A two-phase segmentation approach to the impedance tomography problem. (24th November 2016) - Mendoza, Renier; Keeling, Stephen
Abstract: In electrical impedance tomography, image reconstruction of the conductivity distribution σ in a body Ω is estimated using measured voltages at the boundary ∂ Ω . This is done by solving an inverse problem associated with a generalized Laplacian equation. We approach this problem by using a multi-phase segmentation method. We partition σ into 2 phases according to σ ( x ) = ∑ m = 1 2 σ m ( x ) χ m ( x ), where χ m is the characteristic function of a subdomain Ω m . The subdomains Ω 1 and Ω 2 should partition Ω . We assume that Ω 1 can be a disconnected subset of Ω and that Ω 2 is a connected background. The subdomain Ω 1 can have disjoint connected components but these components should be non-adjacent. The estimated segments are given by the connected components of Ω 1 . The conductivity σ 2 is assumed to be known. Using an optimality condition, the conductivity σ 1 is expressed as a function of χ 1 . The inverse problem is solved by minimizing a cost functional of χ 1, which includes the sum of squared differences between measured and simulated boundary voltages as well as the regularizing total variation of χ 1 . Using a descent method, an update for χ 1 is proposed. Examples using topological derivatives to obtain an initial estimate for χ 1 are also presented. It will be shown that the proposed method can be used to estimate separate inclusions by using only a single phase function, i.e., the number of inclusions need not be known in advance.