Regularized reconstruction in electrical impedance tomography using a variance uniformization constraint.

This paper describes a new approach to reconstruction of the conductivity field in electrical impedance tomography. Our goal is to improve the tradeoff between the quality of the images and the numerical complexity of the reconstruction method. In order to reduce the computational load, we adopt a linearized approximation to the forward problem that describes the relationship between the unknown conductivity and the measurements. In this framework, we focus on finding a proper way to cope with the ill-posed nature of the problem, mainly caused by strong attenuation phenomena; this is done by devising regularization techniques well suited to this particular problem. First, we propose a solution which is based on Tikhonov regularization of the problem. Second, we introduce an original regularized reconstruction method in which the regularization matrix is determined by space-uniformization of the variance of the reconstructed condictivities. Both methods are nonsupervised, i.e., all tuning parameters are automatically determined from the measured data. Tests performed on simulated and real data indicate that Tikhonov regularization provides results similar to those obtained with iterative methods, but with a much smaller amount of computations. Regularization using a variance uniformization constraint yields further improvements, particularly in the central region of the unknown object where attenuation is most severe. We anticipate that the variance uniformization approach could be adapted to iterative methods that preserve the nonlinearity of the forward problem. More generally, it appears as a useful tool for solving other severely ill-posed reconstruction problems such as eddy current tomography.

Practical identification of functional expansions of nonlinear systems submitted to non-Gaussian inputs.

Time-domain identification of nonlinear systems represented by functional expansions is considered. A general framework is defined for the analysis of three identification methods: the widely used cross-correlation method, Korenberg's method, and a suboptimal least-squares method based on a stochastic approximation algorithm. First, the major characteristics of the underlying estimation problem are pointed out. Then, the identification methods are interpreted as approximations to an optimal estimator, which helps gain insight into their internal functioning and to the investigation of their connections and differences. Examination of results previously published and of the simulations reported in this article indicate that stochastic approximation is an interesting alternative to other existing methods. Identification of a biological system stimulated by a non-Gaussian input confirms the practicality of this approach.