# American Institute of Mathematical Sciences

May  2011, 5(2): 485-510. doi: 10.3934/ipi.2011.5.485

## Recovering conductivity at the boundary in three-dimensional electrical impedance tomography

 1 Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810 2 Department of Physics and Mathematics, University of Eastern Finland, FIN-70211 Kuopio, Finland 3 University of Helsinki, Department of Mathematics and Statistics, FI-00014 Helsinki, Finland 4 Department of Mathematics, Graduate School of Engineering, Gunma University, Kiryu 376-8515, Japan

Received  April 2010 Revised  August 2010 Published  May 2011

The aim of electrical impedance tomography (EIT) is to reconstruct the conductivity values inside a conductive object from electric measurements performed at the boundary of the object. EIT has applications in medical imaging, nondestructive testing, geological remote sensing and subsurface monitoring. Recovering the conductivity and its normal derivative at the boundary is a preliminary step in many EIT algorithms; Nakamura and Tanuma introduced formulae for recovering them approximately from localized voltage-to-current measurements in [Recent Development in Theories & Numerics, International Conference on Inverse Problems 2003]. The present study extends that work both theoretically and computationally. As a theoretical contribution, reconstruction formulas are proved in a more general setting. On the computational side, numerical implementation of the reconstruction formulae is presented in three-dimensional cylindrical geometry. These experiments, based on simulated noisy EIT data, suggest that the conductivity at the boundary can be recovered with reasonable accuracy using practically realizable measurements. Further, the normal derivative of the conductivity can also be recovered in a similar fashion if measurements from a homogeneous conductor (dummy load) are available for use in a calibration step.
Citation: Gen Nakamura, Päivi Ronkanen, Samuli Siltanen, Kazumi Tanuma. Recovering conductivity at the boundary in three-dimensional electrical impedance tomography. Inverse Problems & Imaging, 2011, 5 (2) : 485-510. doi: 10.3934/ipi.2011.5.485
##### References:
 [1] A. Adler, R. Guardo, and Y. Berthiaume, Impedance imaging of lung ventilation: Do we need to account for chest expansion?, IEEE Trans. Biomed. Eng., 43 (1996), 414. doi: 10.1109/10.486261. [2] G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements,, J. Diff. Eq., 84 (1990), 252. doi: 10.1016/0022-0396(90)90078-4. [3] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math., 163 (2006), 265. doi: 10.4007/annals.2006.163.265. [4] J. Bikowski, "Electrical Impedance Tomography Reconstructions in two and three Dimensions; From Calderón to Direct Methods,", Ph.D thesis, (2008). [5] R. Blue, "Real-time Three-dimensional Electrical Impedance Tomography,", Ph.D thesis, (1997). [6] L. Borcea, Electrical impedance tomography,, Inverse Problems, 18 (2002). doi: 10.1088/0266-5611/18/6/201. [7] L. Borcea, Addendum to "Electrical impedance tomography",, Inverse Problems, 19 (2002), 997. doi: 10.1088/0266-5611/19/4/501. [8] G. Boverman, D. Isaacson, T-J Kao, G. J. Saulnier and J. C. Newell, "Methods for Direct Image Reconstruction for EIT in Two and Three Dimensions,", in, (2008). [9] R. M. Brown, Recovering the conductivity at the boundary from the Dirichlet to Neumann map: a pointwise result,, J. Inverse and Ill-posed Prob., 9 (2001), 567. [10] R. Brown and R. Torres, Uniqueness in the inverse conductivity problem for conductivities with 3/2 derivatives in $L^p, p>2n,$, J. Fourier Analysis Appl., 9 (2003), 1049. [11] R. M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions,, Comm. Partial Differential Equations, 22 (1997), 1009. doi: 10.1080/03605309708821292. [12] A. P. Calderón, On an inverse boundary value problem,, Seminar on Numerical Analysis and its Applications to Continuum Physics, (1980), 65. [13] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85. doi: 10.1137/S0036144598333613. [14] K-S Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography,, IEEE Transactions on Biomedical Imaging, (1989), 918. [15] R. D. Cook, G. J. Saulnier and J. C. Goble, A phase sensitive voltmeter for a high-speed, high-precision electrical impedance tomograph,, in, (1991), 22. doi: 10.1109/IEMBS.1991.683822. [16] H. Cornean, K. Knudsen and S. Siltanen, Towards a d-bar reconstruction method for three-dimensional EIT,, Journal of Inverse and Ill-Posed Problems, 14 (2006), 111. doi: 10.1515/156939406777571102. [17] R. Courant and D. Hilbert, "Methods of Mathematical Physics,", Interscience Publishers, II (1962). [18] E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge University Press, (1989). doi: 10.1017/CBO9780511566158. [19] B. Gebauer and N. Hyvönen, Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem,, Inverse Probl. Imaging, 2 (2008), 355. doi: 10.3934/ipi.2008.2.355. [20] E. Gersing, B. Hoffman, and M. Osypka, Influence of changing peripheral geometry on electrical impedance tomography measurements,, Medical & Biological Engineering & Computing, 34 (1996), 359. doi: 10.1007/BF02520005. [21] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundlehren der Mathematischen Wissenschaften, 224 (1989). [22] J. Goble, M. Cheney and D. Isaacson, Electrical impedance tomography in three dimensions, Appl. Comput. Electromagn. Soc. J., 7 (1992), 128. [23] A. Greenleaf, M. Lassas and G. Uhlmann, The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction,, Comm. Pure Appl. Math., 56 (2003), 328. doi: 10.1002/cpa.10061. [24] M. Hanke and B. Schappel, The factorization method for electrical impedance tomography in the half-space,, SIAM J. Appl. Math., 68 (2008), 907. doi: 10.1137/06067064X. [25] T. Ide, H. Isozaki, S. Nakata and S. Siltanen, Local detection of three-dimensional inclusions in electrical impedance tomography,, Inverse Problems, 26 (2010), 35001. doi: 10.1088/0266-5611/26/3/035001. [26] D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Reconstructions of chest phantoms by the d-bar method for electrical impedance tomography,, Physiol Meas., 27 (2006), 43. [27] H. Kang and K. Yun, Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann operator,, SIAM J. Math. Anal., 34 (2003), 719. doi: 10.1137/S0036141001395042. [28] R. V. Kohn and M. Vogelius, Determining conductivity by boundary measurements,, Commun. Pure Appl. Math., 37 (1984), 289. doi: 10.1002/cpa.3160370302. [29] R. V. Kohn and M. Vogelius, Determining conductivity by boundary measurements II. Interior results,, Commun. Pure Appl. Math., 38 (1985), 643. doi: 10.1002/cpa.3160380513. [30] V. Kolehmainen, M. Vauhkonen, P. A. Karjalainen and J. P. Kaipio, Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns,, Physiological Measurement, 18 (1997), 289. doi: 10.1088/0967-3334/18/4/003. [31] P. Metherall, D. C. Barber and R. H. Smallwood, Three dimensional electrical impedance tomography,, in, (1995), 510. [32] P. Metherall, D. C. Barber, R. H. Smallwood and B. H. Brown, Three-dimensional electrical impedance tomography,, Nature, 380 (1996), 509. doi: 10.1038/380509a0. [33] P. Metherall, R. H. Smallwood and D. C. Barber, Three dimensional electrical impedance tomography of the human thorax,, in, (1996). [34] J. P. Morucci, M. Granie, M. Lei, M. Chabert and P. M. Marsili, 3D reconstruction in electrical impedance imaging using a direct sensitivity matrix approach,, Physiol. Meas., 16 (1995). doi: 10.1088/0967-3334/16/3A/012. [35] A. I. Nachman, Reconstructions from boundary measurements,, Ann. of Math., 128 (1988), 531. doi: 10.2307/1971435. [36] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math., 143 (1996), 71. doi: 10.2307/2118653. [37] G. Nakamura and K. Tanuma, Local determination of conductivity at the boundary from the Dirichlet-to-Neumann map,, Inverse Problems, 17 (2001), 405. doi: 10.1088/0266-5611/17/3/303. [38] G. Nakamura and K. Tanuma, Direct determination of the derivatives of conductivity at the boundary from the localized Dirichlet to Neumann map,, Comm. Korean Math. Soc., 16 (2001), 415. [39] G. Nakamura and K. Tanuma, Formulas for reconstructing conductivity and its normal derivative at the boundary from the localized Dirichlet to Neumann map,, in, (2003), 192. [40] G. Nakamura, K. Tanuma, S. Siltanen and S. Wang, Numerical recovery of conductivity at the boundary from the localized Dirichlet to Neumann map,, Computing, 75 (2004), 197. doi: 10.1007/s00607-004-0095-x. [41] J. C. Newell, R. S. Blue, D. Isaacson, G. J. Saulnier and A. S. Ross, Phasic three-dimensional impedance imaging of cardiac activity,, Physiol. Meas., 23 (2002), 203. doi: 10.1088/0967-3334/23/1/321. [42] L. Päivärinta, A. Panchenko and G. Uhlmann, Complex geometrical optics for Lipschitz conductivities,, Rev. Mat. Iberoam., 19 (2003), 57. [43] R. L. Robertson, Boundary identifiability of residual stress via the Dirichlet to Neumann map,, Inverse Problems, 13 (1997), 1107. doi: 10.1088/0266-5611/13/4/015. [44] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography,, SIAM J. Appl. Math., 52 (1992), 1023. doi: 10.1137/0152060. [45] G. Strang and G. Fix, "An Analysis of The Finite Element Method,", Prentice Hall, (1973). [46] J. Sylvester, A convergent layer stripping algorithm for the radially symmetric impedance tomography problem,, Comm. PDE, 17 (1992), 1955. doi: 10.1080/03605309208820910. [47] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153. doi: 10.2307/1971291. [48] J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary - continuous dependence,, Comm. Pure Appl. Math., 41 (1988), 197. doi: 10.1002/cpa.3160410205. [49] P. J. Vauhkonen, "Image Reconstruction in Three-Dimensional Electrical Impedance Tomography,", Ph.D thesis, (2004). [50] P. J. Vauhkonen, M. Vauhkonen, T. Savolainen and J. P. Kaipio, Static three-dimensional electrical impedance tomography,, Ann. New York Acad. Sci., 873 (1999), 472. doi: 10.1111/j.1749-6632.1999.tb09496.x. [51] P. J. Vauhkonen, M. Vauhkonen, T. Savolainen and J. P. Kaipio, Three-dimensional electrical impedance tomography based on the complete electrode model,, IEEE Trans. Biomed. Eng., 46 (1999), 1150. doi: 10.1109/10.784147. [52] A. Wexler, Electrical impedance imaging in two and three dimensions,, Clin. Phys. Physiol. Meas., 9 (1988), 29.

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##### References:
 [1] A. Adler, R. Guardo, and Y. Berthiaume, Impedance imaging of lung ventilation: Do we need to account for chest expansion?, IEEE Trans. Biomed. Eng., 43 (1996), 414. doi: 10.1109/10.486261. [2] G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements,, J. Diff. Eq., 84 (1990), 252. doi: 10.1016/0022-0396(90)90078-4. [3] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math., 163 (2006), 265. doi: 10.4007/annals.2006.163.265. [4] J. Bikowski, "Electrical Impedance Tomography Reconstructions in two and three Dimensions; From Calderón to Direct Methods,", Ph.D thesis, (2008). [5] R. Blue, "Real-time Three-dimensional Electrical Impedance Tomography,", Ph.D thesis, (1997). [6] L. Borcea, Electrical impedance tomography,, Inverse Problems, 18 (2002). doi: 10.1088/0266-5611/18/6/201. [7] L. Borcea, Addendum to "Electrical impedance tomography",, Inverse Problems, 19 (2002), 997. doi: 10.1088/0266-5611/19/4/501. [8] G. Boverman, D. Isaacson, T-J Kao, G. J. Saulnier and J. C. Newell, "Methods for Direct Image Reconstruction for EIT in Two and Three Dimensions,", in, (2008). [9] R. M. Brown, Recovering the conductivity at the boundary from the Dirichlet to Neumann map: a pointwise result,, J. Inverse and Ill-posed Prob., 9 (2001), 567. [10] R. Brown and R. Torres, Uniqueness in the inverse conductivity problem for conductivities with 3/2 derivatives in $L^p, p>2n,$, J. Fourier Analysis Appl., 9 (2003), 1049. [11] R. M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions,, Comm. Partial Differential Equations, 22 (1997), 1009. doi: 10.1080/03605309708821292. [12] A. P. Calderón, On an inverse boundary value problem,, Seminar on Numerical Analysis and its Applications to Continuum Physics, (1980), 65. [13] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85. doi: 10.1137/S0036144598333613. [14] K-S Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography,, IEEE Transactions on Biomedical Imaging, (1989), 918. [15] R. D. Cook, G. J. Saulnier and J. C. Goble, A phase sensitive voltmeter for a high-speed, high-precision electrical impedance tomograph,, in, (1991), 22. doi: 10.1109/IEMBS.1991.683822. [16] H. Cornean, K. Knudsen and S. Siltanen, Towards a d-bar reconstruction method for three-dimensional EIT,, Journal of Inverse and Ill-Posed Problems, 14 (2006), 111. doi: 10.1515/156939406777571102. [17] R. Courant and D. Hilbert, "Methods of Mathematical Physics,", Interscience Publishers, II (1962). [18] E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge University Press, (1989). doi: 10.1017/CBO9780511566158. [19] B. Gebauer and N. Hyvönen, Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem,, Inverse Probl. Imaging, 2 (2008), 355. doi: 10.3934/ipi.2008.2.355. [20] E. Gersing, B. Hoffman, and M. Osypka, Influence of changing peripheral geometry on electrical impedance tomography measurements,, Medical & Biological Engineering & Computing, 34 (1996), 359. doi: 10.1007/BF02520005. [21] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundlehren der Mathematischen Wissenschaften, 224 (1989). [22] J. Goble, M. Cheney and D. Isaacson, Electrical impedance tomography in three dimensions, Appl. Comput. Electromagn. Soc. J., 7 (1992), 128. [23] A. Greenleaf, M. Lassas and G. Uhlmann, The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction,, Comm. Pure Appl. Math., 56 (2003), 328. doi: 10.1002/cpa.10061. [24] M. Hanke and B. Schappel, The factorization method for electrical impedance tomography in the half-space,, SIAM J. Appl. Math., 68 (2008), 907. doi: 10.1137/06067064X. [25] T. Ide, H. Isozaki, S. Nakata and S. Siltanen, Local detection of three-dimensional inclusions in electrical impedance tomography,, Inverse Problems, 26 (2010), 35001. doi: 10.1088/0266-5611/26/3/035001. [26] D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Reconstructions of chest phantoms by the d-bar method for electrical impedance tomography,, Physiol Meas., 27 (2006), 43. [27] H. Kang and K. Yun, Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann operator,, SIAM J. Math. Anal., 34 (2003), 719. doi: 10.1137/S0036141001395042. [28] R. V. Kohn and M. Vogelius, Determining conductivity by boundary measurements,, Commun. Pure Appl. Math., 37 (1984), 289. doi: 10.1002/cpa.3160370302. [29] R. V. Kohn and M. Vogelius, Determining conductivity by boundary measurements II. Interior results,, Commun. Pure Appl. Math., 38 (1985), 643. doi: 10.1002/cpa.3160380513. [30] V. Kolehmainen, M. Vauhkonen, P. A. Karjalainen and J. P. Kaipio, Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns,, Physiological Measurement, 18 (1997), 289. doi: 10.1088/0967-3334/18/4/003. [31] P. Metherall, D. C. Barber and R. H. Smallwood, Three dimensional electrical impedance tomography,, in, (1995), 510. [32] P. Metherall, D. C. Barber, R. H. Smallwood and B. H. Brown, Three-dimensional electrical impedance tomography,, Nature, 380 (1996), 509. doi: 10.1038/380509a0. [33] P. Metherall, R. H. Smallwood and D. C. Barber, Three dimensional electrical impedance tomography of the human thorax,, in, (1996). [34] J. P. Morucci, M. Granie, M. Lei, M. Chabert and P. M. Marsili, 3D reconstruction in electrical impedance imaging using a direct sensitivity matrix approach,, Physiol. Meas., 16 (1995). doi: 10.1088/0967-3334/16/3A/012. [35] A. I. Nachman, Reconstructions from boundary measurements,, Ann. of Math., 128 (1988), 531. doi: 10.2307/1971435. [36] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math., 143 (1996), 71. doi: 10.2307/2118653. [37] G. Nakamura and K. Tanuma, Local determination of conductivity at the boundary from the Dirichlet-to-Neumann map,, Inverse Problems, 17 (2001), 405. doi: 10.1088/0266-5611/17/3/303. [38] G. Nakamura and K. Tanuma, Direct determination of the derivatives of conductivity at the boundary from the localized Dirichlet to Neumann map,, Comm. Korean Math. Soc., 16 (2001), 415. [39] G. Nakamura and K. Tanuma, Formulas for reconstructing conductivity and its normal derivative at the boundary from the localized Dirichlet to Neumann map,, in, (2003), 192. [40] G. Nakamura, K. Tanuma, S. Siltanen and S. Wang, Numerical recovery of conductivity at the boundary from the localized Dirichlet to Neumann map,, Computing, 75 (2004), 197. doi: 10.1007/s00607-004-0095-x. [41] J. C. Newell, R. S. Blue, D. Isaacson, G. J. Saulnier and A. S. Ross, Phasic three-dimensional impedance imaging of cardiac activity,, Physiol. Meas., 23 (2002), 203. doi: 10.1088/0967-3334/23/1/321. [42] L. Päivärinta, A. Panchenko and G. Uhlmann, Complex geometrical optics for Lipschitz conductivities,, Rev. Mat. Iberoam., 19 (2003), 57. [43] R. L. Robertson, Boundary identifiability of residual stress via the Dirichlet to Neumann map,, Inverse Problems, 13 (1997), 1107. doi: 10.1088/0266-5611/13/4/015. [44] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography,, SIAM J. Appl. Math., 52 (1992), 1023. doi: 10.1137/0152060. [45] G. Strang and G. Fix, "An Analysis of The Finite Element Method,", Prentice Hall, (1973). [46] J. Sylvester, A convergent layer stripping algorithm for the radially symmetric impedance tomography problem,, Comm. PDE, 17 (1992), 1955. doi: 10.1080/03605309208820910. [47] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153. doi: 10.2307/1971291. [48] J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary - continuous dependence,, Comm. Pure Appl. Math., 41 (1988), 197. doi: 10.1002/cpa.3160410205. [49] P. J. Vauhkonen, "Image Reconstruction in Three-Dimensional Electrical Impedance Tomography,", Ph.D thesis, (2004). [50] P. J. Vauhkonen, M. Vauhkonen, T. Savolainen and J. P. Kaipio, Static three-dimensional electrical impedance tomography,, Ann. New York Acad. Sci., 873 (1999), 472. doi: 10.1111/j.1749-6632.1999.tb09496.x. [51] P. J. Vauhkonen, M. Vauhkonen, T. Savolainen and J. P. Kaipio, Three-dimensional electrical impedance tomography based on the complete electrode model,, IEEE Trans. Biomed. Eng., 46 (1999), 1150. doi: 10.1109/10.784147. [52] A. Wexler, Electrical impedance imaging in two and three dimensions,, Clin. Phys. Physiol. Meas., 9 (1988), 29.
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