Numerische Mathematik

, Volume 117, Issue 2, pp 373–396 | Cite as

Convex backscattering support in electric impedance tomography

  • Martin HankeEmail author
  • Nuutti Hyvönen
  • Stefanie Reusswig


This paper reinvestigates a recently introduced notion of backscattering for the inverse obstacle problem in impedance tomography. Under mild restrictions on the topological properties of the obstacles, it is shown that the corresponding backscatter data are the boundary values of a function that is holomorphic in the exterior of the obstacle(s), which allows to reformulate the obstacle problem as an inverse source problem for the Laplace equation. For general obstacles, the convex backscattering support is then defined to be the smallest convex set that carries an admissible source, i.e., a source that yields the given (backscatter) data as the trace of the associated potential. The convex backscattering support can be computed numerically; numerical reconstructions are included to illustrate the viability of the method.

Mathematics Subject Classification (2000)

35R30 65N21 


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  1. 1.
    Brühl M.: Explicit characterization of inclusions in electrical impedance tomography. SIAM J. Math. Anal. 32, 1327–1341 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bryan K.: Numerical recovery of certain discontinuous electrical conductivities. Inverse Probl. 7, 827–840 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Dautray R., Lions J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 2. Springer, Berlin (1988)Google Scholar
  4. 4.
    Gebauer B.: The factorization method for real elliptic problems. Z. Anal. Anwendungen 25, 81–102 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Haddar H., Kusiak S., Sylvester J.: The convex back-scattering support. SIAM J. Appl. Math. 66, 591–615 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hanke M.: On real-time algorithms for the location search of discontinuous conductivities with one measurement. Inverse Probl. 24, 045005 (2008)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Hanke M., Hyvönen N., Lehn M., Reusswig S.: Source supports in electrostatics. BIT Numer. Math. 48, 245–264 (2008)zbMATHCrossRefGoogle Scholar
  8. 8.
    Hanke M., Hyvönen N., Reusswig S.: Convex source support and its application to electric impedance tomography. SIAM J. Imaging Sci. 1, 364–378 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hanke M., Hyvönen N., Reusswig S.: An inverse backscatter problem for electric impedance tomography. SIAM J. Math. Anal. 41, 1948–1966 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hanke M., Kirsch A.: Samplingmethods. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging, Springer, New York (2010)Google Scholar
  11. 11.
    Henrici P.: Applied and Computational Complex Analysis, vol. 1. Wiley, New York (1974)zbMATHGoogle Scholar
  12. 12.
    Hettlich F., Rundell W.: The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Probl. 14, 67–82 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hyvönen N.: Complete electrode model of electrical impedance tomography: approximation properties and characterization of inclusions. SIAM J. Appl. Math. 64, 902–931 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hyvönen N.: Application of a weaker formulation of the factorization method to the characterization of absorbing inclusions in optical tomography. Inverse Probl. 21, 1331–1343 (2005)zbMATHCrossRefGoogle Scholar
  15. 15.
    Hyvönen N.: Application of the factorization method to the characterization of weak inclusions in electrical impedance tomography. Adv. Appl. Math. 39, 197–221 (2007)zbMATHCrossRefGoogle Scholar
  16. 16.
    Ito K., Kunisch K., Li Z.: Level-set function approach to an inverse interface problem. Inverse Probl. 17, 1225–1242 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kang H., Seo J.K., Sheen D.: Numerical identification of discontinuous conductivity coefficients. Inverse Probl. 13, 113–123 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kirsch A.: The factorization method for a class of inverse elliptic problems. Math. Nachr. 278, 258–277 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kress R.: Inverse Dirichlet problem and conformal mapping. Math. Comput. Simul. 66, 255–265 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kress R., Kühn L.: Linear sampling methods for inverse boundary value problems in potential theory. Appl. Numer. Math. 43, 161–173 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Kress R., Rundell W.: Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Probl. 21, 1207–1223 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Kusiak S., Sylvester J.: The scattering support. Comm. Pure Appl. Math. 56, 1525–1548 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Lions J.-L., Magenes E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin (1972)Google Scholar
  24. 24.
    Rundell W.: Recovering an obstacle and its impedance from Cauchy data. Inverse Probl. 24, 045003 (2008)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Tutschke W., Vasudeva H.L.: An Introduction to Complex Analysis: Classical and Modern Approaches. Chapman & Hall, Boca Raton (2005)zbMATHGoogle Scholar
  26. 26.
    Triebel H.: Interpolation Theory, Function Spaces, Differential Operators. VEB Deutscher Verlag der Wissenschaften, Berlin (1978)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Martin Hanke
    • 1
    Email author
  • Nuutti Hyvönen
    • 2
  • Stefanie Reusswig
    • 1
  1. 1.Institut für MathematikJohannes Gutenberg-Universität MainzMainzGermany
  2. 2.Institute of MathematicsHelsinki University of TechnologyHUTFinland

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