Exame de Qualificação: The Topological Derivate Method Applied to Three-Dimencional Electrical Impedance Tomography

Orientadores:
Lucas dos Santos Fernandez - Laboratório Nacional de Computação Científica - LNCC
Antônio André Novotny - Laboratório Nacional de Computação Científica - LNCC
Ravi Prakash - Laboratório Nacional de Computação Científica - LNCC

Banca Examinadora:
Pablo Javier Blanco - Laboratório Nacional de Computação Científica - LNCC (presidente)
Americo Barbosa da Cunha Junior - Laboratório Nacional de Computação Científica - LNCC
Marcel Duarte da Silva Xavier - Universidade Federal Fluminense - UFF

Suplentes:
Gilson Antônio Giraldi - Laboratório Nacional de Computação Científica - LNCC

Resumo:

Several classes of inverse problems are written in the form of over-determined boundary value prob lems. This difficulty can be overcome by rewriting the inverse problem as an optimization problem. In particular, we are interested in the reconstruction of the support of a set of hidden anomalies embedded into a geometrical domain. The basic idea consists in minimizing a shape functional measuring the misfit of the solution obtained from the model and the data taken from the partial measurements with respect to the parameters of interest. This shape functional is minimized with respect to a set of ball-shaped anomalies by using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting expression is trivially minimized with respect to the parameters under consideration, leading to a non-iterative second order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to noisy data and independent of any initial guess. The proposed method ology is applied to a three-dimensional inverse problem in electrical impedance tomography, where the conductivity values of the anomalies are assumed to be known, while their number, topology, and spatial location remain unknown. The reconstruction of these heterogeneities is performed from partial measurements of the electrical potential collected on the boundary of a reference domain representing a healthy body. Within this framework, we also investigate the sensitivity of the proposed reconstruction procedure with respect to different electrode excitation patterns and varying levels of measurement noise.