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Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results
August  2015, 9(3): 767-789. doi: 10.3934/ipi.2015.9.767

## Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map

 1 Case Western Reserve University, Department of Mathematics, Applied Mathematics, and Statistics, Cleveland, OH 44106 2 University of Auckland, Department of Mathematics, Auckland 3 University of Eastern Finland, Department of Applied Physics, Kuopio

Received  April 2014 Revised  January 2015 Published  July 2015

In [3], the authors discussed the electrical impedance tomography (EIT) problem, in which the computational domain with an unknown conductivity distribution comprises only a portion of the whole conducting body, and a boundary condition along the artificial boundary needs to be set so as to minimally disturbs the estimate in the domain of interest. It was shown that a partial Dirichlet-to-Neumann operator, or Steklov-Poincaré map, provides theoretically a perfect boundary condition. However, since the boundary condition depends on the conductivity in the truncated portion of the conductive body, it is itself an unknown that needs to be estimated along with the conductivity of interest. In this article, we develop further the computational methodology, replacing the unknown integral kernel with a low dimensional approximation. The viability of the approach is demonstrated with finite element simulations as well as with real phantom data.
Citation: Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, Jari P. Kaipio, Erkki Somersalo. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map. Inverse Problems and Imaging, 2015, 9 (3) : 767-789. doi: 10.3934/ipi.2015.9.767
##### References:
 [1] R. Adams and J. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. [2] D. C. Barber and B. H. Brown, Applied potential tomography, J. Phys. E: Sci. Instrum., 17 (1984), 723-733. doi: 10.1088/0022-3735/17/9/002. [3] D. Calvetti, P. J. Hadwin, J. M. J. Huttunen, J. P. Kaipio, D. McGivney, E. Somersalo and J. Volzer, Artificial boundary conditions and domain turncation in electrical impedance tomography. Part I: Theory and preliminary results, Inv. Probl. imaging, 12 (2015). [4] D. Calvetti and E. Somersalo, Statistical compensation of boundary clutter in image debarring, Inverse Problems, 21 (2005), 1697-1714. doi: 10.1088/0266-5611/21/5/012. [5] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101. doi: 10.1137/S0036144598333613. [6] K.-S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Trans. Biomed. Eng., 3 (1989), 918-924. [7] J. Heino, S. Arridge, J. Sikora and E. Somersalo, Anisotropic effects in highly scattering media, Phys. Rev. E, 68 (2003), 031908. doi: 10.1103/PhysRevE.68.031908. [8] I. T. Jolliffe, Principal Component Analysis, Second edition, Springer, New York, 2002. [9] E. Jonsson, Partial Dirichlet to Neumann Maps in the Approximate Reconstruction of Conductivity Distribution, PhD Thesis, Rensselaer Polytechnic Institute, Troy, NY, 1997. [10] E. Jonsson, Electrical conductivity reconstruction using nonlocal boundary conditions, SIAM J. Appl. Math., 59 (1999), 1582-1598. doi: 10.1137/S0036139997327770. [11] J. P. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160, Springer Verlag, New York, 2005. [12] F. Lindgren, H. Rue and J. Lindström, An explicit link between gaussian Markov random fields: The stochastic partial differential equation approach, J. Royal Stat. Soc. B, 73 (2011), 423-498. doi: 10.1111/j.1467-9868.2011.00777.x. [13] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning, The MIT Press, Cambridge, MA, 2006. [14] L. Roininen, J. M. J. Huttunen and S. Lasanen, Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography, Inv. Probl. Imaging, 8 (2014), 561-586. doi: 10.3934/ipi.2014.8.561. [15] S. Salsa, Partial Differential Equations in Action: From Modelling to Theory, Springer Verlag Italia, Milano, 2008. [16] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), 1023-1040. doi: 10.1137/0152060. [17] 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), 1150-1160. doi: 10.1109/10.784147. [18] K. Yosida, Functional Analysis, Springer Verlag, New York, 1980. [19] P. Whittle, Stochastic processes in several dimensions, Bull. Inst. Int. Statist., 40 (1963), 974-994.

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##### References:
 [1] R. Adams and J. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. [2] D. C. Barber and B. H. Brown, Applied potential tomography, J. Phys. E: Sci. Instrum., 17 (1984), 723-733. doi: 10.1088/0022-3735/17/9/002. [3] D. Calvetti, P. J. Hadwin, J. M. J. Huttunen, J. P. Kaipio, D. McGivney, E. Somersalo and J. Volzer, Artificial boundary conditions and domain turncation in electrical impedance tomography. Part I: Theory and preliminary results, Inv. Probl. imaging, 12 (2015). [4] D. Calvetti and E. Somersalo, Statistical compensation of boundary clutter in image debarring, Inverse Problems, 21 (2005), 1697-1714. doi: 10.1088/0266-5611/21/5/012. [5] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101. doi: 10.1137/S0036144598333613. [6] K.-S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Trans. Biomed. Eng., 3 (1989), 918-924. [7] J. Heino, S. Arridge, J. Sikora and E. Somersalo, Anisotropic effects in highly scattering media, Phys. Rev. E, 68 (2003), 031908. doi: 10.1103/PhysRevE.68.031908. [8] I. T. Jolliffe, Principal Component Analysis, Second edition, Springer, New York, 2002. [9] E. Jonsson, Partial Dirichlet to Neumann Maps in the Approximate Reconstruction of Conductivity Distribution, PhD Thesis, Rensselaer Polytechnic Institute, Troy, NY, 1997. [10] E. Jonsson, Electrical conductivity reconstruction using nonlocal boundary conditions, SIAM J. Appl. Math., 59 (1999), 1582-1598. doi: 10.1137/S0036139997327770. [11] J. P. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160, Springer Verlag, New York, 2005. [12] F. Lindgren, H. Rue and J. Lindström, An explicit link between gaussian Markov random fields: The stochastic partial differential equation approach, J. Royal Stat. Soc. B, 73 (2011), 423-498. doi: 10.1111/j.1467-9868.2011.00777.x. [13] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning, The MIT Press, Cambridge, MA, 2006. [14] L. Roininen, J. M. J. Huttunen and S. Lasanen, Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography, Inv. Probl. Imaging, 8 (2014), 561-586. doi: 10.3934/ipi.2014.8.561. [15] S. Salsa, Partial Differential Equations in Action: From Modelling to Theory, Springer Verlag Italia, Milano, 2008. [16] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), 1023-1040. doi: 10.1137/0152060. [17] 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), 1150-1160. doi: 10.1109/10.784147. [18] K. Yosida, Functional Analysis, Springer Verlag, New York, 1980. [19] P. Whittle, Stochastic processes in several dimensions, Bull. Inst. Int. Statist., 40 (1963), 974-994.
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