A flexible statistical and efficient computational approach to object location applied to electrical tomography

Abstract

In electrical tomography, multiple measurements of voltage are taken between electrodes on the boundary of a region with the aim of investigating the electrical conductivity distribution within the region. The relationship between conductivity and voltage is governed by an elliptic partial differential equation derived from Maxwell's equations. Recent statistical approaches, combining Bayesian methods with Markov chain Monte Carlo (MCMC) algorithms, allow to greater flexibility than classical inverse solution approaches and require only the calculation of voltages from a conductivity distribution. However, solution of this forward problem still requires the use of the Finite Difference Method (FDM) or the Finite Element Method (FEM) and many thousands of forward solutions are needed which strains practical feasibility. Many tomographic applications involve locating the perimeter of some homogeneous conductivity objects embedded in a homogeneous background. It is possible to exploit this type of structure using the Boundary Element Method (BEM) to provide a computationally efficient alternative forward solution technique. A geometric model is then used to define the region boundary, with priors on boundary smoothness and on the range of feasible conductivity values. This paper investigates the use of a BEM/MCMC approach for electrical resistance tomography (ERT) data. The efficiency of the boundary-element method coupled with the flexibility of the MCMC technique gives a promising new approach to object identification in electrical tomography. Simulated ERT data are used to illustrate the procedures.