Bayesian Filtering using a Convex Hull Criterion to estimate Multi- Phase Distributions in Electrical Process Tomography

D. Watzenig, G. Steiner, A. Fuchs, H. Zangl, and H. Wegleiter

Institute of Electrical Measurement and Measurement Signal Processing, Graz University of Technology, Kopernikusgasse 24/4, A-8010 Graz, Austria

Email: daniel.watzenig@tugraz.at

ABSTRACT

A key difference between regularization and Bayesian methods to solve inverse ill-posed problems is that Bayesian approaches, instead of giving only single point estimates, present averages over all feasible solutions. The most natural way to solve inverse problems in the framework of Bayesian statistics is the application of computationally expensive Markov chain Monte Carlo methods. However especially in industrial process tomography, e.g. in automatic control of certain process states, the aspect of computational cost plays an important role. Particle filtering – a sequential Monte Carlo method that relies on importance sampling – is able to feature real-time operation while maintaining the advantages of statistical inversion such as yielding robust estimates of properties calculated from the solutions.

In this paper, a particle filter approach to reconstruct inclusions in an otherwise uniform background material from measured electrical capacitance data using a convex hull criterion is presented. This criterion allows for reconstructing both material inclusions, as well as phase interfaces in sedimentation processes without increasing the computational complexity. By taking into account measurement uncertainties via an appropriate measurement noise model, a range of parameters consistent with the measured data can be quantified, from which one can calculate any statistics of interest. The forward map is numerically implemented using finite elements while the inclusion boundaries are modelled by means of parameterized curves. In order to solve the forward problem, the inclusion boundary is mapped on the mesh grid. By prescribing the order of the used Fourier descriptor contour model, prior information is incorporated that leads to smooth boundaries in the image space. Filter convergence is improved due to a fragmented process covariance matrix scaled with different stochastic weights for inclusion position and inclusion shape. The proposed approach is validated for different test cases using measured electrical capacitance data.

Keywords Electrical capacitance tomography, finite elements, particle filtering, Fourier descriptors