Stability and resolution analysis of a linearized problem in electrical impedance tomography

Abstract

The authors consider the linearized inverse problem in electrical impedance tomography in which they seek to determine conductivity perturbations in a medium from electrostatic boundary measurements. The input current patterns considered in this work are sinusoidal, corresponding to maximally distinguishing patterns for circular domain with a centred circular conductivity anomaly. Starting from an integral identity due to Calderon, they first reduce the inverse problem to a moment problem. The latter is solved by expanding the conductivity perturbations in an orthogonal basis involving Zernike polynomials. The expansion leads to a lower triangular system, which can be solved by backsubstitution. This scheme allows them to perform a stability analysis of the problem. They find that the linearized problem is extremely ill-conditioned, and conclude that any stable reconstruction must have limited resolution. The analysis described in this work may be viewed as an inversion procedure. They demonstrate its use in inverting data generated using the finite element method.