A two-phase segmentation approach to the impedance tomography problem

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