Generalized linearization techniques in electrical impedance tomography
Electrical impedance tomography aims at reconstructing the interior electrical conductivity from surface measurements of currents and voltages. As the current–voltage pairs depend nonlinearly on the conductivity, impedance tomography leads to a nonlinear inverse problem. Often, the forward problem is linearized with respect to the conductivity and the resulting linear inverse problem is regarded as a subproblem in an iterative algorithm or as a simple reconstruction method as such. In this paper, we compare this basic linearization approach to linearizations with respect to the resistivity or the logarithm of the conductivity. It is numerically demonstrated that the conductivity linearization often results in compromised accuracy in both forward and inverse computations. Inspired by these observations, we present and analyze a new linearization technique which is based on the logarithm of the Neumann-to-Dirichlet operator. The method is directly applicable to discrete settings, including the complete electrode model. We also consider Fréchet derivatives of the logarithmic operators. Numerical examples indicate that the proposed method is an accurate way of linearizing the problem of electrical impedance tomography.
Mathematics Subject Classification65N21 47B15 35R30
This work was supported by the Academy of Finland (Decision 267789), the Finnish Foundation for Technology Promotion TES, and the Foundation for Aalto University Science and Technology.
- 1.Adler, A., Arnold, J.H., Bayford, R., Borsic, A., Brown, B., Dixon, P., Faes, T.J.C., Frerichs, I., Gagnon, H., Gärber, Y., Grychtol, B., Hahn, G., Lionheart, W.R.B., Malik, A., Patterson, R.P., Stocks, J., Tizzard, A., Weiler, N., Wolf, G.K.: GREIT: a unified approach to 2D linear EIT reconstruction of lung images. Physiol. Meas. 30, S35–S55 (2009)CrossRefGoogle Scholar
- 3.Calderón, A.-P.: On an inverse boundary value problem. In: Seminar on Numerical Analysis and its Applications to Continuum Physics, pp. 65–73. Sociedade Brasileira de Matematica, Rio de Janeiro (1980)Google Scholar
- 28.Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin. Translated from French by P. Kenneth (1973)Google Scholar