A Note on Reconstructing the Conductivity in Impedance Tomography by Elastic Perturbation

Conference paper
Part of the Mathematics for Industry book series (MFI, volume 1)


We give a short review on the hybrid inverse problem of reconstructing the conductivity in a medium in \(\mathbf {R}^n, n = 2,3,\) from the knowledge of the pointwise values of the energy densities associated with imposed boundary voltages. We show that given \(n\) boudary voltages, the associated voltage potentials solve an elliptic system of PDE’s in the subregions where they define a diffeomorphism, from which stability estimates can be obtained.


Inverse conductivity Calderón’s problem Hybrid methods Stability 


  1. 1.
    Alessandrini, G.: Stable determination of conductivity by boundary measurements. App. Anal. 27, 153172 (1988)MathSciNetGoogle Scholar
  2. 2.
    Alessandrini, G., Nesi, V.: Univalent \(\sigma \)-harmonic mappings. Arch. Rational Mech. Anal. 158, 155–171 (2001)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Ammari, H., Bonnetier, E., Capdeboscq, Y., Tanter, M., Fink, M.: Electrical impedance tomography by elastic deformation. SIAM J. Appl. Math. 68(6), 1557–1573 (2008)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bal, G.: Cauchy problem for ultrasound modulated EIT. To appear in analysis and PDE (2013).Google Scholar
  5. 5.
    Bal, G., Bonnetier, E., Monard, F., Triki, F.: Inverse diffusion from knowledge of power densities. Inverse Probl. Imag. 7(2), 353–375 (2013)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bal, G., Monard, F.: Inverse diffusion problem with redundant internal information. Inverse Probl. 29(8), 084001 (2012)MathSciNetGoogle Scholar
  7. 7.
    Bal, G., Ren, K., Uhlmann, G., Zhou, T.: Quantitative thermoacoustics and related problems. Inverse Probl. 27, 055007 (2011)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Bal, G., Schotland, J.C.: Inverse scattering and acousto-optics imaging. Phys. Rev. Lett. 104, 043902 (2010)CrossRefGoogle Scholar
  9. 9.
    Bal, G., Uhlmann, G.: Reconstructions for some coupled-physics inverse problems. Appl. Math. Lett. 25–7, 1030–1033 (2012)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Briane, M., Milton, G.W., Nesi, V.: Change of sign of the correctors determinant for homogenization in three-dimensional conductivity. Arch. Rational Mech. Anal. 173, 133–150 (2004)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Capdeboscq, Y., Fehrenbach, J., de Gournay, F., Kavian, O.: Imaging by modification: numerical reconstruction of local conductivities from corresponding power density measurements. SIAM J. Imag. Sci. 2, 1003–1030 (2009)CrossRefMATHGoogle Scholar
  12. 12.
    Gebauer, B., Scherzer, O.: Impedance-acoustic tomography. SIAM J. Appl. Math. 69–2, 565–576 (2008)Google Scholar
  13. 13.
    Li, C., Wang, L.V.: Photoacoustic tomogtaphy and sensing in biomedicine. Phys. Med. Biol. 54, R59–R97 (2009)CrossRefGoogle Scholar
  14. 14.
    Pride, S.R.: Governing equations for the coupled electro-magnetics and acoustics of porous media. Phys. Rev. B 50, 15678–15696 (1994)CrossRefGoogle Scholar
  15. 15.
    Stefanov, p., Uhlmann, G.: Multi-wave methods via ultrasound, vol. 60, pp. 271–324. In Inside Out II, MSRI Publications (2012).Google Scholar
  16. 16.
    Uhlmann, G.: Electrical impedance tomography and Calderón’s problem. Inverse Probl. 25, 123011 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Laboratoire Jean KuntzmannUniversité Grenoble-Alpes and CNRSGrenobleFrance

Personalised recommendations