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 & 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.

[2]

L. Borcea, Electrical impedance tomography,, Inverse Problems, 18 (2002). 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.

[4]

M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Rev., 41 (1999), 85. 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. doi: 10.1109/10.35300.

[6]

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

[7]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,'', Springer-Verlag, (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, (1988).

[10]

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

[11]

R. Griesmaier and N. Hyvönen, A regularized Newton method for locating thin tubular conductivity inhomogeneities,, Inverse Problems, 27 (2011). 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. 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. doi: 10.1088/0957-0233/13/12/308.

[14]

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

[15]

F. Hettlich, Erratum: Fréchet derivatives in inverse obstacle scattering,, Inverse Problems, 14 (1998), 209. 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. 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, (2011).

[18]

N. Hyvönen, Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions,, SIAM J. Appl. Math, 64 (2004), 902. 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. 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. 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. doi: 10.1088/0266-5611/16/5/321.

[22]

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

[23]

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

[24]

V. Kolehmainen, M. Lassas and P. Ola, The inverse conductivity problem with an imperfectly known boundary,, SIAM J. Appl. Math, 66 (2005), 365. 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.

[26]

A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography: A numerical study,, Inverse Problems, 22 (2006), 1967. 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). doi: 10.1088/0266-5611/24/6/065009.

[28]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,'', \textbfI, (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). 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). 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. doi: 10.1109/TMI.2010.2073716.

[32]

J. Nocedal and S. J. Wright, "Numerical Optimization,'', Springer-Verlag, (1999).

[33]

M. Soleimani, C. Gómez-Laberge and A. Adler, Imaging of conductivity changes and electrode movement in EIT,, Physiol. Meas., 27 (2006). 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. doi: 10.1137/0152060.

[35]

G. Uhlmann, Electrical impedance tomography and Calderón's problem,, Inverse Problems, 25 (2009). 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. 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.

[2]

L. Borcea, Electrical impedance tomography,, Inverse Problems, 18 (2002). 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.

[4]

M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Rev., 41 (1999), 85. 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. doi: 10.1109/10.35300.

[6]

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

[7]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,'', Springer-Verlag, (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, (1988).

[10]

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

[11]

R. Griesmaier and N. Hyvönen, A regularized Newton method for locating thin tubular conductivity inhomogeneities,, Inverse Problems, 27 (2011). 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. 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. doi: 10.1088/0957-0233/13/12/308.

[14]

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

[15]

F. Hettlich, Erratum: Fréchet derivatives in inverse obstacle scattering,, Inverse Problems, 14 (1998), 209. 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. 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, (2011).

[18]

N. Hyvönen, Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions,, SIAM J. Appl. Math, 64 (2004), 902. 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. 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. 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. doi: 10.1088/0266-5611/16/5/321.

[22]

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

[23]

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

[24]

V. Kolehmainen, M. Lassas and P. Ola, The inverse conductivity problem with an imperfectly known boundary,, SIAM J. Appl. Math, 66 (2005), 365. 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.

[26]

A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography: A numerical study,, Inverse Problems, 22 (2006), 1967. 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). doi: 10.1088/0266-5611/24/6/065009.

[28]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,'', \textbfI, (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). 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). 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. doi: 10.1109/TMI.2010.2073716.

[32]

J. Nocedal and S. J. Wright, "Numerical Optimization,'', Springer-Verlag, (1999).

[33]

M. Soleimani, C. Gómez-Laberge and A. Adler, Imaging of conductivity changes and electrode movement in EIT,, Physiol. Meas., 27 (2006). 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. doi: 10.1137/0152060.

[35]

G. Uhlmann, Electrical impedance tomography and Calderón's problem,, Inverse Problems, 25 (2009). 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. doi: 10.1088/0957-0233/13/12/307.

[1]

Nuutti Hyvönen, Harri Hakula, Sampsa Pursiainen. Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (2) : 299-317. doi: 10.3934/ipi.2007.1.299

[2]

Fabrice Delbary, Rainer Kress. Electrical impedance tomography using a point electrode inverse scheme for complete electrode data. Inverse Problems & Imaging, 2011, 5 (2) : 355-369. doi: 10.3934/ipi.2011.5.355

[3]

Nicolay M. Tanushev, Luminita Vese. A piecewise-constant binary model for electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (2) : 423-435. doi: 10.3934/ipi.2007.1.423

[4]

Bastian Gebauer. Localized potentials in electrical impedance tomography. Inverse Problems & Imaging, 2008, 2 (2) : 251-269. doi: 10.3934/ipi.2008.2.251

[5]

Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, Jari P. Kaipio, Erkki Somersalo. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map. Inverse Problems & Imaging, 2015, 9 (3) : 767-789. doi: 10.3934/ipi.2015.9.767

[6]

Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, David Isaacson, Jari P. Kaipio, Debra McGivney, Erkki Somersalo, Joseph Volzer. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results. Inverse Problems & Imaging, 2015, 9 (3) : 749-766. doi: 10.3934/ipi.2015.9.749

[7]

Ville Kolehmainen, Matti Lassas, Petri Ola, Samuli Siltanen. Recovering boundary shape and conductivity in electrical impedance tomography. Inverse Problems & Imaging, 2013, 7 (1) : 217-242. doi: 10.3934/ipi.2013.7.217

[8]

Kari Astala, Jennifer L. Mueller, Lassi Päivärinta, Allan Perämäki, Samuli Siltanen. Direct electrical impedance tomography for nonsmooth conductivities. Inverse Problems & Imaging, 2011, 5 (3) : 531-549. doi: 10.3934/ipi.2011.5.531

[9]

Hideo Takaoka. Energy transfer model for the derivative nonlinear Schrödinger equations on the torus. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5819-5841. doi: 10.3934/dcds.2017253

[10]

Liliana Borcea, Fernando Guevara Vasquez, Alexander V. Mamonov. Study of noise effects in electrical impedance tomography with resistor networks. Inverse Problems & Imaging, 2013, 7 (2) : 417-443. doi: 10.3934/ipi.2013.7.417

[11]

Dong liu, Ville Kolehmainen, Samuli Siltanen, Anne-maria Laukkanen, Aku Seppänen. Estimation of conductivity changes in a region of interest with electrical impedance tomography. Inverse Problems & Imaging, 2015, 9 (1) : 211-229. doi: 10.3934/ipi.2015.9.211

[12]

Gen Nakamura, Päivi Ronkanen, Samuli Siltanen, Kazumi Tanuma. Recovering conductivity at the boundary in three-dimensional electrical impedance tomography. Inverse Problems & Imaging, 2011, 5 (2) : 485-510. doi: 10.3934/ipi.2011.5.485

[13]

Kimmo Karhunen, Aku Seppänen, Jari P. Kaipio. Adaptive meshing approach to identification of cracks with electrical impedance tomography. Inverse Problems & Imaging, 2014, 8 (1) : 127-148. doi: 10.3934/ipi.2014.8.127

[14]

J. Alberto Conejero, Marko Kostić, Pedro J. Miana, Marina Murillo-Arcila. Distributionally chaotic families of operators on Fréchet spaces. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1915-1939. doi: 10.3934/cpaa.2016022

[15]

Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825

[16]

Lassi Roininen, Janne M. J. Huttunen, Sari Lasanen. Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography. Inverse Problems & Imaging, 2014, 8 (2) : 561-586. doi: 10.3934/ipi.2014.8.561

[17]

Helmut Harbrecht, Thorsten Hohage. A Newton method for reconstructing non star-shaped domains in electrical impedance tomography. Inverse Problems & Imaging, 2009, 3 (2) : 353-371. doi: 10.3934/ipi.2009.3.353

[18]

Sarah Jane Hamilton, Andreas Hauptmann, Samuli Siltanen. A data-driven edge-preserving D-bar method for electrical impedance tomography. Inverse Problems & Imaging, 2014, 8 (4) : 1053-1072. doi: 10.3934/ipi.2014.8.1053

[19]

Melody Alsaker, Sarah Jane Hamilton, Andreas Hauptmann. A direct D-bar method for partial boundary data electrical impedance tomography with a priori information. Inverse Problems & Imaging, 2017, 11 (3) : 427-454. doi: 10.3934/ipi.2017020

[20]

Henrik Garde, Kim Knudsen. 3D reconstruction for partial data electrical impedance tomography using a sparsity prior. Conference Publications, 2015, 2015 (special) : 495-504. doi: 10.3934/proc.2015.0495

2016 Impact Factor: 1.094

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (17)

[Back to Top]