August  2012, 6(3): 399-421. doi: 10.3934/ipi.2012.6.399

Fine-tuning electrode information in electrical impedance tomography

1. 

Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076 Aalto, Finland, Finland, Finland, Finland

Received  July 2011 Revised  May 2012 Published  September 2012

Electrical impedance tomography is a noninvasive imaging technique for recovering the admittivity distribution inside a body from boundary measurements of current and voltage. In practice, impedance tomography suffers from inaccurate modelling of the measurement setting: The exact electrode locations and the shape of the imaged object are not necessarily known precisely. In this work, we tackle the problem with imperfect electrode information by introducing the Fréchet derivative of the boundary measurement map of impedance tomography with respect to the electrode shapes and locations. This enables us to include the fine-tuning of the information on the electrode positions as a part of a Newton-type output least squares reconstruction algorithm; we demonstrate that this approach is feasible via a two-dimensional numerical example based on simulated data. The impedance tomography measurements are modelled by the complete electrode model, which is in good agreement with real-life electrode measurements.
Citation: Jérémi Dardé, Harri Hakula, Nuutti Hyvönen, Stratos Staboulis. Fine-tuning electrode information in electrical impedance tomography. Inverse Problems and Imaging, 2012, 6 (3) : 399-421. doi: 10.3934/ipi.2012.6.399
References:
[1]

D. C. Barber and B. H. Brown, Errors in reconstruction of resistivity images using a linear reconstruction technique, Clin. Phys. Physiol. Meas., 9 (1988), 101-104.

[2]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136. doi: 10.1088/0266-5611/18/6/201.

[3]

W. Breckon and M. Pidcock, Data errors and reconstruction algorithms in electrical impedance tomography, Clin. Phys. Physiol. Meas., 9 (1988), 105-109.

[4]

M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101. doi: 10.1137/S0036144598333613.

[5]

K.-S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Trans. Biomed. Eng., 36 (1989), 918-924. doi: 10.1109/10.35300.

[6]

D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,'' Wiley, New York, 1983.

[7]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,'' Springer-Verlag, Berlin, 1992.

[8]

J. Dardé, N. Hyvönen, A. Seppänen and S. Staboulis, Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography,, submitted., (). 

[9]

R. Dautray and J. L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,'' Vol. 2, Springer-Verlag, Berlin, 1988.

[10]

M. C. Delfour and J. P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus and Optimization,'' SIAM, Philadelphia, 2001.

[11]

R. Griesmaier and N. Hyvönen, A regularized Newton method for locating thin tubular conductivity inhomogeneities, Inverse Problems, 27 (2011), 115008. doi: 10.1088/0266-5611/27/11/115008.

[12]

M. Hanke, B. Harrach and N. Hyvönen, Justification of point electrode models in electrical impedance tomography, Math. Models Methods Appl. Sci., 21 (2011), 1395-1413. doi: 10.1142/S0218202511005362.

[13]

L. M. Heikkinen, T. Vilhunen, R. M. West and M. Vauhkonen, Simultaneous reconstruction of electrode contact impedances and internal electrical properties: II. Laboratory experiments, Meas. Sci. Technol., 13 (2002), 1855-1861. doi: 10.1088/0957-0233/13/12/308.

[14]

F. Hettlich, Fréchet derivatives in inverse obstacle scattering, Inverse Problems, 11 (1995), 371-382. doi: 10.1088/0266-5611/11/2/007.

[15]

F. Hettlich, Erratum: Fréchet derivatives in inverse obstacle scattering, Inverse Problems, 14 (1998), 209-210. doi: 10.1088/0266-5611/14/1/017.

[16]

F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Problems, 14 (1998), 67-82. doi: 10.1088/0266-5611/14/1/008.

[17]

L. Horesh, E. Haber and L. Tenorio, Optimal experimental design for the large-scale nonlinear ill-posed problem of impedance imaging, in "Large-scale Inverse Problems and Quantification of Uncertainty'' (eds. L. Biegler, G. Biros, O. Ghattas, M. Heinkenschloss, D. Keyes, B. Mallick, Y. Marzouk, L. Tenorio, B. Van Bloemen Waanders and K. Willcox), Wiley, Chichester, 2011.

[18]

N. Hyvönen, Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions, SIAM J. Appl. Math, 64 (2004), 902-931. doi: 10.1137/S0036139903423303.

[19]

N. Hyvönen, Fréchet derivative with respect to the shape of a strongly convex nonscattering region in optical tomography, Inverse Problems, 23 (2007), 2249-2270. doi: 10.1088/0266-5611/23/5/026.

[20]

N. Hyvönen, K. Karhunen and A. Seppänen, Fréchet derivative with respect to the shape of an internal electrode in electrical impedance tomography, SIAM J. Appl. Math, 70 (2010), 1878-1898. doi: 10.1137/09075929X.

[21]

J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography, Inverse Problems, 16 (2000), 1487-1522. doi: 10.1088/0266-5611/16/5/321.

[22]

J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,'' Springer-Verlag, New York, 2005.

[23]

A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9 (1993), 81-96.

[24]

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.

[25]

V. Kolehmainen, M. Vauhkonen, P. A. Karjalainen and J. P. Kaipio, Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns, Physiol. Meas., 18 (1997), 289-303.

[26]

A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography: A numerical study, Inverse Problems, 22 (2006), 1967-1987. doi: 10.1088/0266-5611/22/6/004.

[27]

A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography: Convergence by local injectivity, Inverse Problems, 24 (2008), 065009. doi: 10.1088/0266-5611/24/6/065009.

[28]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,'' \textbfI, Springer-Verlag, New York-Heidelberg, 1972.

[29]

A. Nissinen, L. M. Heikkinen and J. P. Kaipio, The Bayesian approximation error approach for electrical impedance tomography - experimental results, Meas. Sci. Technol., 19 (2008), 015501. doi: 10.1088/0957-0233/19/1/015501.

[30]

A. Nissinen, L. M. Heikkinen, V. Kolehmainen and J. P. Kaipio, Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography, Meas. Sci. Technol., 20 (2009), 105504. doi: 10.1088/0957-0233/20/10/105504.

[31]

A. Nissinen, V. Kolehmainen and J. P. Kaipio, Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography, IEEE Trans. Med. Imag., 30 (2011), 231-242. doi: 10.1109/TMI.2010.2073716.

[32]

J. Nocedal and S. J. Wright, "Numerical Optimization,'' Springer-Verlag, New York, 1999.

[33]

M. Soleimani, C. Gómez-Laberge and A. Adler, Imaging of conductivity changes and electrode movement in EIT, Physiol. Meas., 27 (2006), S103-S113. doi: 10.1088/0967-3334/27/5/S09.

[34]

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.

[35]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.

[36]

T. Vilhunen, J. P. Kaipio, P. J. Vauhkonen, T. Savolainen and M. Vauhkonen, Simultaneous reconstruction of electrode contact impedances and internal electrical properties: I. Theory, Meas. Sci. Technol., 13 (2002), 1848-1854. doi: 10.1088/0957-0233/13/12/307.

show all references

References:
[1]

D. C. Barber and B. H. Brown, Errors in reconstruction of resistivity images using a linear reconstruction technique, Clin. Phys. Physiol. Meas., 9 (1988), 101-104.

[2]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136. doi: 10.1088/0266-5611/18/6/201.

[3]

W. Breckon and M. Pidcock, Data errors and reconstruction algorithms in electrical impedance tomography, Clin. Phys. Physiol. Meas., 9 (1988), 105-109.

[4]

M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101. doi: 10.1137/S0036144598333613.

[5]

K.-S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Trans. Biomed. Eng., 36 (1989), 918-924. doi: 10.1109/10.35300.

[6]

D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,'' Wiley, New York, 1983.

[7]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,'' Springer-Verlag, Berlin, 1992.

[8]

J. Dardé, N. Hyvönen, A. Seppänen and S. Staboulis, Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography,, submitted., (). 

[9]

R. Dautray and J. L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,'' Vol. 2, Springer-Verlag, Berlin, 1988.

[10]

M. C. Delfour and J. P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus and Optimization,'' SIAM, Philadelphia, 2001.

[11]

R. Griesmaier and N. Hyvönen, A regularized Newton method for locating thin tubular conductivity inhomogeneities, Inverse Problems, 27 (2011), 115008. doi: 10.1088/0266-5611/27/11/115008.

[12]

M. Hanke, B. Harrach and N. Hyvönen, Justification of point electrode models in electrical impedance tomography, Math. Models Methods Appl. Sci., 21 (2011), 1395-1413. doi: 10.1142/S0218202511005362.

[13]

L. M. Heikkinen, T. Vilhunen, R. M. West and M. Vauhkonen, Simultaneous reconstruction of electrode contact impedances and internal electrical properties: II. Laboratory experiments, Meas. Sci. Technol., 13 (2002), 1855-1861. doi: 10.1088/0957-0233/13/12/308.

[14]

F. Hettlich, Fréchet derivatives in inverse obstacle scattering, Inverse Problems, 11 (1995), 371-382. doi: 10.1088/0266-5611/11/2/007.

[15]

F. Hettlich, Erratum: Fréchet derivatives in inverse obstacle scattering, Inverse Problems, 14 (1998), 209-210. doi: 10.1088/0266-5611/14/1/017.

[16]

F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Problems, 14 (1998), 67-82. doi: 10.1088/0266-5611/14/1/008.

[17]

L. Horesh, E. Haber and L. Tenorio, Optimal experimental design for the large-scale nonlinear ill-posed problem of impedance imaging, in "Large-scale Inverse Problems and Quantification of Uncertainty'' (eds. L. Biegler, G. Biros, O. Ghattas, M. Heinkenschloss, D. Keyes, B. Mallick, Y. Marzouk, L. Tenorio, B. Van Bloemen Waanders and K. Willcox), Wiley, Chichester, 2011.

[18]

N. Hyvönen, Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions, SIAM J. Appl. Math, 64 (2004), 902-931. doi: 10.1137/S0036139903423303.

[19]

N. Hyvönen, Fréchet derivative with respect to the shape of a strongly convex nonscattering region in optical tomography, Inverse Problems, 23 (2007), 2249-2270. doi: 10.1088/0266-5611/23/5/026.

[20]

N. Hyvönen, K. Karhunen and A. Seppänen, Fréchet derivative with respect to the shape of an internal electrode in electrical impedance tomography, SIAM J. Appl. Math, 70 (2010), 1878-1898. doi: 10.1137/09075929X.

[21]

J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography, Inverse Problems, 16 (2000), 1487-1522. doi: 10.1088/0266-5611/16/5/321.

[22]

J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,'' Springer-Verlag, New York, 2005.

[23]

A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9 (1993), 81-96.

[24]

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.

[25]

V. Kolehmainen, M. Vauhkonen, P. A. Karjalainen and J. P. Kaipio, Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns, Physiol. Meas., 18 (1997), 289-303.

[26]

A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography: A numerical study, Inverse Problems, 22 (2006), 1967-1987. doi: 10.1088/0266-5611/22/6/004.

[27]

A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography: Convergence by local injectivity, Inverse Problems, 24 (2008), 065009. doi: 10.1088/0266-5611/24/6/065009.

[28]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,'' \textbfI, Springer-Verlag, New York-Heidelberg, 1972.

[29]

A. Nissinen, L. M. Heikkinen and J. P. Kaipio, The Bayesian approximation error approach for electrical impedance tomography - experimental results, Meas. Sci. Technol., 19 (2008), 015501. doi: 10.1088/0957-0233/19/1/015501.

[30]

A. Nissinen, L. M. Heikkinen, V. Kolehmainen and J. P. Kaipio, Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography, Meas. Sci. Technol., 20 (2009), 105504. doi: 10.1088/0957-0233/20/10/105504.

[31]

A. Nissinen, V. Kolehmainen and J. P. Kaipio, Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography, IEEE Trans. Med. Imag., 30 (2011), 231-242. doi: 10.1109/TMI.2010.2073716.

[32]

J. Nocedal and S. J. Wright, "Numerical Optimization,'' Springer-Verlag, New York, 1999.

[33]

M. Soleimani, C. Gómez-Laberge and A. Adler, Imaging of conductivity changes and electrode movement in EIT, Physiol. Meas., 27 (2006), S103-S113. doi: 10.1088/0967-3334/27/5/S09.

[34]

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.

[35]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.

[36]

T. Vilhunen, J. P. Kaipio, P. J. Vauhkonen, T. Savolainen and M. Vauhkonen, Simultaneous reconstruction of electrode contact impedances and internal electrical properties: I. Theory, Meas. Sci. Technol., 13 (2002), 1848-1854. doi: 10.1088/0957-0233/13/12/307.

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