Adaptive Trajectory Correction for Improved Kalman Filter Convergence in Dynamic Process Imaging

The objective in electrical capacitance tomography (ECT) is to acquire information of a potentially time-variant process in an inaccessible region by measuring inter-electrode capacitances at the boundary. Based on the measured data, the two-dimensional or three-dimensional distribution of physical parameters can be determined by solving the underlying ill-posed inverse problem. In various industrial processes the material distribution is time-dependent, for instance, when multi-phase flows have to be investigated and controlled. Such non-stationary problems are usually treated as state estimation problems and solved by means of Kalman filtering (KF). A main drawback is the strong correlation between the initial filter state and the estimation result. Furthermore, the overall sensor sensitivity, i.e. high parameter sensitivity at the boundary close to the electrodes and low sensitivity in the center of the pipe, works against the filter convergence. In this article, an adaptive correction of the filter tracking trajectory is proposed that significantly improves filter convergence of the applied extended KF. By monitoring the measured capacitances, the filter trajectory is recalculated and adjusted. As a remarkable result, the estimation result is not biased by the initial filter state. Since the KF provides regularization only in temporal direction, spatial regularization is incorporated via the state-space model. The resultant improved filter performance is demonstrated for a tracking example of a circle-shaped phantom given measured data.