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Computing Volume Bounds of Inclusions by Eit Measurements

  • Giovanni Alessandrini
  • Antonio Bilotta
  • Antonino Morassi
  • Edi Rosset
  • Emilio Turco
Article

Abstract

The size estimates approach for Electrical Impedance Tomography (EIT) allows for estimating the size (area or volume) of an unknown inclusion in an electrical conductor by means of one pair of boundary measurements of voltage and current. In this paper we show by numerical simulations how to obtain such bounds for practical application of the method. The computations are carried out both in a 2-D and a 3-D setting.

Keywords

Size estimates Electrical impedance tomography 

References

  1. 1.
    Alessandrini, G.: Stable determination of conductivity by boundary measurements. Appl. Anal. 27, 153–172 (1988) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Mandache, N.: Exponential instability in an inverse problem for the Schrödinger equation. Inverse Probl. 17, 1435–1444 (2001) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alessandrini, G., Vessella, S.: Lipschitz stability for the inverse conductivity problem. Adv. Appl. Math. 35, 207–241 (2005) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Friedman, A.: Detection of mines by electric measurements. SIAM J. Appl. Math. 47, 201–212 (1987) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Friedman, A., Gustafsson, B.: Identification of the conductivity coefficient in an elliptic equation. SIAM J. Math. Anal. 18, 777–787 (1987) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Friedman, A., Isakov, V.: On the uniqueness in the inverse conductivity problem with one measurement. Indiana Univ. Math. J. 38, 563–579 (1989) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Alessandrini, G., Isakov, V.: Analyticity and uniqueness for the inverse conductivity problem. Rend. Ist. Mat. Univ. Trieste 28, 351–370 (1996) MATHMathSciNetGoogle Scholar
  8. 8.
    Fabes, E., Kang, H., Seo, J.K.: Inverse conductivity problem with one measurement: error estimates and approximate identification for perturbed disks. SIAM J. Math. Anal. 30, 699–720 (1999) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Alessandrini, G., Isakov, V., Powell, J.: Local uniqueness in the inverse conductivity problem with one measurement. Trans. Am. Math. Soc. 347, 3031–3041 (1995) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Isakov, V.: On uniqueness of recovery of a discontinuous conductivity coefficient. Commun. Pure Appl. Math. 41, 865–877 (1988) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Isakov, V.: Inverse Problems for Partial Differential Equations. Springer, New York (1998) MATHGoogle Scholar
  12. 12.
    Di Cristo, M., Rondi, L.: Examples of exponential instability for inverse inclusion and scattering problems. Inverse Probl. 19, 685–701 (2003) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Alessandrini, G., Rosset, E.: The inverse conductivity problem with one measurement: bounds on the size of the unknown object. SIAM J. Appl. Math. 58, 1060–1071 (1998) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kang, H., Seo, J.K., Sheen, D.: The inverse conductivity problem with one measurement: stability and estimation of size. SIAM J. Math. Anal. 28, 1389–1405 (1997) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Alessandrini, G., Rosset, E., Seo, J.K.: Optimal size estimates for the inverse conductivity problem with one measurement. Proc. Am. Math. Soc. 128, 53–64 (2000) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Alessandrini, G., Morassi, A., Rosset, E.: Size estimates. In: Alessandrini, G., Uhlmann, G. (eds.) Inverse Problems: Theory and Applications. Contemp. Math., vol. 333, pp. 1–33. American Mathematical Society, Providence (2003) Google Scholar
  17. 17.
    Ikehata, M.: Size estimation of inclusion. J. Inverse Ill-Posed Probl. 6, 127–140 (1998) MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Alessandrini, G., Morassi, A., Rosset, E.: Detecting an inclusion in an elastic body by boundary measurements. SIAM Rev. 46, 477–498 (2004). Revised and updated version of SIAM J. Math. Anal. 3, 1247–1268 (2002) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Alessandrini, G., Bilotta, A., Formica, G., Morassi, A., Rosset, E., Turco, E.: Numerical size estimates of inclusions in elastic bodies. Inverse Probl. 21, 133–151 (2005) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Cheng, K.S., Isaacson, D., Newell, J.C., Gisser, D.G.: Electrode models for electric current computed tomography. IEEE Trans. Biomed. Eng. 36, 918–924 (1989) CrossRefGoogle Scholar
  21. 21.
    Paulson, K., Breckon, W., Pidcock, M.: Electrode modelling in electrical impedance tomography. SIAM J. Appl. Math. 52, 1012–1022 (1992) MATHCrossRefGoogle Scholar
  22. 22.
    Somersalo, E., Cheney, M., Isaacson, D.: Existence and uniqueness for the electrode models for electric current computed tomography. SIAM J. Appl. Math. 52, 1023–1040 (1992) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Alessandrini, G., Rosset, E.: Volume bounds of inclusions from physical EIT measurements. Inverse Probl. 20, 575–588 (2004) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Aristodemo, M.: A high-continuity finite element model for two-dimensional elastic problems. Comput. Struct. 21, 987–993 (1985) MATHCrossRefGoogle Scholar
  25. 25.
    Bilotta, A., Formica, G., Turco, E.: Performances of a high-continuity finite element in three-dimensional elasticity. Report LabMeC No. 26, www.labmec.unical.it, 2003; submitted to Computer and Structures
  26. 26.
    Surowiec, A.J., Stuchly, S.S., Barr, J.R., Swarup, A.: Dielectric properties of breast carcinoma and the surrounding tissues. IEEE Trans. Biomed. Eng. 35, 257–263 (1988) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Giovanni Alessandrini
    • 1
  • Antonio Bilotta
    • 2
  • Antonino Morassi
    • 3
  • Edi Rosset
    • 1
  • Emilio Turco
    • 4
  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di TriesteTriesteItaly
  2. 2.Dipartimento di StruttureUniversità della CalabriaRende (CS)Italy
  3. 3.Dipartimento di Georisorse e TerritorioUniversità degli Studi di UdineUdineItaly
  4. 4.Dipartimento di Architettura e PianificazioneUniversità degli Studi di SassariAlgheroItaly

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