Sampling Conductivity Images via MCMC Colin Fox and Geoff Nicholls Mathematics Department, Auckland University, Private Bag 92019, Auckland, New Zealand May 1997 Abstract Electrical impedance tomography (EIT) is a technique for imaging the conductivity of material inside an object, using cur- rent/voltage measurements at its surface. We demonstrate Bayesian inference from EIT data. A prior probability distribution modeling the unknown conductivity distribution is given. A MCMC algorithm is specified which samples the posterior prob- ability for the conductivity given the prior and the EIT data. In order to compute the likelihood of a conductivity distribution it is necessary to solve a second order linear partial differential equation (PDE). This would appear to make the sampling problem computationally intractable. However by using a Markov chain with Metropolis-Hastings dynamics, and treating the conductivity update as a small perturbation, we are able to avoid solving the PDE for those updates which are rejected by the MCMC sampling process. For real applications the likelihood will need to be sensitive to very small changes in the state, so that the posterior distribution may be sharply peaked as well as multi-modal. The details of the Metropolis-Hastings dynamics are chosen so that ergodic behavior is displayed on useful time scales. We show that the sampling problem is tractable, and illustrate inference from a simple synthetic data set. 1 Introduction Conductivity imaging has been a promising new imaging technique for some time now. The equipment (essentially a current source and a voltmeter) is cheap, portable and non-invasive. Electrodes are attached to the surface of an object, and a small current is applied. The electrical potential is measured (in volts) at points over the surface. This surface potential depends on the unknown conductivity of materials inside the object. In EIT we attempt to reconstruct the conductivity at each point in the object using the known currents and the observed potentials. Consider an open two-dimensional region, with boundary . The conductivity (units ohms ) is an unknown function of position, , in the region. Electrodes producing fixed current source distributions for and for are applied at the boundary and in the interior of the region. The dimensions of and are current per unit length and current per unit area respectively. The electrical potential, , is measured along the boundary. When measurements are made at low frequency, Ohm's law (current is proportional to potential gradient), and Kirchoff's law (current is conserved) together imply that within the region the potential satisfies the partial differential equation (1) Many materials (for example, animal muscle) conduct better along some axes than others. We restrict attention to isotropicly conducting materials, so that is a scalar. If the region is otherwise insulated, the current crossing into is just . Hence if is the unit outward normal on the boundary , the boundary current density is (2) This is just Ohm's Law, and corresponds to a Neumann boundary condition at . Note that, since we are assuming conservation of current, there is a consistency condition on the applied currents. Let be the measure of the element of area at and let be the measure of the element of length at . The applied currents also satisfy Kirchoff's law, (3)