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:
[1]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math. (2), 163 (2006), 265. doi: 10.4007/annals.2006.163.265.

[2]

K. Astala and L. Päivärinta, "A Boundary Integral Equation for Calderón's Inverse Conductivity Problem,", Proc. 7th Internat. Conference on Harmonic Analysis, (2006).

[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. doi: 10.1016/j.acha.2009.08.001.

[4]

K. Astala, T. Iwaniec and G. Martin, "Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane,", Princeton Mathematical Series, 48 (2009).

[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.

[6]

J. Bikowski and J. L. Mueller, 2D EIT reconstructions using Calderón's method,, Inverse Problems and Imaging, 2 (2008), 43.

[7]

J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms,, Inverse Problems, 27 (2011).

[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, (2008).

[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.

[10]

A. P. Calderón, On an inverse boundary value problem,, in, (1980), 65.

[11]

M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85. doi: 10.1137/S0036144598333613.

[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.

[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.

[14]

P. Daripa, A fast algorithm to solve the Beltrami equation with applications to quasiconformal mappings,, Journal of Computational Physics, 106 (1993), 355.

[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. doi: 10.1088/0967-3334/31/2/008.

[16]

D. Gaydashev and D. Khmelev, On numerical algorithms for the solution of a Beltrami equation,, , (2005).

[17]

M. Huhtanen and A. Perämäki, Numerical solution of the $\R$-linear Beltrami equation,, to appear in Math. Comp., ().

[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. doi: 10.1109/TMI.2004.827482.

[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). doi: 10.1088/0967-3334/27/5/S04.

[20]

K. Knudsen, "On the Inverse Conductivity Problem,", Ph.D. thesis, (2002).

[21]

K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane,, Physiol. Meas., 24 (2003), 391. doi: 10.1088/0967-3334/24/2/351.

[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).

[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. doi: 10.1137/060656930.

[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.

[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. doi: 10.1016/j.jcp.2004.01.028.

[26]

K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane,, Comm. Partial Differential Equations, 29 (2004), 361.

[27]

J. L. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements,, SIAM J. Sci. Comp., 24 (2003), 1232. doi: 10.1137/S1064827501394568.

[28]

J. L. Mueller, S. Siltanen and D. Isaacson, A direct reconstruction algorithm for electrical impedance tomography,, IEEE Trans. Med. Im., 21 (2002), 555.

[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). doi: 10.1088/0967-3334/28/7/S08.

[30]

A. I. Nachman, Global uniqueness for a two dimensional inverse boundary value problem,, Ann. of Math. (2), 143 (1996), 71. doi: 10.2307/2118653.

[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. doi: 10.1137/0907058.

[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. doi: 10.1088/0266-5611/16/3/310.

[33]

S. Siltanen, J. Mueller and D. Isaacson, "Reconstruction of High Contrast 2-D Conductivities by the Algorithm of A. Nachman,", In, 278 (2000), 241.

[34]

S. Siltanen and J. Tamminen, Reconstructing conductivities with boundary corrected D-bar method,, Submitted manuscript., ().

[35]

G. Strang and G. Fix, "An Analysis of The Finite Element Method,", Prentice-Hall, (1973).

[36]

G. Vainikko, Fast solvers of the Lippmann-Schwinger equation,, in, 5 (1997), 423.

show all references

References:
[1]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math. (2), 163 (2006), 265. doi: 10.4007/annals.2006.163.265.

[2]

K. Astala and L. Päivärinta, "A Boundary Integral Equation for Calderón's Inverse Conductivity Problem,", Proc. 7th Internat. Conference on Harmonic Analysis, (2006).

[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. doi: 10.1016/j.acha.2009.08.001.

[4]

K. Astala, T. Iwaniec and G. Martin, "Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane,", Princeton Mathematical Series, 48 (2009).

[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.

[6]

J. Bikowski and J. L. Mueller, 2D EIT reconstructions using Calderón's method,, Inverse Problems and Imaging, 2 (2008), 43.

[7]

J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms,, Inverse Problems, 27 (2011).

[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, (2008).

[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.

[10]

A. P. Calderón, On an inverse boundary value problem,, in, (1980), 65.

[11]

M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85. doi: 10.1137/S0036144598333613.

[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.

[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.

[14]

P. Daripa, A fast algorithm to solve the Beltrami equation with applications to quasiconformal mappings,, Journal of Computational Physics, 106 (1993), 355.

[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. doi: 10.1088/0967-3334/31/2/008.

[16]

D. Gaydashev and D. Khmelev, On numerical algorithms for the solution of a Beltrami equation,, , (2005).

[17]

M. Huhtanen and A. Perämäki, Numerical solution of the $\R$-linear Beltrami equation,, to appear in Math. Comp., ().

[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. doi: 10.1109/TMI.2004.827482.

[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). doi: 10.1088/0967-3334/27/5/S04.

[20]

K. Knudsen, "On the Inverse Conductivity Problem,", Ph.D. thesis, (2002).

[21]

K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane,, Physiol. Meas., 24 (2003), 391. doi: 10.1088/0967-3334/24/2/351.

[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).

[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. doi: 10.1137/060656930.

[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.

[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. doi: 10.1016/j.jcp.2004.01.028.

[26]

K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane,, Comm. Partial Differential Equations, 29 (2004), 361.

[27]

J. L. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements,, SIAM J. Sci. Comp., 24 (2003), 1232. doi: 10.1137/S1064827501394568.

[28]

J. L. Mueller, S. Siltanen and D. Isaacson, A direct reconstruction algorithm for electrical impedance tomography,, IEEE Trans. Med. Im., 21 (2002), 555.

[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). doi: 10.1088/0967-3334/28/7/S08.

[30]

A. I. Nachman, Global uniqueness for a two dimensional inverse boundary value problem,, Ann. of Math. (2), 143 (1996), 71. doi: 10.2307/2118653.

[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. doi: 10.1137/0907058.

[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. doi: 10.1088/0266-5611/16/3/310.

[33]

S. Siltanen, J. Mueller and D. Isaacson, "Reconstruction of High Contrast 2-D Conductivities by the Algorithm of A. Nachman,", In, 278 (2000), 241.

[34]

S. Siltanen and J. Tamminen, Reconstructing conductivities with boundary corrected D-bar method,, Submitted manuscript., ().

[35]

G. Strang and G. Fix, "An Analysis of The Finite Element Method,", Prentice-Hall, (1973).

[36]

G. Vainikko, Fast solvers of the Lippmann-Schwinger equation,, in, 5 (1997), 423.

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