost surely true near known conductivities.

Local recovery is possible on domains with corners using boundary measurements.

The method applies to known polygonal decompositions with corners.

Abstract

We study the local recovery of an unknown piecewise constant anisotropic conductivity in EIT (electric impedance tomography) on certain bounded Lipschitz domains $\Omega$ in $\mathbb{R}^2$ with corners. The measurement is conducted on a connected open subset of the boundary $\partial\Omega$ of $\Omega$ containing corners and is given as a localized Neumann-to-Dirichlet map. The above unknown conductivity is defined via a decomposition of $\Omega$ into polygonal cells. Specifically, we consider a parallelogram-based decomposition and a trapezoid-based decomposition. We assume that the decomposition is known, but the conductivity on each cell is unknown. We prove that the local recovery is almost surely true near a known piecewise constant anisotropic conductivity $\gamma_0$. We do so by proving that the injectivity of the Fr\'echet derivative $F'(\gamma_0)$ of the forward map $F$, say, at $\gamma_0$ is almost surely true. The proof presented, here, involves defining different classes of decompositions for $\gamma_0$ and a perturbation or contrast $H$ in a proper way so that we can find in the interior of a cell for $\gamma_0$ exposed single or double corners of a cell of $\mbox{supp}H$ for the former decomposition and latter decomposition, respectively. Then, by adapting the usual proof near such corners, we establish the aforementioned injectivity.