# 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. Google Scholar [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. Google Scholar [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. Google Scholar [4] J. Bikowski, "Electrical Impedance Tomography Reconstructions in two and three Dimensions; From Calderón to Direct Methods,", Ph.D thesis, (2008). Google Scholar [5] R. Blue, "Real-time Three-dimensional Electrical Impedance Tomography,", Ph.D thesis, (1997). Google Scholar [6] L. Borcea, Electrical impedance tomography,, Inverse Problems, 18 (2002). doi: 10.1088/0266-5611/18/6/201. Google Scholar [7] L. Borcea, Addendum to "Electrical impedance tomography",, Inverse Problems, 19 (2002), 997. doi: 10.1088/0266-5611/19/4/501. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] A. P. Calderón, On an inverse boundary value problem,, Seminar on Numerical Analysis and its Applications to Continuum Physics, (1980), 65. Google Scholar [13] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85. doi: 10.1137/S0036144598333613. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] R. Courant and D. Hilbert, "Methods of Mathematical Physics,", Interscience Publishers, II (1962). Google Scholar [18] E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge University Press, (1989). doi: 10.1017/CBO9780511566158. Google Scholar [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. Google Scholar [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. Google Scholar [21] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundlehren der Mathematischen Wissenschaften, 224 (1989). Google Scholar [22] J. Goble, M. Cheney and D. Isaacson, Electrical impedance tomography in three dimensions, Appl. Comput. Electromagn. Soc. J., 7 (1992), 128. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [28] R. V. Kohn and M. Vogelius, Determining conductivity by boundary measurements,, Commun. Pure Appl. Math., 37 (1984), 289. doi: 10.1002/cpa.3160370302. Google Scholar [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. Google Scholar [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. Google Scholar [31] P. Metherall, D. C. Barber and R. H. Smallwood, Three dimensional electrical impedance tomography,, in, (1995), 510. Google Scholar [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. Google Scholar [33] P. Metherall, R. H. Smallwood and D. C. Barber, Three dimensional electrical impedance tomography of the human thorax,, in, (1996). Google Scholar [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. Google Scholar [35] A. I. Nachman, Reconstructions from boundary measurements,, Ann. of Math., 128 (1988), 531. doi: 10.2307/1971435. Google Scholar [36] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math., 143 (1996), 71. doi: 10.2307/2118653. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [42] L. Päivärinta, A. Panchenko and G. Uhlmann, Complex geometrical optics for Lipschitz conductivities,, Rev. Mat. Iberoam., 19 (2003), 57. Google Scholar [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. Google Scholar [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. Google Scholar [45] G. Strang and G. Fix, "An Analysis of The Finite Element Method,", Prentice Hall, (1973). Google Scholar [46] J. Sylvester, A convergent layer stripping algorithm for the radially symmetric impedance tomography problem,, Comm. PDE, 17 (1992), 1955. doi: 10.1080/03605309208820910. Google Scholar [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. Google Scholar [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. Google Scholar [49] P. J. Vauhkonen, "Image Reconstruction in Three-Dimensional Electrical Impedance Tomography,", Ph.D thesis, (2004). Google Scholar [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. Google Scholar [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. Google Scholar [52] A. Wexler, Electrical impedance imaging in two and three dimensions,, Clin. Phys. Physiol. Meas., 9 (1988), 29. Google Scholar

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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. Google Scholar [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. Google Scholar [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. Google Scholar [4] J. Bikowski, "Electrical Impedance Tomography Reconstructions in two and three Dimensions; From Calderón to Direct Methods,", Ph.D thesis, (2008). Google Scholar [5] R. Blue, "Real-time Three-dimensional Electrical Impedance Tomography,", Ph.D thesis, (1997). Google Scholar [6] L. Borcea, Electrical impedance tomography,, Inverse Problems, 18 (2002). doi: 10.1088/0266-5611/18/6/201. Google Scholar [7] L. Borcea, Addendum to "Electrical impedance tomography",, Inverse Problems, 19 (2002), 997. doi: 10.1088/0266-5611/19/4/501. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] A. P. Calderón, On an inverse boundary value problem,, Seminar on Numerical Analysis and its Applications to Continuum Physics, (1980), 65. Google Scholar [13] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85. doi: 10.1137/S0036144598333613. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] R. Courant and D. Hilbert, "Methods of Mathematical Physics,", Interscience Publishers, II (1962). Google Scholar [18] E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge University Press, (1989). doi: 10.1017/CBO9780511566158. Google Scholar [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. Google Scholar [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. Google Scholar [21] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundlehren der Mathematischen Wissenschaften, 224 (1989). Google Scholar [22] J. Goble, M. Cheney and D. Isaacson, Electrical impedance tomography in three dimensions, Appl. Comput. Electromagn. Soc. J., 7 (1992), 128. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [28] R. V. Kohn and M. Vogelius, Determining conductivity by boundary measurements,, Commun. Pure Appl. Math., 37 (1984), 289. doi: 10.1002/cpa.3160370302. Google Scholar [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. Google Scholar [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. Google Scholar [31] P. Metherall, D. C. Barber and R. H. Smallwood, Three dimensional electrical impedance tomography,, in, (1995), 510. Google Scholar [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. Google Scholar [33] P. Metherall, R. H. Smallwood and D. C. Barber, Three dimensional electrical impedance tomography of the human thorax,, in, (1996). Google Scholar [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. Google Scholar [35] A. I. Nachman, Reconstructions from boundary measurements,, Ann. of Math., 128 (1988), 531. doi: 10.2307/1971435. Google Scholar [36] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math., 143 (1996), 71. doi: 10.2307/2118653. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [42] L. Päivärinta, A. Panchenko and G. Uhlmann, Complex geometrical optics for Lipschitz conductivities,, Rev. Mat. Iberoam., 19 (2003), 57. Google Scholar [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. Google Scholar [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. Google Scholar [45] G. Strang and G. Fix, "An Analysis of The Finite Element Method,", Prentice Hall, (1973). Google Scholar [46] J. Sylvester, A convergent layer stripping algorithm for the radially symmetric impedance tomography problem,, Comm. PDE, 17 (1992), 1955. doi: 10.1080/03605309208820910. Google Scholar [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. Google Scholar [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. Google Scholar [49] P. J. Vauhkonen, "Image Reconstruction in Three-Dimensional Electrical Impedance Tomography,", Ph.D thesis, (2004). Google Scholar [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. Google Scholar [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. Google Scholar [52] A. Wexler, Electrical impedance imaging in two and three dimensions,, Clin. Phys. Physiol. Meas., 9 (1988), 29. Google Scholar
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