A fully Bayesian approach to shape estimation of objects from tomography data using MFS forward solutions

摘要

It is possible to characterize the aim of many practical inverse geometric problems as one of identifying the shape of an object within some domain of interest using non-intrusive measurements collected on the boundary of the domain. In the problem considered here the object is a rigid inclusion within a homogeneous background medium of constant conductivity, and the data are potential and current flux measurements made on the boundary of the region. The rigid inclusion is described using a geometric parametrization in terms of a star-shaped object. A Bayesian modelling approach is used to combine data likelihood and prior information, and posterior estimation is based on a Markov chain Monte Carlo algorithm which provides measures of uncertainty, as well as point estimates. This means that the inverse problem is never solved directly, but the cost is that instead the forwar solution must be found many thousands of times. The forward problem is solved using the method of fundamental solutions (MFS) which is an efficient meshless alternative to the more common finite element or boundary element methods. This paper is the first to apply Bayesian modelling to a problem using the MFS, with numerical results demonstrating that for appropriate choices of prior distributions accurate results are possible. Further, it demonstrates that a fully Bayesian approach is possible where all prior smoothing parameters are estimated. It is important to note that the geometric modelling and statistical estimation approach are not limited to this example and hence the general technique can be easily applied to other inverse problems. A great benefit of the approach is that it allows an intuitive model description and directly interpretable output. The methods are illustrated using numerical simulations.