First-Order System Least Squares and Electrical Impedance Tomography

Hugh MacMillan, Steve McCormick, and Tom Manteuffel
Department of Applied Mathematics
Campus Box 526
Colorado University-Boulder
Boulder, CO 80309-0526


Abstract

In Electrical impedance tomography (EIT), an image is created by estimating interior spatial variations of an electromagnetic parameter such that it is consistent with a set of boundary data. The standard approach to EIT is output least squares (OLS). Given a set of applied normal boundary currents, one minimizes the defect between the measured and computed boundary voltages associated, respectively, with the exact impedance and its approximation. In minimizing aboundary functional, OLS implicitly imposes the governing Poisson equation as an optimization constraint. We introduce a new first-order system least squares (FOSLS) formulation that incorporates the elliptic PDE as an interior functional in a global minimization scheme. We then establish equivalence of our functional to OLS and to an existing least-squares interior functional due to Kohn and Vogelius. That the latter maybe viewed as a FOSLL* formulation suggests FOSLS as a unifying framework for EIT.

The limited capacityfor resolution in EIT, due to the necessarily finite set of inexact boundary data and the diffusive nature of current flow into the interior, traditionally leads to the conclusion that reconstructing the interior impedance is an ill-posed problem. EIT inherits this difflculty from the simplified inverse problem of reconstructing the electrical conductivity. Since quantifying the limited capacityfor resolution is the focus of our theory, we begin with the static assumption and consider the reconstruction of conductivity only, leaving that of the impedance as future work. We show that each functional in the FOSLS framework is equivalent to a natural norm on the error of the approximate conductivity. We analyze the topology induced by this norm to reveal the qualities of the exact conductivity that we should, in practice, expect to recover. In this paper, we do not present preliminary numerical results for the FOSLS formulation, though we have observed that they are faithful to our theory.

Our approach here represents a significant departure from conventionin that we do not rely on a generic regularization term. Rather, we accept and incorporate the underlying physics, albeit inhibiting. Problem-specific information, which otherwise might be used to "regularize" the "ill-posed"problem, can be included by either introducing an additional term to the functional or supplementing the space of admissible conductivity.