Bayesian resolution of the non linear inverse problem of Electrical Impedance Tomography with Finite Element modeling

Resistivity distribution estimation, widely known as Electrical Impedance Tomography (EIT is a non linear ill-posed inverse problem. However, the partial derivative equation ruling this experiment yields no analytical solution for arbitrary conductivity distribution. Thus, solving the forward problem requires an approximation. The Finite Element Method (FEM) provides us with a computationally cheap forward model which preserves the non linear image-data relation and also reveals sufficiently accurate for the inversion. Within the Bayesian approach, Markovian priors on the log-conductivity distribution are introduced for regularization. The neighborhood system is directly derived from the FEM triangular mesh structure. We first propose a maximum a posteriori (MAP) estimation with a Huber-Markov prior which favours smooth distributions while preserving locally discontinuous features. The resulting criterion is minimized with the pseudo-conjugate gradient method. Simulation results reveal significant improvements in terms of robustness to noise, computation rapidity and ability to estimate discontinuous and highly contrasted distribution. Then, we give a constrained bilinear presentation of the EIT problem. Taking advantage of such a structure as well as the high sparsity of the MEF modeling, we propose a stochastic estimation method of the log-conductivity posterior mean (PM). Actually, the sought expectation is obtained from the convergence of a Gibbs sampled cloud of points that are averaged with an Importance Sampling weighting.