volume | PIER Journals

Abstract

In electromagnetic tomography and resistivity survey a linearized model approximation is often used, in the context of regularized regression, to image the conductivity distribution in a domain of interest. Due to the error introduced by the simplified model, quantitative image reconstruction becomes challenging unless the conductivity is sufficiently close to a constant. We derive a closed form expression of the linearization error in electrical impedance tomography based on the complete electrode model. The error term is expressed in an integral form involving the gradient of the perturbed electric potential and renders itself readily available for analytical or numerical computation. For real isotropic conductivity changes with piecewise uniform characteristic functions the perturbed potential field can be shown to satisfy Poisson's equation with Robin boundary conditions and interior point sources positioned at the interfaces of the inhomogeneities. Simulation experiments using a finite element method have been performed to validate these results.

1. Assenheimer, M., O. Laver-Moskovitz, D. Malonek, D. Manor, U. Nahaliel, R. Nitzan, and A. Saad, "The T-SCAN technology: Electrical impedance as a diagnostic tool for breast cancer detection," Physiological Measurement, Vol. 22, 18, 2001.
doi:10.1088/0967-3334/22/1/301 Google Scholar

2. Borcea, L., "Electrical impedance tomography: Topical review," Inverse Problems, Vol. 18, No. 6, 99-136, 2002.
doi:10.1088/0266-5611/18/6/201 Google Scholar

3. Breckon, W. R., Image reconstruction in electrical impedance tomography, PhD thesis, Oxford Polytechnic, 1990. On-line copy at http://www.maths.manchester.ac.uk/~bl..

4. Calderon, A. P., "On an inverse boundary value problem," Computational and Applied Mathematics, Vol. 25, No. 2-3, 133-138, 2006 (Reprint of original paper). Google Scholar

5. Holder, D. H., Electrical Impedance Tomography: Methods, History and Applications, Institute of Physics, Bristol, 2002.

6. Kaipio, J. and E. Somersalo, Statistical and Computational Inverse Problems, Springer, 2004.

7. Paulson, K. S., W. R. Breckon, and M. K. Pidcock, "Electrode modelling in electrical impedance tomography," SIAM Journal of Applied Mathematics, Vol. 52, 1012-1022, 1992.
doi:10.1137/0152059 Google Scholar

8. Pidcock, M. K., M. Kuzuoglu, and K. Leblebicioglu, "Analytic and semi-analytic solutions in electrical impedance tomography. II. Three-dimensional problems," Physiological Measurement, Vol. 16, 91-110, 1995.
doi:10.1088/0967-3334/16/2/002 Google Scholar

9. Polydorides, N. and W. R. B. Lionheart, "A MATLAB based toolkit for three-dimensional electrical impedance tomography: A contribution to the EIDORS project," Measurement Science and Technology, Vol. 13, No. 12, 1871-1883, 2002.
doi:10.1088/0957-0233/13/12/310 Google Scholar

10. Seo, J. K., O. Kwon, H. Ammari, and E. J.Woo, "A mathematical model for breast cancer lesion estimation: Electrical impedance technique using TS2000 commercial system," IEEE Transactions on Biomedical Engineering, Vol. 51, No. 11, 1898-1906, 2004.
doi:10.1109/TBME.2004.834261 Google Scholar

11. Silvester, J. and G. Uhlmann, "A global uniqueness theorem for an inverse boundary valued problem," Annals of Mathematics, Vol. 125, 153-169, 1987.
doi:10.2307/1971291 Google Scholar

12. Somersalo, E., M. Cheney, and D. Isaacson, "Existence and uniqueness for electrode models for electric current computed tomography," SIAM Journal on Applied Mathematics, Vol. 52, No. 4, 1023-1040, 1992.
doi:10.1137/0152060 Google Scholar

13. Vauhkonen, M., W. R. B. Lionheart, L. M. Heikkinen, P. J. Vauhkonen, and J. P. Kaipio, "A MATLAB package for the EIDORS project to reconstruct two-dimensional EIT images," Physiological Measurement, Vol. 22, 107-111, 2001.
doi:10.1088/0967-3334/22/1/314 Google Scholar

14. Yorkey, T. J., Comparing reconstruction methods for electrical impedance tomography, PhD thesis, University of Wisconsin, Madison, 1986.

15. Brandstatter, B., "Jacobian calculation for electrical impedance tomography based on the reciprocity principle," IEEE Transactions on Magnetics, Vol. 39, No. 3, 1309-1312, 2003.
doi:10.1109/TMAG.2003.810390 Google Scholar

16. Soleimani, M., C. N. Mitchell, R. Banasiak, R. Wajman, and A. Adler, "Four-dimensional electrical capacitance tomography imaging using experimental data," Progress In Electromagnetics Research, Vol. 90, 171-186, 2009.
doi:10.2528/PIER09010202 Google Scholar

Dr. Nick Polydorides