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Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results
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 |
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. |
show all references
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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