Local recovery of a piecewise constant anisotropic conductivity in EIT on domains with exposed corners. (1st February 2023) - de Hoop, Maarten V; Furuya, Takashi; Lin, Ching-Lung; Nakamura, Gen; Vashisth, Manmohan
Abstract: We study the local recovery of an unknown piecewise constant anisotropic conductivity in electric impedance tomography on certain bounded Lipschitz domains Ω in R 2 with corners. The measurement is conducted on a connected open subset of the boundary ∂ Ω of Ω containing corners and is given as a localized Neumann-to-Dirichlet map. The above unknown conductivity is defined via a decomposition of Ω 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 γ 0 . We do so by proving that the injectivity of the Fréchet derivative F ′ ( γ 0 ) of the forward map F, say, at γ 0 is almost surely true. The proof presented, here, involves defining different classes of decompositions for γ 0 and a perturbation or contrast H in a proper way so that we can find in the interior of a cell for γ 0 exposed single or double corners of a cell of supp H for the former decomposition and latter decomposition, respectively. Then, by adapting the usual proof near such corners, we establish the aforementioned injectivity.