American Institute of Mathematical Sciences

August  2011, 5(3): 531-549. doi: 10.3934/ipi.2011.5.531

Direct electrical impedance tomography for nonsmooth conductivities

 1 University of Helsinki, Department of Mathematics and Statistics, FI-00014 Helsinki, Finland, Finland 2 Colorado State University, Department of Mathematics and School of Biomedical Engineering, Fort Collins, CO 80523-1874, United States 3 Aalto University, Institute of Mathematics, P.O.Box 1100, FI-00076 Aalto, Finland

Received  August 2010 Revised  June 2011 Published  August 2011

A new reconstruction algorithm is presented for eit in dimension two, based on the constructive uniqueness proof given by Astala and Päivärinta in [Ann. of Math. 163 (2006)]. The method is non-iterative, provides a noise-robust solution of the full nonlinear eit problem, and applies to more general conductivities than previous approaches. In particular, the new algorithm applies to piecewise smooth conductivities. Reconstructions from noisy and non-noisy simulated data from conductivity distributions representing a cross-sections of a chest and a layered medium such as stratified flow in a pipeline are presented. The results suggest that the new method can recover useful and reasonably accurate eit images from data corrupted by realistic amounts of measurement noise. In particular, the dynamic range in medium-contrast conductivities is reconstructed remarkably well.
Citation: Kari Astala, Jennifer L. Mueller, Lassi Päivärinta, Allan Perämäki, Samuli Siltanen. Direct electrical impedance tomography for nonsmooth conductivities. Inverse Problems & Imaging, 2011, 5 (3) : 531-549. doi: 10.3934/ipi.2011.5.531
References:
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Ruiz., Stability of calderón inverse conductivity problem in the plane, Journal de Mathématiques Pures et Appliqués, 88 (2007), 522-556. Google Scholar [6] J. Bikowski and J. L. Mueller, 2D EIT reconstructions using Calderón's method, Inverse Problems and Imaging, 2 (2008), 43-61.  Google Scholar [7] J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms, Inverse Problems, 27 (2011), 19 pp.  Google Scholar [8] G. Boverman, D. Isaacson, T.-J. Kao, G. Saulnier and J. C. Newell, "Methods for Direct Image Reconstruction for EIT in Two and Three Dimensions," Proceedings of the 2008 Electrical Impedance Tomography Conference at Dartmouth College, Hanover, New Hampshire, June 16 to 18, 2008. Google Scholar [9] R. M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009-1027.  Google Scholar [10] A. P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasileira de Matemàtica, Rio de Janeiro, (1980), 65-73.  Google Scholar [11] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Review, 41 (1999), 85-101. doi: 10.1137/S0036144598333613.  Google Scholar [12] A. Clop, D. Faraco and A. Ruiz., Integral stability of calderón inverse conductivity problem in the plane, Inverse Problems and Imaging, 4 (2010), 49-91. Google Scholar [13] R. D. Cook, G. J. Saulnier and J. C. Goble, "A Phase Sensitive Voltmeter for a High-Speed, High-Precision Electrical Impedance Tomograph," Proc. Annu. Int. Conf. IEEE Engineering in Medicine and Biology Soc., (1991), 22-23. Google Scholar [14] P. Daripa, A fast algorithm to solve the Beltrami equation with applications to quasiconformal mappings, Journal of Computational Physics, 106 (1993), 355-365.  Google Scholar [15] M. DeAngelo and J. L. Mueller, D-bar reconstructions of human chest and tank data using an improved approximation to the scattering transform, Physiological Measurement, 31 (2010), 221-232. doi: 10.1088/0967-3334/31/2/008.  Google Scholar [16] D. Gaydashev and D. Khmelev, On numerical algorithms for the solution of a Beltrami equation, arXiv:math/0510516, 2005. Google Scholar [17] M. Huhtanen and A. Perämäki, Numerical solution of the $\R$-linear Beltrami equation,, to appear in Math. Comp., ().   Google Scholar [18] D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography, IEEE Trans. Med. Im., 23 (2004), 821-828. doi: 10.1109/TMI.2004.827482.  Google Scholar [19] D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography, Physiological Measurement, 27 (2006), S43-S50. doi: 10.1088/0967-3334/27/5/S04.  Google Scholar [20] K. Knudsen, "On the Inverse Conductivity Problem," Ph.D. thesis, Department of Mathematical Sciences, Aalborg University, Denmark, 2002. Google Scholar [21] K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane, Physiol. Meas., 24 (2003), 391-401. doi: 10.1088/0967-3334/24/2/351.  Google Scholar [22] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, "Reconstructions of Piecewise Constant Conductivities by the D-Bar Method for Electrical Impedance Tomography," Proceedings of the 4th AIP International Conference and the 1st Congress of the IPIA, Vancouver, Journal of Physics: Conference Series, 124, 2008. Google Scholar [23] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, D-bar method for electrical impedance tomography with discontinuous conductivities, SIAM J. Appl. Math., 67 (2007), 893-913. doi: 10.1137/060656930.  Google Scholar [24] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), 599-624.  Google Scholar [25] K. Knudsen, J. L. Mueller and S. Siltanen, Numerical solution method for the dbar-equation in the plane, J. Comp. Phys., 198 (2004), 500-517. doi: 10.1016/j.jcp.2004.01.028.  Google Scholar [26] K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane, Comm. Partial Differential Equations, 29 (2004), 361-381.  Google Scholar [27] J. L. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM J. Sci. Comp., 24 (2003), 1232-1266. doi: 10.1137/S1064827501394568.  Google Scholar [28] J. L. Mueller, S. Siltanen and D. Isaacson, A direct reconstruction algorithm for electrical impedance tomography, IEEE Trans. Med. Im., 21 (2002), 555-559. Google Scholar [29] E. Murphy, J. L. Mueller and J. C. Newell, Reconstruction of conductive and insulating targets using the D-bar method on an elliptical domain, Physiol. Meas., 28 (2007), S101-S114. doi: 10.1088/0967-3334/28/7/S08.  Google Scholar [30] A. I. Nachman, Global uniqueness for a two dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), 71-96. doi: 10.2307/2118653.  Google Scholar [31] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), 856-869. doi: 10.1137/0907058.  Google Scholar [32] S. Siltanen, J. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem, Inverse Problems, 16 (2000), 681-699, (Erratum: Inverse problems, 17, 1561-1563). doi: 10.1088/0266-5611/16/3/310.  Google Scholar [33] S. Siltanen, J. Mueller and D. Isaacson, "Reconstruction of High Contrast 2-D Conductivities by the Algorithm of A. Nachman," In "Radon Transforms and Tomography" (South Hadley, MA, 2000), 241-254, Contemporary Mathematics, 278, Amer. Math. Soc., Providence, RI, 2001.  Google Scholar [34] S. Siltanen and J. Tamminen, Reconstructing conductivities with boundary corrected D-bar method,, Submitted manuscript., ().   Google Scholar [35] G. Strang and G. Fix, "An Analysis of The Finite Element Method," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973.  Google Scholar [36] G. Vainikko, Fast solvers of the Lippmann-Schwinger equation, in "Direct and Inverse Problems of Mathematical Physics" (Newark, DE, 1997), 423-440, Int. Soc. Anal. Appl. Comput., 5, Kluwer Acad. Publ, Dordrecht, 2000.  Google Scholar

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References:
 [1] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.  Google Scholar [2] K. Astala and L. Päivärinta, "A Boundary Integral Equation for Calderón's Inverse Conductivity Problem," Proc. 7th Internat. Conference on Harmonic Analysis, Collectanea Mathematica, 2006. Google Scholar [3] K. Astala, J. L. Mueller, L. Päivärinta and S. Siltanen, Numerical computation of complex geometrical optics solutions to the conductivity equation, Applied and Computational Harmonic Analysis, 29 (2010), 2-17. doi: 10.1016/j.acha.2009.08.001.  Google Scholar [4] K. Astala, T. Iwaniec and G. Martin, "Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane," Princeton Mathematical Series, 48, Princeton University Press, Princeton, NJ, 2009.  Google Scholar [5] T. Barceló, D. Faraco, and A. Ruiz., Stability of calderón inverse conductivity problem in the plane, Journal de Mathématiques Pures et Appliqués, 88 (2007), 522-556. Google Scholar [6] J. Bikowski and J. L. Mueller, 2D EIT reconstructions using Calderón's method, Inverse Problems and Imaging, 2 (2008), 43-61.  Google Scholar [7] J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms, Inverse Problems, 27 (2011), 19 pp.  Google Scholar [8] G. Boverman, D. Isaacson, T.-J. Kao, G. Saulnier and J. C. Newell, "Methods for Direct Image Reconstruction for EIT in Two and Three Dimensions," Proceedings of the 2008 Electrical Impedance Tomography Conference at Dartmouth College, Hanover, New Hampshire, June 16 to 18, 2008. Google Scholar [9] R. M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009-1027.  Google Scholar [10] A. P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasileira de Matemàtica, Rio de Janeiro, (1980), 65-73.  Google Scholar [11] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Review, 41 (1999), 85-101. doi: 10.1137/S0036144598333613.  Google Scholar [12] A. Clop, D. Faraco and A. Ruiz., Integral stability of calderón inverse conductivity problem in the plane, Inverse Problems and Imaging, 4 (2010), 49-91. Google Scholar [13] R. D. Cook, G. J. Saulnier and J. C. Goble, "A Phase Sensitive Voltmeter for a High-Speed, High-Precision Electrical Impedance Tomograph," Proc. Annu. Int. Conf. IEEE Engineering in Medicine and Biology Soc., (1991), 22-23. Google Scholar [14] P. Daripa, A fast algorithm to solve the Beltrami equation with applications to quasiconformal mappings, Journal of Computational Physics, 106 (1993), 355-365.  Google Scholar [15] M. DeAngelo and J. L. Mueller, D-bar reconstructions of human chest and tank data using an improved approximation to the scattering transform, Physiological Measurement, 31 (2010), 221-232. doi: 10.1088/0967-3334/31/2/008.  Google Scholar [16] D. Gaydashev and D. Khmelev, On numerical algorithms for the solution of a Beltrami equation, arXiv:math/0510516, 2005. Google Scholar [17] M. Huhtanen and A. Perämäki, Numerical solution of the $\R$-linear Beltrami equation,, to appear in Math. Comp., ().   Google Scholar [18] D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography, IEEE Trans. Med. Im., 23 (2004), 821-828. doi: 10.1109/TMI.2004.827482.  Google Scholar [19] D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography, Physiological Measurement, 27 (2006), S43-S50. doi: 10.1088/0967-3334/27/5/S04.  Google Scholar [20] K. Knudsen, "On the Inverse Conductivity Problem," Ph.D. thesis, Department of Mathematical Sciences, Aalborg University, Denmark, 2002. Google Scholar [21] K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane, Physiol. Meas., 24 (2003), 391-401. doi: 10.1088/0967-3334/24/2/351.  Google Scholar [22] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, "Reconstructions of Piecewise Constant Conductivities by the D-Bar Method for Electrical Impedance Tomography," Proceedings of the 4th AIP International Conference and the 1st Congress of the IPIA, Vancouver, Journal of Physics: Conference Series, 124, 2008. Google Scholar [23] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, D-bar method for electrical impedance tomography with discontinuous conductivities, SIAM J. Appl. Math., 67 (2007), 893-913. doi: 10.1137/060656930.  Google Scholar [24] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), 599-624.  Google Scholar [25] K. Knudsen, J. L. Mueller and S. Siltanen, Numerical solution method for the dbar-equation in the plane, J. Comp. Phys., 198 (2004), 500-517. doi: 10.1016/j.jcp.2004.01.028.  Google Scholar [26] K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane, Comm. Partial Differential Equations, 29 (2004), 361-381.  Google Scholar [27] J. L. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM J. Sci. Comp., 24 (2003), 1232-1266. doi: 10.1137/S1064827501394568.  Google Scholar [28] J. L. Mueller, S. Siltanen and D. Isaacson, A direct reconstruction algorithm for electrical impedance tomography, IEEE Trans. Med. Im., 21 (2002), 555-559. Google Scholar [29] E. Murphy, J. L. Mueller and J. C. Newell, Reconstruction of conductive and insulating targets using the D-bar method on an elliptical domain, Physiol. Meas., 28 (2007), S101-S114. doi: 10.1088/0967-3334/28/7/S08.  Google Scholar [30] A. I. Nachman, Global uniqueness for a two dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), 71-96. doi: 10.2307/2118653.  Google Scholar [31] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), 856-869. doi: 10.1137/0907058.  Google Scholar [32] S. Siltanen, J. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem, Inverse Problems, 16 (2000), 681-699, (Erratum: Inverse problems, 17, 1561-1563). doi: 10.1088/0266-5611/16/3/310.  Google Scholar [33] S. Siltanen, J. Mueller and D. Isaacson, "Reconstruction of High Contrast 2-D Conductivities by the Algorithm of A. Nachman," In "Radon Transforms and Tomography" (South Hadley, MA, 2000), 241-254, Contemporary Mathematics, 278, Amer. Math. Soc., Providence, RI, 2001.  Google Scholar [34] S. Siltanen and J. Tamminen, Reconstructing conductivities with boundary corrected D-bar method,, Submitted manuscript., ().   Google Scholar [35] G. Strang and G. Fix, "An Analysis of The Finite Element Method," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973.  Google Scholar [36] G. Vainikko, Fast solvers of the Lippmann-Schwinger equation, in "Direct and Inverse Problems of Mathematical Physics" (Newark, DE, 1997), 423-440, Int. Soc. Anal. Appl. Comput., 5, Kluwer Acad. Publ, Dordrecht, 2000.  Google Scholar
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