| Electrical impedance tomography (EIT) belongs to a family of imaging techniques that attempt to distinguish spatial variations in an internal electromagnetic parameter. 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 a boundary 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 may be viewed as a FOSLL* formulation suggests FOSLS as a unifying framework for EIT.; The limited capacity for 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 difficulty from the simplified inverse problem of reconstructing the electrical conductivity. Since quantifying the limited capacity is the focus of our theory, we begin with the static assumption and consider the reconstruction of conductivity, 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. Finally, we present preliminary numerical results for the FOSLS formulation and observe that they are faithful to our theory.; Our approach represents a significant departure from convention in 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. |
