FEM convergence of a segmentation approach to the electrical impedance tomography problem

In Electrical Impedance Tomography (EIT), different current patterns are injected to the unknown object through the electrodes attached at the boundary ∂ Ω of Ω. The corresponding voltages V are then measured on its boundary surface. Based on these measured voltages, the image reconstruction of the conductivity distribution σ is done by solving an inverse problem of a generalized Laplace equation subject to a homogeneous Neumann boundary condition. In other words, with known V, we seek to solve for the typically piecewise values of σ, from which the geometry of internal objects may be inferred. We approach this problem by using a multi-phase segmentation method. We express σ as σ(x)=∑m=1Mσm(x)χm(x), where χm is the characteristic function of a subdomain Ωm such that Ωm ∩ Ωn = O, m ≠ n and Ω=∪m=1MΩm. The expected number of phases for Ω is M, where M = 2 for this work. The number of segments is the number of connected components of the subdomains. Using a calculated optimality condition, the conductivity va...