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 Classification

65N21 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. 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
  2. 2.
    Brühl, M.: Explicit characterization of inclusions in electrical impedance tomography. SIAM J. Math. Anal. 32, 1327–1341 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 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
  4. 4.
    Cheney, M., Isaacson, D., Newell, J.C., Simske, S., Goble, J.: NOSER: an algorithm for solving the inverse conductivity problem. Int. J. Imaging Syst. Technol. 2, 66–75 (1990)CrossRefGoogle Scholar
  5. 5.
    Cheng, K.-S., Isaacson, D., Newell, J.C., Gisser, D.G.: Electrode models for electric current computed tomography. IEEE Trans. Biomed. Eng. 36, 918–924 (1989)CrossRefGoogle Scholar
  6. 6.
    Dardé, J., Hyvönen, N., Seppänen, A., Staboulis, S.: Simultaneous recovery of admittivity and body shape in electrical impedance tomography: an experimental evaluation. Inverse Prob. 29, 085004 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dardé, J., Staboulis, S.: Electrode modelling: the effect of contact impedance. ESAIM Math. Model. Numer. Anal. 50, 415–431 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology: Functional and Variational Methods, vol. 2. Springer, Berlin (1988)CrossRefMATHGoogle Scholar
  9. 9.
    Demmel, J.W.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)CrossRefMATHGoogle Scholar
  10. 10.
    Engl, H., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)CrossRefMATHGoogle Scholar
  11. 11.
    Garde, H., Knudsen, K.: Distinguishability revisited: depth dependent bounds on reconstruction quality in electrical impedance tomography. SIAM J. Appl. Math. 77, 697–720 (2017)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, London (1985)MATHGoogle Scholar
  13. 13.
    Hanke, M., Hyvönen, N., Reusswig, S.: An inverse backscatter problem for electric impedance tomography. SIAM J. Math. Anal. 41, 1948–1966 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Harhanen, L., Hyvönen, N., Majander, H., Staboulis, S.: Edge-enhancing reconstruction algorithm for three-dimensional electrical impedance tomography. SIAM J. Sci. Comput. 37, B60–B78 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Harrach, B., Seo, J.K.: Exact shape-reconstruction by one-step linearization in electrical impedance tomography. SIAM J. Math. Anal. 42, 1505–1518 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Henrici, P.: Applied and Computational Complex Analysis, vol. 1. Wiley, New York (1974)MATHGoogle Scholar
  17. 17.
    Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)CrossRefMATHGoogle Scholar
  18. 18.
    Hyvönen, N., Mustonen, L.: Smoothened complete electrode model. SIAM J. Appl. Math. 77, 2250–2271 (2017)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Isaacson, D.: Distinguishability of conductivities by electric current computed tomography. IEEE Trans. Med. Imaging 5, 91–95 (1986)CrossRefGoogle Scholar
  20. 20.
    Kaipio, J.P., Kolehmainen, V., Somersalo, E., Vauhkonen, M.: Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography. Inverse Prob. 16, 1487–1522 (2000)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kaipio, J .P., Somersalo, E.: Statistical and Computational Inverse Problems. Springer, Berlin (2004)MATHGoogle Scholar
  22. 22.
    Karhunen, K., Lehikoinen, A., Monteiro, P.J.M., Kaipio, J.P.: Electrical resistance tomography imaging of concrete. Cem. Concrete Res. 40, 137–145 (2010)CrossRefGoogle Scholar
  23. 23.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995)CrossRefMATHGoogle Scholar
  24. 24.
    Kourunen, J., Savolainen, T., Lehikoinen, A., Vauhkonen, M., Heikkinen, L.M.: Suitability of a PXI platform for an electrical impedance tomography system. Meas. Sci. Technol. 20, 015503 (2009)CrossRefGoogle Scholar
  25. 25.
    Kriegl, A., Michor, P.W., Rainer, A.: Denjoy–Carleman differentiable perturbation of polynomials and unbounded operators. Integral Equ. Oper. Theory 71, 407–416 (2011)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Lechleiter, A., Rieder, A.: Newton regularizations for impedance tomography: convergence by local injectivity. Inverse Prob. 24, 065009 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Leinonen, M., Hakula, H., Hyvönen, N.: Application of stochastic Galerkin FEM to the complete electrode model of electrical impedance tomography. J. Comput. Phys. 269, 181–200 (2014)MathSciNetCrossRefMATHGoogle Scholar
  28. 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
  29. 29.
    Somersalo, E., Cheney, M., Isaacson, D.: Existence and uniqueness for electrode models for electric current computed tomography. SIAM J. Appl. Math. 52, 1023–1040 (1992)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Uhlmann, G.: Electrical impedance tomography and Calderón’s problem. Inverse Prob. 25, 123011 (2009)CrossRefMATHGoogle Scholar
  31. 31.
    Yosida, K.: Functional Analysis. Springer, Berlin (1980)MATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Systems AnalysisAalto UniversityAaltoFinland
  2. 2.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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