A two-phase segmentation approach to the impedance tomography problem

In electrical impedance tomography, image reconstruction of the conductivity distribution sigma in a body Omega is estimated using measured voltages at the boundary partial derivative Omega. 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 sigma into 2 phases according to sigma(x) = Sigma(2)(m)(=1) sigma(m)(x)chi(m)(x), where chi(m) is the characteristic function of a sub-domain chi(m). The subdomains Omega(1) and Omega(2) should partition Omega. We assume that Omega(1) can be a disconnected subset of Omega and that Omega(2) is a connected background. The subdomain Omega(1) can have disjoint connected components but these components should be non-adjacent. The estimated segments are given by the connected components of Omega(1). The conductivity sigma(2) is assumed to be known. Using an optimality condition, the conductivity sigma(1) is expressed as a function of chi(1). The inverse problem is solved by minimizing a cost functional of chi(1), which includes the sum of squared differences between measured and simulated boundary voltages as well as the regularizing total variation of chi(1). Using a descent method, an update for chi(1) is proposed. Examples using topological derivatives to obtain an initial estimate for chi(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.