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 & 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.

[2]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", vol. \textbf{93} of Applied Mathematical Sciences, 93 (1998).

[3]

F. Delbary and R. Kress, Electrical impedance tomography with point electrodes,, J. Integral Equations Appl., 22 (2010), 193. 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. 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. 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., ().

[7]

N. Hyvönen, Approximating idealized boundary data of electric impedance tomography by electrode measurements,, Math. Models and Meth. in Appl. Sciences, 19 (2009), 1185. 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. doi: 10.1137/040612737.

[9]

R. Kress, "Linear Integral Equations,", vol. 82 of Applied Mathematical Sciences, 82 (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. 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. 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. doi: 10.1006/jmaa.2001.7615.

show all references

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.

[2]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", vol. \textbf{93} of Applied Mathematical Sciences, 93 (1998).

[3]

F. Delbary and R. Kress, Electrical impedance tomography with point electrodes,, J. Integral Equations Appl., 22 (2010), 193. 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. 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. 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., ().

[7]

N. Hyvönen, Approximating idealized boundary data of electric impedance tomography by electrode measurements,, Math. Models and Meth. in Appl. Sciences, 19 (2009), 1185. 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. doi: 10.1137/040612737.

[9]

R. Kress, "Linear Integral Equations,", vol. 82 of Applied Mathematical Sciences, 82 (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. 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. 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. doi: 10.1006/jmaa.2001.7615.

[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]

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

[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]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

Hiroshi Isozaki. Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (1) : 107-134. doi: 10.3934/ipi.2007.1.107

[13]

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

[14]

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

[15]

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

[16]

María J. Rivera, Juan A. López Molina, Macarena Trujillo, Enrique J. Berjano. Theoretical modeling of RF ablation with internally cooled electrodes: Comparative study of different thermal boundary conditions at the electrode-tissue interface. Mathematical Biosciences & Engineering, 2009, 6 (3) : 611-627. doi: 10.3934/mbe.2009.6.611

[17]

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

[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]

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

[20]

Melody Dodd, Jennifer L. Mueller. A real-time D-bar algorithm for 2-D electrical impedance tomography data. Inverse Problems & Imaging, 2014, 8 (4) : 1013-1031. doi: 10.3934/ipi.2014.8.1013

2016 Impact Factor: 1.094

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]