A Bayesian filtering approach to layer stripping for electrical impedance tomography

Abstract

Layer stripping is a method for solving inverse boundary value problems for elliptic PDEs, originally proposed in the literature for solving the Calderón problem of electrical impedance tomography (EIT), where the data consist of the Neumann-to-Dirichlet operator on the boundary. Defining a tangent–normal coordinate system near the boundary, the data are extended to a family of boundary operators on tangential surfaces inside the body, and it is shown that the operators satisfy a non-linear Riccati type differential equation with respect to the normal coordinate. The layer stripping process consists of a sequence of two alternating steps: the conductivity near the current boundary is estimated from the spatial high-frequency limit of the boundary data, and the boundary operator is propagated through a thin layer further into the domain via the Riccati equation. This way, the unknown conductivity in the interior of the domain is estimated layer by layer starting from the boundary and moving inward. The ill-posedness of the EIT problem manifests itself in such high sensitivity of the backwards Riccati equation to errors in the boundary data to cause the solutions to blow up in finite time, thus requiring regularization. In this article, we formulate the layer stripping process in the framework of Bayesian inverse problems, and we revisit the implementation in the light of Bayesian filtering. More specifically, we recast the related inverse boundary value problem as a state estimation problem, and propose an algorithm for its numerical solution based on ensemble Kalman filtering (EnKF). The new Bayesian layer stripping approach that we propose is quite robust, derivative-free and intrinsically suited for the quantification of uncertainties in the estimate. Furthermore, we show that the algorithm can be extended to realistic data collected by using a finite number of contact electrodes.