Abstract
Electrical impedance tomography (EIT) or electrical resistivity tomography (ERT) current and measure voltages at the boundary of a domain through electrodes.Significance: The movement or incorrect placement of electrodes may lead to modelling errors that result in significant reconstructed image artifacts. These errors may be accounted for by allowing for electrode position estimates in the model. Movement may be reconstructed through a first-order approximation, the electrode position Jacobian. A reconstruction that incorporates electrode position estimates and conductivity can significantly reduce image artifacts. Conversely, if electrode position is ignored it can be difficult to distinguish true conductivity changes from reconstruction artifacts which may increase the risk of a flawed interpretation.Objective: In this work, we aim to determine the fastest, most accurate approach for estimating the electrode position Jacobian.Approach: Four methods of calculating the electrode position Jacobian were evaluated on a homogeneous halfspace.Main results: Results show that Fréchet derivative and rank-one update methods are competitive in computational efficiency but achieve different solutions for certain values of contact impedance and mesh density.
Highlights
Calculations to estimate the electrode position Jacobian by direct perturbation of an Finite Element Method (FEM) mesh requires multiple forward solutions: one per electrode and position dimension. (In three dimensions, a 32 electrode array would require 96 additional forward solutions per Gauss-Newton iteration.) The method is a direct implementation of equation (4): the partial derivatives are replaced with a small perturbation δxj of electrode j’s location
The electrode position Jacobian was calculated for each electrode using our implementation of the naıve perturbation, minimal perturbation, rank-one update, and Frechet derivative
For a “complete” stimulus set where all possible combinations of stimulus electrodes are used, that no further forward solutions are required than have already been computed to that point in the Gauss-Newton algorithm. Such a computational saving is hard to quantify in a limited test bench such as the one presented here, but could reduce the incremental cost of calculating a electrode position Jacobian to a few multiply-and-accumulate operations if the forward solution nodal voltages are already available from the conductivity Jacobian
Summary
The electrode position Jacobian can be used to reconstruct electrode position This is practical when the interior conductivity distribution is known and the forward model is otherwise accurate. The interior conductivity is not known and a joint reconstruction of electrode position and conductivity can significantly reduce the occurrence of artifacts (Soleimani et al 2006, Gomez-Laberge and Adler 2008, Jehl et al 2015). An update (δσ, δx) is calculated (1) based on the combined Jacobian J, the inverse noise covariance of the measurements W, regularization R scaled by hyperparameter λ, the measurement misfit b, and the prior misfit c. For J = Jσ ; Jx where the Jacobian is calculated on a prior estimate of conductivity and electrode position. Joint electrode position and conductivity reconstructions are an on-going topic of research in both the biomedical Electrical Impedance Tomography (EIT) and geophysics Electrical Resistivity Tomography (ERT) fields (Jehl et al 2015, Wilkinson et al 2016)
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