Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-25T22:54:14.461Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical determination of anomalies in multifrequency electrical impedance tomography

Published online by Cambridge University Press:  17 May 2018

HABIB AMMARI ,
FAOUZI TRIKI  and
CHUN-HSIANG TSOU
HABIB AMMARI
Affiliation:
Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland email: habib.ammari@math.ethz.ch
FAOUZI TRIKI
Affiliation:
Laboratoire Jean Kuntzmann, UMR CNRS 5224, Université Grenoble-Alpes, 700 Avenue Centrale, 38401 Saint-Martin-d'Hères, France email: faouzi.triki@univ-grenoble-alpes.fr
CHUN-HSIANG TSOU
Affiliation:
Laboratoire Jean Kuntzmann, UMR CNRS 5224, Université Grenoble-Alpes, 700 Avenue Centrale, 38401 Saint-Martin-d'Hères, France email: chun-hsiang.tsou@univ-grenoble-alpes.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The multifrequency electrical impedance tomography consists in retrieving the conductivity distribution of a sample by injecting a finite number of currents with multiple frequencies. In this paper, we consider the case where the conductivity distribution is piecewise constant, takes a constant value outside a single smooth anomaly, and a frequency dependent function inside the anomaly itself. Using an original spectral decomposition of the solution of the forward conductivity problem in terms of Poincaré variational eigenelements, we retrieve the Cauchy data corresponding to the extreme case of a perfect conductor, and the conductivity profile. We then reconstruct the anomaly from the Cauchy data. The numerical experiments are conducted using gradient descent optimization algorithms.

Keywords

Inverse problemsmultifrequency electric impedance tomographyanomalies reconstruction
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 3 , June 2019 , pp. 481 - 504
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01).

References

[1] Alberti, G. S., Ammari, H., Jin, B., Seo, J.-K. & Zhang, W. (2016) The linearized inverse problem in multifrequency electrical impedance tomography. SIAM J. Imag. Sci. 9, 1525–1551.Google Scholar
[2] Ammari, H., Boulier, T. & Garnier, J. (2013) Modeling active electrolocation in weakly electric fish. SIAM J. Imaging Sci 6, 285–321.Google Scholar
[3] Ammari, H., Boulier, T., Garnier, J. & Wang, H. (2014) Shape recognition and classification in electro-sensing. Proc. Natl. Acad. Sci. USA 111, 11652–11657.Google Scholar
[4] Ammari, H., Chow, Y. T., Liu, K. & Zou, J. (2015) Optimal shape design by partial spectral data. SIAM J. Sci. Comput. 37, B855–B883.Google Scholar
[5] Ammari, H., Deng, Y. & Millien, P. (2016) Surface plasmon resonance of nanoparticles and applications in imaging. Arch. Ration. Mech. Anal. 220, 109–153.Google Scholar
[6] Ammari, H. & Kang, H. (2004) Reconstruction of Small Inhomogeneities from Boundary Measurements Lecture Notes in Mathematics, Vol. 1846, Springer-Verlag, Berlin.Google Scholar
[7] Ammari, H., Garnier, J., Giovangigli, L., Jing, W. & Seo, J. K. (2016) Spectroscopic imaging of a dilute cell suspension. J. Math. Pures Appl. 105, 603–661.Google Scholar
[8] Ammari, H. & Kang, H. (2004) Reconstruction of Small Inhomogeneities from Boundary Measurements Lecture Notes in Mathematics, Vol. 1846, Springer-Verlag, Berlin.Google Scholar
[9] Ammari, H. & Seo, J. K. (2003) An accurate formula for the reconstruction of conductivity inhomogeneities. Adv. Appl. Math. 30, 679–705.Google Scholar
[10] Ammari, H. & Triki, F. (2017) Identification of an Inclusion in Multifrequency Electric Impedance Tomography. Communications in Partial Differential Equations, Vol. 42(1), Taylor & Francis, pp. 159–177.Google Scholar
[11] Ammari, H., Kang, H., Lim, M. & Zribi, H. (2010) Conductivity interface problems. Part I: small perturbations of an interface. Trans. Am. Math. Soc. 362 (5), 2435–2449.Google Scholar
[12] Ammari, H., Millien, P., Ruiz, M. & Zhang, H. (2017) Mathematical analysis of plasmonic nanoparticles: the scalar case. Arch. Ration. Mech. Anal. 224, 597–658.Google Scholar
[13] Ando, K. & Kang, H. (2016) Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operator. J. Math. Anal. Appl. 435, 162–178.Google Scholar
[14] Ando, K., Kang, H., & Liu, H. (2016) Plasmon resonance with finite frequencies: A validation of the quasi-static approximation for diametrically small inclusions. SIAM J. Appl. Math. 76, 731–749.Google Scholar
[15] Borcea, L. (2002) Electrical impedance tomography. Inverse Problems 18 (6), R99R136.Google Scholar
[16] Bonnetier, E. & Triki, F. (2013) On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D. Arch. Ration. Mech. Anal. 209, 541–567.Google Scholar
[17] Colton, D. & Kress, R. (1998) Inverse Acoustic and Electromagnetic Scattering Theory, Springer Verlag.Google Scholar
[18] Gabriel, C., Peyman, A. & Grant, E. H. (2009) Electrical conductivity of tissue at frequencies below 1 MHz. Phys. Med. Biol. 54, 4863–4878.Google Scholar
[19] Griesmaier, R. & Hanke, H. (2015) Multifrequency impedance imaging with multiple signal classification. SIAM J. Imaging Sci. 8, 939–967.Google Scholar
[20] Hecht, F. (2012) New development in FreeFem++. J. Numer. Math. 20 (3–4), 251–265.Google Scholar
[21] Jang, J. & Seo, J. K. (2015) Detection of admittivity anomaly on high-contrast heterogeneous backgrounds using frequency difference EIT. Phys. Meas. 36, 1179–1192.Google Scholar
[22] Ando, K., Kang, H. & Miyanishi, Y. (2016) Exponential decay estimates of the eigenvalues for the Neumann-Poincaré operator on analytic boundaries in two dimensions, to appear in J. Integral Equations Applications (2018).Google Scholar
[23] Kang, H., Kim, K. & Lee, H. (2016) Spectral properties of the Neumann Poincaré operator and uniformity of estimates for the conductivity equation with complex coefficients. J. Lond. Math. Soc. 93, 519–545.Google Scholar
[24] Klibanov, M. V. & Santosa, F. (1991) A computational quasi-reversibility method for Cauchy problems for Laplace's equation. SIAM J. Appl. Math. 51, 1653–1675.Google Scholar
[25] Lattés, R. & Lions, J. L. (1969) The Method of Quasi-reversibility. Applications to Partial Differential Equations, American Elsevier, New York.Google Scholar
[26] Malone, E., Sato dos Santos, G., Holder, D. & Arridge, S. (2014) Multifrequency electrical impedance tomography using spectral constraints. IEEE Trans. Med. Imag. 33, 340–350.Google Scholar
[27] Mayergoyz, I. D., Fredkin, D. R. & Zhang, Z. (2005) Electrostatic (plasmon) resonances in nanoparticles. Phys. Rev. B 72, 155412.Google Scholar
[28] Milton, G. W. & Nicorovici, N.-A. P. (2006) On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. A 462, 3027–3059.Google Scholar
[29] Miyanishi, Y. & Suzuki, T. (2017) Eigenvalues and eigenfunctions of double layer potentials. Transactions of the American Mathematical Society 369 (11), 8037–8059.Google Scholar
[30] Perfekt, K.-M. & Putinar, M. (2014) Spectral bounds for the Neumann-Poincaré operator on planar domains with corners. J. Anal. Math. 124, 39–57.Google Scholar
[31] Rundell, W. (2009) Recovering an obstacle using integral equations. Inverse Problems Imaging 3/2, 319–332.Google Scholar