# American Institute of Mathematical Sciences

May  2011, 5(2): 355-369. doi: 10.3934/ipi.2011.5.355

## Electrical impedance tomography using a point electrode inverse scheme for complete electrode data

 1 Department of Informatics and Mathematical Modelling, Technical University of Denmark, 2800 Kongens Lyngby, Denmark 2 Institut für Numerische und Angewandte Mathematik, Universität Göttingen, 37083 Göttingen

Received  May 2010 Revised  September 2010 Published  May 2011

For the two dimensional inverse electrical impedance problem in the case of piecewise constant conductivities with the currents injected at adjacent point electrodes and the resulting voltages measured between the remaining electrodes, in [3] the authors proposed a nonlinear integral equation approach that extends a method that has been suggested by Kress and Rundell [10] for the case of perfectly conducting inclusions. As the main motivation for using a point electrode method we emphasized on numerical difficulties arising in a corresponding approach by Eckel and Kress [4, 5] for the complete electrode model. Therefore, the purpose of the current paper is to illustrate that the inverse scheme based on point electrodes can be successfully employed when synthetic data from the complete electrode model are used.
Citation: Fabrice Delbary, Rainer Kress. Electrical impedance tomography using a point electrode inverse scheme for complete electrode data. Inverse Problems and Imaging, 2011, 5 (2) : 355-369. doi: 10.3934/ipi.2011.5.355
##### References:
 [1] A. Baba and M. J. Burke, Measurement of the electrical properties of ungelled ECG electrodes, Int. J. Biol. Biomed. Eng., 2 (2008), 89-97. [2] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," vol. 93 of Applied Mathematical Sciences, Springer-Verlag, Berlin, second ed., 1998. [3] F. Delbary and R. Kress, Electrical impedance tomography with point electrodes, J. Integral Equations Appl., 22 (2010), 193-216. doi: 10.1216/JIE-2010-22-2-193. [4] H. Eckel and R. Kress, Nonlinear integral equations for the inverse electrical impedance problem, Inverse Problems, 23 (2007), 475-491. doi: 10.1088/0266-5611/23/2/002. [5] H. Eckel and R. Kress, Non-linear integral equations for the complete electrode model in inverse impedance tomography, Appl. Anal., 87 (2008), 1267-1288. doi: 10.1080/00036810802032151. [6] M. Hanke, B. Harrach and N. Hyvönen, Justification of point electrode models in electrical impedance tomography, Math. Models Methods Appl. Sci., accepted. [7] N. Hyvönen, Approximating idealized boundary data of electric impedance tomography by electrode measurements, Math. Models and Meth. in Appl. Sciences, 19 (2009), 1185-1202. doi: 10.1142/S0218202509003759. [8] V. Kolehmainen, M. Lassas and P. Ola, The inverse conductivity problem with an imperfectly known boundary, SIAM J. Appl. Math., 66 (2005), 365-383. doi: 10.1137/040612737. [9] R. Kress, "Linear Integral Equations," vol. 82 of Applied Mathematical Sciences, Springer-Verlag, New York, second ed., 1999. [10] R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 1207-1223. doi: 10.1088/0266-5611/21/4/002. [11] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), 1023-1040. doi: 10.1137/0152060. [12] O. Steinbach and W. L. Wendland, On C. Neumann's method for second-order elliptic systems in domains with non-smooth boundaries, J. Math. Anal. Appl., 262 (2001), 733-748. doi: 10.1006/jmaa.2001.7615.

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##### References:
 [1] A. Baba and M. J. Burke, Measurement of the electrical properties of ungelled ECG electrodes, Int. J. Biol. Biomed. Eng., 2 (2008), 89-97. [2] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," vol. 93 of Applied Mathematical Sciences, Springer-Verlag, Berlin, second ed., 1998. [3] F. Delbary and R. Kress, Electrical impedance tomography with point electrodes, J. Integral Equations Appl., 22 (2010), 193-216. doi: 10.1216/JIE-2010-22-2-193. [4] H. Eckel and R. Kress, Nonlinear integral equations for the inverse electrical impedance problem, Inverse Problems, 23 (2007), 475-491. doi: 10.1088/0266-5611/23/2/002. [5] H. Eckel and R. Kress, Non-linear integral equations for the complete electrode model in inverse impedance tomography, Appl. Anal., 87 (2008), 1267-1288. doi: 10.1080/00036810802032151. [6] M. Hanke, B. Harrach and N. Hyvönen, Justification of point electrode models in electrical impedance tomography, Math. Models Methods Appl. Sci., accepted. [7] N. Hyvönen, Approximating idealized boundary data of electric impedance tomography by electrode measurements, Math. Models and Meth. in Appl. Sciences, 19 (2009), 1185-1202. doi: 10.1142/S0218202509003759. [8] V. Kolehmainen, M. Lassas and P. Ola, The inverse conductivity problem with an imperfectly known boundary, SIAM J. Appl. Math., 66 (2005), 365-383. doi: 10.1137/040612737. [9] R. Kress, "Linear Integral Equations," vol. 82 of Applied Mathematical Sciences, Springer-Verlag, New York, second ed., 1999. [10] R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 1207-1223. doi: 10.1088/0266-5611/21/4/002. [11] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), 1023-1040. doi: 10.1137/0152060. [12] O. Steinbach and W. L. Wendland, On C. Neumann's method for second-order elliptic systems in domains with non-smooth boundaries, J. Math. Anal. Appl., 262 (2001), 733-748. doi: 10.1006/jmaa.2001.7615.
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