Skip to main content
Log in

Computing Volume Bounds of Inclusions by Eit Measurements

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alessandrini, G.: Stable determination of conductivity by boundary measurements. Appl. Anal. 27, 153–172 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  2. Mandache, N.: Exponential instability in an inverse problem for the Schrödinger equation. Inverse Probl. 17, 1435–1444 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alessandrini, G., Vessella, S.: Lipschitz stability for the inverse conductivity problem. Adv. Appl. Math. 35, 207–241 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Friedman, A.: Detection of mines by electric measurements. SIAM J. Appl. Math. 47, 201–212 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Friedman, A., Gustafsson, B.: Identification of the conductivity coefficient in an elliptic equation. SIAM J. Math. Anal. 18, 777–787 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  6. Friedman, A., Isakov, V.: On the uniqueness in the inverse conductivity problem with one measurement. Indiana Univ. Math. J. 38, 563–579 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  7. Alessandrini, G., Isakov, V.: Analyticity and uniqueness for the inverse conductivity problem. Rend. Ist. Mat. Univ. Trieste 28, 351–370 (1996)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  10. Isakov, V.: On uniqueness of recovery of a discontinuous conductivity coefficient. Commun. Pure Appl. Math. 41, 865–877 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  11. Isakov, V.: Inverse Problems for Partial Differential Equations. Springer, New York (1998)

    MATH  Google Scholar 

  12. Di Cristo, M., Rondi, L.: Examples of exponential instability for inverse inclusion and scattering problems. Inverse Probl. 19, 685–701 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Ikehata, M.: Size estimation of inclusion. J. Inverse Ill-Posed Probl. 6, 127–140 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  21. Paulson, K., Breckon, W., Pidcock, M.: Electrode modelling in electrical impedance tomography. SIAM J. Appl. Math. 52, 1012–1022 (1992)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  23. Alessandrini, G., Rosset, E.: Volume bounds of inclusions from physical EIT measurements. Inverse Probl. 20, 575–588 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Aristodemo, M.: A high-continuity finite element model for two-dimensional elastic problems. Comput. Struct. 21, 987–993 (1985)

    Article  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Emilio Turco.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alessandrini, G., Bilotta, A., Morassi, A. et al. Computing Volume Bounds of Inclusions by Eit Measurements. J Sci Comput 33, 293–312 (2007). https://doi.org/10.1007/s10915-007-9153-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-007-9153-9

Keywords

Navigation