In electrical capacitance tomography (ECT) the main focus is on the reconstruction of distinct objects with sharp transitions between different phases. Being inherently ill-posed, the reconstruction algorithm requires some sort of regularization to stabilize the solution of the inverse problem. However, introducing regularization may counteract the reconstruction of well-defined contours for grid-based methods. Level set propagation approaches which also rely on regularization are able to model sharp phase boundaries but suffer from high computational demands. In this contribution, two different state-space representations of closed contours based on B-splines and on Fourier descriptors are investigated. Both approaches allow us to describe the problem with only a small set of state-space variables. Regularization is incorporated implicitly which can be directly interpreted in the object domain as it relates to smooth contours. To solve the inverse problem, statistical inversion is performed by means of particle filtering providing the opportunity to conveniently incorporate prior information and to take measurement uncertainties into account. The proposed particle filter approach is compared to an extended Kalman filter realization in terms of complexity, computation time and estimation accuracy.

Bayesian inference applied to electrical capacitance tomography, or other inverse problems, provides a framework for quantified model fitting. Estimation of unknown quantities of interest is based on the posterior distribution over the unknown permittivity and unobserved data, conditioned on measured data. Key components in this framework are a prior model requiring a parametrization of the permittivity and a normalizable prior density, the likelihood function that follows from a decomposition of measurements into deterministic and random parts, and numerical simulation of noise-free measurements. Uncertainty in recovered permittivities arises from measurement noise, measurement sensitivities, model inaccuracy, discretization error and a priori uncertainty; each of these sources may be accounted for and in some cases taken advantage of. Estimates or properties of the permittivity can be calculated as summary statistics over the posterior distribution using Markov chain Monte Carlo sampling. Several modified Metropolis–Hastings algorithms are available to speed up this computationally expensive step. The bias in estimates that is induced by the representation of unknowns may be avoided by design of a prior density. The differing purpose of applications means that there is no single ‘Bayesian’ analysis. Further, differing solutions will use different modelling choices, perhaps influenced by the need for computational efficiency. We solve a reference problem of recovering the unknown shape of a constant permittivity inclusion in an otherwise uniform background. Statistics calculated in the reference problem give accurate estimates of inclusion area, and other properties, when using measured data. The alternatives available for structuring inferential solutions in other applications are clarified by contrasting them against the choice we made in our reference solution.

Electrical capacitance tomography (ECT) is a technique for reconstructing information about the spatial distribution of the contents of closed pipes by measuring variations in the dielectric properties of the material inside the pipe. In this paper, we propose a method that solves the non-linear reconstruction problem directly leading to less iterations and higher accuracy than linear back projection algorithms currently in use in most ECT systems.

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