TOPOLOGICAL DERIVATIVE METHOD FOR ELECTRICAL IMPEDANCE TOMOGRAPHY PROBLEMS

In the field of shape and topology optimization the new concept is the topological derivative of a given shape functional. The asymptotic analysis is applied in order to determine the topological derivative of shape functionals for elliptic problems. The topological derivative (TD) is a tool to measure the influence on the specific shape functional of insertion of small defect into a geometrical domain for the elliptic boundary value problem (BVP) under considerations. The domain with the small defect stands for perturbed domain by topological variations. This means that given the topological derivative, we have in hand the first order approximation with respect to the small parameter which governs the volume of the defect for the shape functional evaluated in the perturbed domain. TD is a function defined in the original (unperturbed) domain which can be evaluated from the knowledge of solutions to BVP in such a domain. This means that we can evaluate TD by solving only the BVP in the intact domain. One can consider the first and the second order topological derivatives as well, which furnish the approximation of the shape functional with better precision compared to the first order TD expansion in perturbed domain. In this work the topological derivative is applied in the context of Electrical Impedance Tomography (EIT). In particular, we are interested in reconstructing a number of anomalies embedded within a medium subject to a set of current fluxes, from measurements of the corresponding electrical potentials on its boundary. The basic idea consists in minimize a functional measuring the misfit between the boundary measurements and the electrical potentials obtained from the model with respect to a set of ball-shaped anomalies. The first and second order topological derivatives are used, leading to a non-iterative second order reconstruction algorithm. Finally, a numerical experiment is presented, showing that the resulting reconstruction algorithm is very robust with respect to noisy data.