Fixed‐lag smoothing and state estimation in dynamic electrical impedance tomography

Abstract

SUMMARYIn electrical impedance tomography (EIT), an approximation for the internal resistivity distribution is computed based on the knowledge of the injected currents and measured voltages on the surface of the body. The conventional approach is to inject several di erent current patterns and use the associated data for the reconstruction of a single distribution. This is an ill-posed inverse problem. In some applications the resistivity changes may be so fast that the target changes between the injection of the current patterns and thus the data do not correspond to the same target distribution. In these cases traditional reconstruction methods yield severely blurred resistivity estimates. We have earlier proposed to formulate the EIT problem as an augmented system theoretical state estimation problem. The reconstruction problem can then be solved with Kalman ÿlter and Kalman smoother algorithms. In this paper, we use the so-called ÿxed-lag smoother to solve the dynamic EIT reconstruction problem. We show that data storage di culties that are associated with the previously used ÿxed-interval smoother can be avoided using the ÿxed-lag smoother. The proposed methods are compared with simulated measurements and real data.

Recursive penalized least squares solution for dynamical inverse problems of EEG generation

“…For the reconstruction of current distributions in the EEG inverse problem, or, more generally, the EEG/MEG inverse problem, other algorithms for solving the same large-scale least squares problem as mentioned above have been developed (Schmidt et al 2001;Schmidt and Louis 2002a,b). Inverse problems arising in the analysis of data obtained by Electrical Impedance Tomography (EIT) and Single Photon Emission Tomography (SPET) have been formulated as state estimation problems (Karjalainen et al 1997;Kaipio et al 1999;Vauhkonen et al 2001) , and the use of Kalman filtering and Kalman smoothing has been suggested for the purpose of obtaining estimates of the state. Phillips et al (2002) have suggested to introduce a temporal constraint into the EEG/MEG inverse problem by employing a time window and Gaussian kernels.…”

Efficient algorithms for the regularization of dynamic inverse problems: II. Applications

“…It should be annotated, that there is another approach called fixed-lag smoothing [19] which has less memory consumption than the (fixed-interval) Kalman smoother used here. The extended Kalman filter studied in [20], is an analogue to our Gauß-Newton approach.…”

State space models in process tomography — approximation of state noise covariance

“…These have yet to be investigated in order to assess the reliability of the estimates in such real data cases in which there is no access to true sample data. However, we have previously shown preliminary results that when the flow is not extremely fast when compared to measurement speed, even the random walk model gives qualitatively feasible results [18,24].…”