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