Spatial-temporal Modelling for Electrical Impedance Imaging of a Mixing Process

Robert M West1, Sha Meng2, Robert G Aykroyd2 and Richard A Williams3 1Nuffield Institute, University of Leeds, UK

2Department of Statistics, University of Leeds, UK

3Department of Mining & Mineral Engineering, University of Leeds, UK

ABSTRACT

Many imaging problems can be shown to be inverse problems, being ill-posed or ill-determined. As a consequence, image reconstruction based on measured data alone is unstable, particularly if the reconstruction is also nonlinear. To deliver a practical stable solution it is necessary to make considerable use of prior information. The rôle of a Bayesian approach is therefore of fundamental importance, and when coupled with Markov chain Monte Carlo (MCMC) sampling, it can provide valuable information about solution behaviour.

For many applications of image analysis, spatially smoothing priors have been well justified and seen to produce much improved stability of reconstruction. An example is considered here of the use of EIT to image a tank within which a mixing process occurs. Nonlinearity increases the difficulty of tomographic imaging so that reconstructions can have artifacts including blurring, masking, shadowing and distortions. These defects could lead to reticence in utilizing automatic monitoring and control within an industrial environment.

This example serves well to illustrate the use of relevant prior information. In circumstances where mixing is believed to be poor, such as the initialization of mixing, less reliance can be made on spatially smoothing priors. Therefore an evolution model is incorporated to gain benefit from temporal smoothing. As mixing progresses however the balance between spatial and temporal components is shifted. Once it is seen that mixing has progressed, then it is more appropriate to use spatial smoothing. A predetermined progression from temporal to spatial smoothing is used and is justified when good knowledge of the progression of mixing is available. When less is known about mixing, then greater benefit will be gained by adaptive imaging.

Quality of mixing can be ascertained from the MCMC samples and so it is possible to judge when sufficient mixing has occurred with high probability. In this way mixing time can be optimized to yield efficient use of resources.

Keywords Electrical impedance tomography; Bayesian modelling; Markov chain Monte Carlo