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Accelerated Markov chain Monte Carlo sampling in electrical capacitance tomography

Daniel Watzenig (Institute of Electrical Measurement and Measurement Signal Processing, Graz University of Technology, Graz, Austria)
Markus Neumayer (Institute of Electrical Measurement and Measurement Signal Processing, Graz University of Technology, Graz, Austria)
Colin Fox (Department of Physics, University of Otago, Dunedin, New Zealand)



The purpose of this paper is to establish a cheap but accurate approximation of the forward map in electrical capacitance tomography in order to approach robust real‐time inversion in the framework of Bayesian statistics based on Markov chain Monte Carlo (MCMC) sampling.


Existing formulations and methods to reduce the order of the forward model with focus on electrical tomography are reviewed and compared. In this work, the problem of fast and robust estimation of shape and position of non‐conducting inclusions in an otherwise uniform background is considered. The boundary of the inclusion is represented implicitly using an appropriate interpolation strategy based on radial basis functions. The inverse problem is formulated as Bayesian inference, with MCMC sampling used to efficiently explore the posterior distribution. An affine approximation to the forward map built over the state space is introduced to significantly reduce the reconstruction time, while maintaining spatial accuracy. It is shown that the proposed approximation is unbiased and the variance of the introduced additional model error is even smaller than the measurement error of the tomography instrumentation. Numerical examples are presented, avoiding all inverse crimes.


Provides a consistent formulation of the affine approximation with application to imaging of binary mixtures in electrical tomography using MCMC sampling with Metropolis‐Hastings‐Green dynamics.

Practical implications

The proposed cheap approximation indicates that accurate real‐time inversion of capacitance data using statistical inversion is possible.


The proposed approach demonstrates that a tolerably small increase in posterior uncertainty of relevant parameters, e.g. inclusion area and contour shape, is traded for a huge reduction in computing time without introducing bias in estimates. Furthermore, the proposed framework – approximated forward map combined with statistical inversion – can be applied to all kinds of soft‐field tomography problems.



Watzenig, D., Neumayer, M. and Fox, C. (2011), "Accelerated Markov chain Monte Carlo sampling in electrical capacitance tomography", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 30 No. 6, pp. 1842-1854.



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