Local recovery of a piecewise constant anisotropic conductivity in EIT on domains with exposed corners. (1st February 2023)

• Record Type: Journal Article
• Authors: de Hoop, Maarten V; Furuya, Takashi; Lin, Ching-Lung; Nakamura, Gen; Vashisth, Manmohan
• Is Part Of: Inverse problems. Volume 39:Number 2(2023)
• Publication Date: 2023-02-01
• Subjects: EIT -- domain decomposition -- corners -- anisotropy -- local recovery -- finite measurements; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/acb008
• Languages: English
• ISSNs: 0266-5611

Abstract

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.