Abstract

Electrical impedance tomography is an imaging modality for extracting information on the interior structure of a physical body from boundary measurements of current and voltage. This work studies a new robust way of modeling the contact electrodes used for driving current patterns into the examined object and measuring the resulting voltages. The idea is to not define the electrodes as strict geometric objects on the measurement boundary but only to assume approximate knowledge about their whereabouts and let a boundary admittivity function determine the actual locations of the current inputs. Such an approach enables reconstructing the boundary admittivity, i.e., the locations and strengths of the contacts, at the same time and with analogous methods as the interior admittivity. The functionality of the new model is verified by two-dimensional numerical experiments based on water tank data.

Keywords

  1. electrical impedance tomography
  2. electrode models
  3. varying contact admittance
  4. Bayesian inversion
  5. extended electrodes

MSC codes

  1. 35R30
  2. 35J25
  3. 65N21

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
D. C. Barber and B. H. Brown, Applied potential tomography, J. Phys. E Sci. Instrum., 17 (1984), pp. 723--733.
2.
D. C. Barber and B. H. Brown, Errors in reconstruction of resistivity images using a linear reconstruction technique, Clin. Phys. Physiol. Meas., 9 (1988), pp. 101--104.
3.
L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), pp. R99--R136.
4.
W. Breckon and M. Pidcock, Data errors and reconstruction algorithms in electrical impedance tomography, Clin. Phys. Physiol. Meas., 9 (1988), pp. 105--109.
5.
V. Candiani, A. Hannukainen, and N. Hyvönen, Computational framework for applying electrical impedance tomography to head imaging, SIAM J. Sci. Comput., 41 (2019), pp. B1034--B1060.
6.
M. Cheney, D. Isaacson, and J. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), pp. 85--101.
7.
K.-S. Cheng, D. Isaacson, J. S. Newell, and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Trans. Biomed. Eng., 36 (1989), pp. 918--924.
8.
P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, 2013.
9.
J. Dardé, H. Hakula, N. Hyvönen, and S. Staboulis, Fine-tuning electrode information in electrical impedance tomography, Inverse Probl. Imag., 6 (2012), pp. 399--421.
10.
J. Dardé, N. Hyvönen, A. Seppänen, and S. Staboulis, Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography, SIAM J. Imaging Sci., 6 (2013), pp. 176--198.
11.
J. Dardé, N. Hyvönen, A. Seppänen, and S. Staboulis, Simultaneous recovery of admittivity and body shape in electrical impedance tomography: An experimental evaluation, Inverse Problems, 29 (2013), 085004.
12.
J. Dardé and S. Staboulis, Electrode modelling: The effect of contact impedance, ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 415--431.
13.
M. Hanke, B. Harrach, and N. Hyvönen, Justification of point electrode models in electrical impedance tomography, Math. Models Methods Appl. Sci., 21 (2011), pp. 1395--1413.
14.
B. Harrach, An introduction to finite element methods for inverse coefficient problems in elliptic PDEs, Jahresber. Dtsch. Math.-Ver., 123 (2021), pp. 183--210.
15.
N. Hyvönen, Approximating idealized boundary data of electric impedance tomography by electrode measurements, Math. Models Methods Appl. Sci., 19 (2009), pp. 1185--1202.
16.
N. Hyvönen, V. Kaarnioja, L. Mustonen, and S. Staboulis, Polynomial collocation for handling an inaccurately known measurement configuration in electrical impedance tomography, SIAM J. Appl. Math., 77 (2017), pp. 202--223.
17.
N. Hyvönen, H. Majander, and S. Staboulis, Compensation for geometric modeling errors by positioning of electrodes in electrical impedance tomography, Inverse Problems, 33 (2017), 035006.
18.
N. Hyvönen and L. Mustonen, Smoothened electrode model, SIAM J. Appl. Math., 77 (2017), pp. 2250--2271.
19.
N. Hyvönen and L. Mustonen, Generalized linearization techniques in electrical impedance tomography, Numer. Math., 140 (2018), pp. 95--120.
20.
J. Jauhiainen, P. Kuusela, A. Seppänen, and T. Valkonen, Relaxed Gauss--Newton methods with applications to electrical impedance tomography, SIAM J. Imaging Sci., 13 (2020), pp. 1415--1445, https://doi.org/10.1137/20M1321711.
21.
J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer, New York, 2005.
22.
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), pp. 1487--1522.
23.
V. Kolehmainen, M. Lassas, and P. Ola, Inverse conductivity problem with an imperfectly known boundary, SIAM J. Appl. Math., 66 (2005), pp. 365--383.
24.
V. Kolehmainen, M. Lassas, and P. Ola, The inverse conductivity problem with an imperfectly known boundary in three dimensions, SIAM J. Appl. Math., 67 (2007), pp. 1440--1452.
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), pp. 289--303.
26.
J. Kourunen, T. Savolainen, A. Lehikoinen, M. Vauhkonen, and L. M. Heikkinen, Suitability of a PXI platform for an electrical impedance tomography system, Meas. Sci. Technol., 20 (2009), 015503.
27.
A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography: A numerical study, Inverse Problems, 22 (2006), pp. 1967--1987.
28.
J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer Monogr. Math., Springer, Heidelberg, 2012, https://doi.org/10.1007/978-3-642-10455-8.
29.
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), pp. 231--242.
30.
A. Nissinen, V. Kolehmainen, and J. P. Kaipio, Reconstruction of domain boundary and conductivity in electrical impedance tomography using the approximation error approach, Int. J. Uncertain. Quantif., 1 (2011), pp. 203--222.
31.
J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, 1999.
32.
S. Salzo and S. Villa, Convergence analysis of a proximal Gauss--Newton method, Comput. Optim. Appl., 53 (2012), pp. 557--589, https://doi.org/10.1007/s10589-012-9476-9.
33.
J. R. Shewchuk, Triangle: Engineering a 2D quality mesh generator and delaunay triangulator, in Applied Computational Geometry: Towards Geometric Engineering, Lecture Notes in Comput. Sci. 1148, M. C. Lin and D. Manocha, eds., Springer, Cham, Switzerland, pp. 203--222.
34.
M. Soleimani, C. Gómez-Laberge, and A. Adler, Imaging of conductivity changes and electrode movement in EIT, Physiol. Meas., 27 (2006), pp. S103--S113.
35.
E. Somersalo, M. Cheney, and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), pp. 1023--1040.
36.
G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011.
37.
T. Valkonen, Regularisation, optimisation, subregularity, Inverse Problems, 37 (2021), 045010, https://doi.org/10.1088/1361-6420/abe4aa.
38.
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), pp. 1848--1854.

Information & Authors

Information

Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 427 - 449
ISSN (online): 1095-712X

History

Submitted: 3 February 2021
Accepted: 19 October 2021
Published online: 15 March 2022

Keywords

  1. electrical impedance tomography
  2. electrode models
  3. varying contact admittance
  4. Bayesian inversion
  5. extended electrodes

MSC codes

  1. 35R30
  2. 35J25
  3. 65N21

Authors

Affiliations

Funding Information

Embassy of France in Finland
French Ministry of Higher Education, Research, and Innovation
Finnish Society of Sciences and Letters
Academy of Finland https://doi.org/10.13039/501100002341 : 312124, 314701, 320022
Institut Français de Finlande https://doi.org/10.13039/501100004589
Suomalainen Tiedeakatemia https://doi.org/10.13039/501100002342

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

View Options

View options

PDF

View PDF

Figures

Tables

Media

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media