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

• Published in: AIP conference proceedings Vol. 1707; no. 1
• Main Authors: Mendoza, Renier, Keeling, Stephen
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Melville American Institute of Physics 11.02.2016
• Subjects: Boundary conditions; Characteristic functions; Convergence; Electrical impedance; Electrical resistivity; Finite element method; Formulations; Image reconstruction; Image segmentation; Inverse problems; Laplace equation; Mathematical analysis; Tomography
• ISSN: 0094-243X; 1551-7616
• DOI: 10.1063/1.4940841

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 = 1 M σ m ( x ) χ m ( x ) , where χ m is the characteristic function of a subdomain Ω m such that Ω m ∩ Ω n = Ø, m ≠ n and Ω = ∪ m = 1 M Ω m . 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 value σ m is expressed as a function of χ m . The total variation of χ m is then introduced to regularize the resulting cost functional. Using a descent method, an update for χ m is proposed. In this work, the Finite Element Method (FEM) convergence of all the resulting variational formulations are studied. A real analytic mollification of χ m is introduced to guarantee convergence.