Abstract
Electrical impedance tomography (EIT) is a medical imaging technique in which an image of the conductivity (or permittivity) of part of the body is determined from electrical surface measurements. Typically, conducting electrodes are attached to the skin of the subject and small alternating currents are applied to some or all of the electrodes. The resulting electrical potentials are measured, and the process may be repeated for numerous different configurations of applied currents.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
If the domain is simply connected.
- 2.
Provided γ, f, and u are smooth enough.
- 3.
Here, L 2(D)2 denotes the space of vector-valued functions \(D \rightarrow {\mathbb{R}}^{2}\) such that both components are in L 2(D).
- 4.
Here we make use of the assumption that \(B \setminus \overline{D}\) is connected.
- 5.
Take a sequence (x j ) in X such that Ax j  → y. Then \(0 = {({A}^{{_\ast}}y,{x}_{j})}_{X} = {(y,A{x}_{j})}_{Y } \rightarrow {(y,y)}_{Y }\); that is, y = 0.
References
R.A. Adams and J. Fournier. Sobolev Spaces. Academic Press, 2nd, repr. edition, 2005.
K. Astala and L. Päivärinta. Calderón’s inverse conductivity problem in the plane. Ann. Math., 163:265–299, 2006.
L. Borcea. Electrical impedance tomography. Inverse Problems, 18:R99–R136, 2002.
A.P. Calderón. On an inverse boundary value problem. In Seminar on Numerical Analysis and its Applications to Continuum Mechanics, pages 65–73, Rio de Janerio, 1980. Soc. Brasileira de Matemática.
M. Cheney, D. Isaacson, and J.C. Newell. Electrical impedance tomography. SIAM Review, 41:85–101, 1999.
D. Colton and R. Kress. Integral Equation Methods in Scattering Theory. Wiley-Interscience, New York, 1983.
M. Hanke and M. Brühl. Recent progress in electrical impedance tomography. Inverse Problems, 19:S65–S90, 2003.
D. Isaacson and M. Cheney. Effects of measurement precision and finite number of electrodes on linear impedance imaging algorithms. SIAM J. Appl. Math., 51:1705–1731, 1991.
S. Järvenpäa and E. Somersalo. Impedance imaging and electrode models. In Inverse Problems in Medical Imaging and Nondestructive Testing, pages 65–74, Vienna, 1996. Springer. Proceedings of the Conference in Oberwolfach.
A. Lechleiter. A regularization technique for the factorization method. Inverse Problems, 22:1605–1625, 2006.
W. McLean. Strongly Elliptic Systems and Boundary Integral Operators. Cambridge University Press, Cambridge, UK, 2000.
E. Somersalo, M. Cheney, and D. Isaacson. Existence and uniqueness for electrode models for electric current computed tomography. SIAM J. Appl. Math., 52:1023–1040, 1992.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Kirsch, A. (2011). An Inverse Problem in Electrical Impedance Tomography. In: An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences, vol 120. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8474-6_5
Download citation
DOI: https://doi.org/10.1007/978-1-4419-8474-6_5
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-8473-9
Online ISBN: 978-1-4419-8474-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)