Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem

Previous Article
Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem
IPI Home
This Issue
Next Article
A real-time D-bar algorithm for 2-D electrical impedance tomography data

November 2014, 8(4): 991-1012. doi: 10.3934/ipi.2014.8.991

Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem

1.	University of Genoa, Department of Mathematics, Via Dodecaneso 35, 16146 Genoa, Italy
2.	Danmarks Tekniske Universitet, Department of Applied Mathematics and Computer Science, Matematiktorvet, Building 303 B, DK - 2800 Kgs. Lyngby

Received May 2014 Revised October 2014 Published November 2014

Abstract
Related Papers

The Calderón problem is the mathematical formulation of the inverse problem in Electrical Impedance Tomography and asks for the uniqueness and reconstruction of an electrical conductivity distribution in a bounded domain from the knowledge of the Dirichlet-to-Neumann map associated to the generalized Laplace equation. The 3D problem was solved in theory in late 1980s using complex geometrical optics solutions and a scattering transform. Several approximations to the reconstruction method have been suggested and implemented numerically in the literature, but here, for the first time, a complete computer implementation of the full nonlinear algorithm is given. First a boundary integral equation is solved by a Nyström method for the traces of the complex geometrical optics solutions, second the scattering transform is computed and inverted using fast Fourier transform, and finally a boundary value problem is solved for the conductivity distribution. To test the performance of the algorithm highly accurate data is required, and to this end a boundary element method is developed and implemented for the forward problem. The numerical reconstruction algorithm is tested on simulated data and compared to the simpler approximations. In addition, convergence of the numerical solution towards the exact solution of the boundary integral equation is proved.

Keywords: Calderón problem, singular boundary integral equation., numerical solution, electrical impedance tomography, reconstruction algorithm.

Mathematics Subject Classification: Primary: 35R30; Secondary: 35J25, 65N21, 65R3.

Citation: Fabrice Delbary, Kim Knudsen. Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem. Inverse Problems & Imaging, 2014, 8 (4) : 991-1012. doi: 10.3934/ipi.2014.8.991

References:

[1]	A. Abubakar, T. M. Habashy, M. Li and J. Liu, Inversion algorithms for large-scale geophysical electromagnetic,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/12/123012.
[2]	G. Alessandrini, Stable determination of conductivity by boundary measurements,, Appl. Anal., 27 (1988), 153. doi: 10.1080/00036818808839730.
[3]	G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements,, J. Differential Equations, 84 (1990), 252. doi: 10.1016/0022-0396(90)90078-4.
[4]	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.
[5]	E. Anderson, Z. Bai, C. Bischof, L. S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney and D. Sorensen, LAPACK Users' Guide,, 3rd edition, (1999). doi: 10.1137/1.9780898719604.
[6]	J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/1/015002.
[7]	G. Boverman, T.-J. Kao, D. Isaacson and G. J. Saulnier, An implementation of Calderón's Method for 3-D Limited-View EIT,, IEEE Trans. Med. Imaging, 28 (2009), 1073. doi: 10.1109/tmi.2009.2012892.
[8]	R. M. Brown and G. A. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions,, Comm. Partial Differential Equations, 22 (1997), 1009. doi: 10.1080/03605309708821292.
[9]	Alberto-P. Calderón, On an inverse boundary value problem,, In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, (1980), 65.
[10]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences,, 3rd edition, (2013). doi: 10.1007/978-1-4614-4942-3.
[11]	H. Cornean, K. Knudsen and S. Siltanen, Towards a $d$-bar reconstruction method for three-dimensional EIT,, J. Inverse Ill-Posed Probl., 14 (2006), 111. doi: 10.1515/156939406777571102.
[12]	T. A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method,, ACM Transactions on Mathematical Software, 30 (2004), 165. doi: 10.1145/992200.992205.
[13]	T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal method for sparse LU factorization,, SIAM Journal on Matrix Analysis and Applications, 18 (1997), 140. doi: 10.1137/S0895479894246905.
[14]	T. A. Davis and I. S. Duff, A combined unifrontal/multifrontal method for unsymmetric sparse matrices,, ACM Transactions on Mathematical Software, 25 (1999), 1. doi: 10.1145/305658.287640.
[15]	T. A. Davis and I. S. Duff, Algorithm 832: Umfpack v4.3-an unsymmetric-pattern multifrontal method,, ACM Transactions on Mathematical Software, 30 (2004), 196. doi: 10.1145/992200.992206.
[16]	F. Delbary, P. C. Hansen and K. Knudsen, Electrical impedance tomography: 3D reconstructions using scattering transforms,, Appl. Anal., 91 (2012), 737. doi: 10.1080/00036811.2011.598863.
[17]	F. Delbary and K. Knudsen, Regularized electrical impedance tomography using truncated scattering transforms,, In preparation., ().
[18]	F. Delbary and R. Kress, Electrical impedance tomography with point electrodes,, J. Integral Equations Appl., 22 (2010), 193. doi: 10.1216/JIE-2010-22-2-193.
[19]	F. Delbary and R. Kress, Electrical impedance tomography using a point electrode inverse scheme for complete electrode data,, Inverse Probl. Imaging, 5 (2011), 355. doi: 10.3934/ipi.2011.5.355.
[20]	M. Frigo and S. G. Johnson, The design and implementation of FFTW3,, Proceedings of the IEEE, 93 (2005), 1. doi: 10.1109/JPROC.2004.840301.
[21]	M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth and F. Rossi, GNU Scientific Library Reference Manual,, 3rd edition, (2009).
[22]	M. Ganesh and I. G. Graham, A high-order algorithm for obstacle scattering in three dimensions,, J. Comput. Phys., 198 (2004), 211. doi: 10.1016/j.jcp.2004.01.007.
[23]	C. Geuzaine and R. J.-F., Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities,, International Journal for Numerical Methods in Engineering, 79 (2009), 1309. doi: 10.1002/nme.2579.
[24]	K. Hesse and I. H. Sloan, Cubature over the sphere $S^2$ in Sobolev spaces of arbitrary order,, J. Approx. Theory, 141 (2006), 118. doi: 10.1016/j.jat.2006.01.004.
[25]	K. Hesse and I. H. Sloan, Hyperinterpolation on the sphere,, in Frontiers in interpolation and approximation, (2007), 213.
[26]	D. S. Holder, Electrical Impedance Tomography: Methods, History and Applications,, Series in Medical Physics and Biomedical Engineering, (2004). doi: 10.1201/9781420034462.
[27]	O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phase-less far field data,, J. Comput. Phys., 230 (2011), 3443. doi: 10.1016/j.jcp.2011.01.038.
[28]	K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane,, Physiological Measurement, 24 (2003), 391. doi: 10.1088/0967-3334/24/2/351.
[29]	K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem,, Inverse Probl. Imaging, 3 (2009), 599. doi: 10.3934/ipi.2009.3.599.
[30]	K. Knudsen and J. L. Mueller, The Born approximation and Calderón's method for reconstruction of conductivities in 3-D,, Discrete Contin. Dyn. Syst., II (2011), 844.
[31]	K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane,, Comm. Partial Differential Equations, 29 (2004), 361. doi: 10.1081/pde-120030401.
[32]	R. Kress, Linear Integral Equations, vol. 82 of Applied Mathematical Sciences,, 3rd edition, (2014). doi: 10.1007/978-1-4614-9593-2.
[33]	N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation,, Inverse Problems, 17 (2001), 1435. doi: 10.1088/0266-5611/17/5/313.
[34]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000).
[35]	J. L. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements,, SIAM J. Sci. Comput., 24 (2003), 1232. doi: 10.1137/S1064827501394568.
[36]	J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications,, Computational Science & Engineering, (2012). doi: 10.1137/1.9781611972344.
[37]	A. Nachman, J. Sylvester and G. Uhlmann, An n-dimensional Borg-Levinson theorem,, Comm. Math. Phys., 115 (1988), 595. doi: 10.1007/BF01224129.
[38]	A. I. Nachman, Reconstructions from boundary measurements,, Ann. of Math. (2), 128 (1988), 531. doi: 10.2307/1971435.
[39]	A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math. (2), 143 (1996), 71. doi: 10.2307/2118653.
[40]	R. G. Newton, Inverse Schrödinger Scattering in Three Dimensions,, Texts and Monographs in Physics, (1989). doi: 10.1007/978-3-642-83671-8.
[41]	R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta\psi +(v(x)-Eu(x))\psi=0$,, Funktsional. Anal. i Prilozhen., 22 (1988), 11. doi: 10.1007/bf01077418.
[42]	S. Siltanen, J. Mueller and D. Isaacson, Erratum: "An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem [Inverse Problems 16 (2000), 681-699; MR:1766226 (2001g:35269)],, Inverse Problems, 17 (2001), 1561. doi: 10.1088/0266-5611/17/5/501.
[43]	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.
[44]	O. Steinbach and W. L. Wendland, On C. Neumann's method for second-order elliptic systems in domains with non-smooth boundaries,, J. Math. Anal. Appl., 262 (2001), 733. doi: 10.1006/jmaa.2001.7615.
[45]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math. (2), 125 (1987), 153. doi: 10.2307/1971291.
[46]	G. Uhlmann, Electrical impedance tomography and Calderón's problem,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/12/123011.
[47]	L. Wienert, Die Numerische Approximation von Randintegraloperatoren Für die Helmholtzgleichung im $\mathbbR^3$, 1990,, Thesis (Ph.D.)-Georg-August-Universität Göttingen., ().
[48]	T. A. York, \doititle{Status of electrical tomography in industrial applications},, J. Electron. Imaging., 10 (2001), 608. doi: 10.1117/1.1377308.

show all references

References:

[1]	A. Abubakar, T. M. Habashy, M. Li and J. Liu, Inversion algorithms for large-scale geophysical electromagnetic,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/12/123012.
[2]	G. Alessandrini, Stable determination of conductivity by boundary measurements,, Appl. Anal., 27 (1988), 153. doi: 10.1080/00036818808839730.
[3]	G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements,, J. Differential Equations, 84 (1990), 252. doi: 10.1016/0022-0396(90)90078-4.
[4]	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.
[5]	E. Anderson, Z. Bai, C. Bischof, L. S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney and D. Sorensen, LAPACK Users' Guide,, 3rd edition, (1999). doi: 10.1137/1.9780898719604.
[6]	J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/1/015002.
[7]	G. Boverman, T.-J. Kao, D. Isaacson and G. J. Saulnier, An implementation of Calderón's Method for 3-D Limited-View EIT,, IEEE Trans. Med. Imaging, 28 (2009), 1073. doi: 10.1109/tmi.2009.2012892.
[8]	R. M. Brown and G. A. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions,, Comm. Partial Differential Equations, 22 (1997), 1009. doi: 10.1080/03605309708821292.
[9]	Alberto-P. Calderón, On an inverse boundary value problem,, In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, (1980), 65.
[10]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences,, 3rd edition, (2013). doi: 10.1007/978-1-4614-4942-3.
[11]	H. Cornean, K. Knudsen and S. Siltanen, Towards a $d$-bar reconstruction method for three-dimensional EIT,, J. Inverse Ill-Posed Probl., 14 (2006), 111. doi: 10.1515/156939406777571102.
[12]	T. A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method,, ACM Transactions on Mathematical Software, 30 (2004), 165. doi: 10.1145/992200.992205.
[13]	T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal method for sparse LU factorization,, SIAM Journal on Matrix Analysis and Applications, 18 (1997), 140. doi: 10.1137/S0895479894246905.
[14]	T. A. Davis and I. S. Duff, A combined unifrontal/multifrontal method for unsymmetric sparse matrices,, ACM Transactions on Mathematical Software, 25 (1999), 1. doi: 10.1145/305658.287640.
[15]	T. A. Davis and I. S. Duff, Algorithm 832: Umfpack v4.3-an unsymmetric-pattern multifrontal method,, ACM Transactions on Mathematical Software, 30 (2004), 196. doi: 10.1145/992200.992206.
[16]	F. Delbary, P. C. Hansen and K. Knudsen, Electrical impedance tomography: 3D reconstructions using scattering transforms,, Appl. Anal., 91 (2012), 737. doi: 10.1080/00036811.2011.598863.
[17]	F. Delbary and K. Knudsen, Regularized electrical impedance tomography using truncated scattering transforms,, In preparation., ().
[18]	F. Delbary and R. Kress, Electrical impedance tomography with point electrodes,, J. Integral Equations Appl., 22 (2010), 193. doi: 10.1216/JIE-2010-22-2-193.
[19]	F. Delbary and R. Kress, Electrical impedance tomography using a point electrode inverse scheme for complete electrode data,, Inverse Probl. Imaging, 5 (2011), 355. doi: 10.3934/ipi.2011.5.355.
[20]	M. Frigo and S. G. Johnson, The design and implementation of FFTW3,, Proceedings of the IEEE, 93 (2005), 1. doi: 10.1109/JPROC.2004.840301.
[21]	M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth and F. Rossi, GNU Scientific Library Reference Manual,, 3rd edition, (2009).
[22]	M. Ganesh and I. G. Graham, A high-order algorithm for obstacle scattering in three dimensions,, J. Comput. Phys., 198 (2004), 211. doi: 10.1016/j.jcp.2004.01.007.
[23]	C. Geuzaine and R. J.-F., Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities,, International Journal for Numerical Methods in Engineering, 79 (2009), 1309. doi: 10.1002/nme.2579.
[24]	K. Hesse and I. H. Sloan, Cubature over the sphere $S^2$ in Sobolev spaces of arbitrary order,, J. Approx. Theory, 141 (2006), 118. doi: 10.1016/j.jat.2006.01.004.
[25]	K. Hesse and I. H. Sloan, Hyperinterpolation on the sphere,, in Frontiers in interpolation and approximation, (2007), 213.
[26]	D. S. Holder, Electrical Impedance Tomography: Methods, History and Applications,, Series in Medical Physics and Biomedical Engineering, (2004). doi: 10.1201/9781420034462.
[27]	O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phase-less far field data,, J. Comput. Phys., 230 (2011), 3443. doi: 10.1016/j.jcp.2011.01.038.
[28]	K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane,, Physiological Measurement, 24 (2003), 391. doi: 10.1088/0967-3334/24/2/351.
[29]	K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem,, Inverse Probl. Imaging, 3 (2009), 599. doi: 10.3934/ipi.2009.3.599.
[30]	K. Knudsen and J. L. Mueller, The Born approximation and Calderón's method for reconstruction of conductivities in 3-D,, Discrete Contin. Dyn. Syst., II (2011), 844.
[31]	K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane,, Comm. Partial Differential Equations, 29 (2004), 361. doi: 10.1081/pde-120030401.
[32]	R. Kress, Linear Integral Equations, vol. 82 of Applied Mathematical Sciences,, 3rd edition, (2014). doi: 10.1007/978-1-4614-9593-2.
[33]	N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation,, Inverse Problems, 17 (2001), 1435. doi: 10.1088/0266-5611/17/5/313.
[34]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000).
[35]	J. L. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements,, SIAM J. Sci. Comput., 24 (2003), 1232. doi: 10.1137/S1064827501394568.
[36]	J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications,, Computational Science & Engineering, (2012). doi: 10.1137/1.9781611972344.
[37]	A. Nachman, J. Sylvester and G. Uhlmann, An n-dimensional Borg-Levinson theorem,, Comm. Math. Phys., 115 (1988), 595. doi: 10.1007/BF01224129.
[38]	A. I. Nachman, Reconstructions from boundary measurements,, Ann. of Math. (2), 128 (1988), 531. doi: 10.2307/1971435.
[39]	A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math. (2), 143 (1996), 71. doi: 10.2307/2118653.
[40]	R. G. Newton, Inverse Schrödinger Scattering in Three Dimensions,, Texts and Monographs in Physics, (1989). doi: 10.1007/978-3-642-83671-8.
[41]	R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta\psi +(v(x)-Eu(x))\psi=0$,, Funktsional. Anal. i Prilozhen., 22 (1988), 11. doi: 10.1007/bf01077418.
[42]	S. Siltanen, J. Mueller and D. Isaacson, Erratum: "An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem [Inverse Problems 16 (2000), 681-699; MR:1766226 (2001g:35269)],, Inverse Problems, 17 (2001), 1561. doi: 10.1088/0266-5611/17/5/501.
[43]	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.
[44]	O. Steinbach and W. L. Wendland, On C. Neumann's method for second-order elliptic systems in domains with non-smooth boundaries,, J. Math. Anal. Appl., 262 (2001), 733. doi: 10.1006/jmaa.2001.7615.
[45]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math. (2), 125 (1987), 153. doi: 10.2307/1971291.
[46]	G. Uhlmann, Electrical impedance tomography and Calderón's problem,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/12/123011.
[47]	L. Wienert, Die Numerische Approximation von Randintegraloperatoren Für die Helmholtzgleichung im $\mathbbR^3$, 1990,, Thesis (Ph.D.)-Georg-August-Universität Göttingen., ().
[48]	T. A. York, \doititle{Status of electrical tomography in industrial applications},, J. Electron. Imaging., 10 (2001), 608. doi: 10.1117/1.1377308.

[1]	Ville Kolehmainen, Matti Lassas, Petri Ola, Samuli Siltanen. Recovering boundary shape and conductivity in electrical impedance tomography. Inverse Problems & Imaging, 2013, 7 (1) : 217-242. doi: 10.3934/ipi.2013.7.217
[2]	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
[3]	Bastian Gebauer. Localized potentials in electrical impedance tomography. Inverse Problems & Imaging, 2008, 2 (2) : 251-269. doi: 10.3934/ipi.2008.2.251
[4]	Henrik Garde, Kim Knudsen. 3D reconstruction for partial data electrical impedance tomography using a sparsity prior. Conference Publications, 2015, 2015 (special) : 495-504. doi: 10.3934/proc.2015.0495
[5]	Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems & Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021
[6]	Melody Dodd, Jennifer L. Mueller. A real-time D-bar algorithm for 2-D electrical impedance tomography data. Inverse Problems & Imaging, 2014, 8 (4) : 1013-1031. doi: 10.3934/ipi.2014.8.1013
[7]	Nuutti Hyvönen, Harri Hakula, Sampsa Pursiainen. Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (2) : 299-317. doi: 10.3934/ipi.2007.1.299
[8]	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 & Imaging, 2015, 9 (3) : 767-789. doi: 10.3934/ipi.2015.9.767
[9]	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
[10]	Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, David Isaacson, Jari P. Kaipio, Debra McGivney, Erkki Somersalo, Joseph Volzer. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results. Inverse Problems & Imaging, 2015, 9 (3) : 749-766. doi: 10.3934/ipi.2015.9.749
[11]	Melody Alsaker, Sarah Jane Hamilton, Andreas Hauptmann. A direct D-bar method for partial boundary data electrical impedance tomography with a priori information. Inverse Problems & Imaging, 2017, 11 (3) : 427-454. doi: 10.3934/ipi.2017020
[12]	Hiroshi Isozaki. Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (1) : 107-134. doi: 10.3934/ipi.2007.1.107
[13]	Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems & Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026
[14]	Roman Chapko, B. Tomas Johansson. On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach. Inverse Problems & Imaging, 2012, 6 (1) : 25-38. doi: 10.3934/ipi.2012.6.25
[15]	Liliana Borcea, Fernando Guevara Vasquez, Alexander V. Mamonov. Study of noise effects in electrical impedance tomography with resistor networks. Inverse Problems & Imaging, 2013, 7 (2) : 417-443. doi: 10.3934/ipi.2013.7.417
[16]	Dong liu, Ville Kolehmainen, Samuli Siltanen, Anne-maria Laukkanen, Aku Seppänen. Estimation of conductivity changes in a region of interest with electrical impedance tomography. Inverse Problems & Imaging, 2015, 9 (1) : 211-229. doi: 10.3934/ipi.2015.9.211
[17]	Nicolay M. Tanushev, Luminita Vese. A piecewise-constant binary model for electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (2) : 423-435. doi: 10.3934/ipi.2007.1.423
[18]	Nuutti Hyvönen, Lassi Päivärinta, Janne P. Tamminen. Enhancing D-bar reconstructions for electrical impedance tomography with conformal maps. Inverse Problems & Imaging, 2018, 12 (2) : 373-400. doi: 10.3934/ipi.2018017
[19]	Kimmo Karhunen, Aku Seppänen, Jari P. Kaipio. Adaptive meshing approach to identification of cracks with electrical impedance tomography. Inverse Problems & Imaging, 2014, 8 (1) : 127-148. doi: 10.3934/ipi.2014.8.127
[20]	Jérémi Dardé, Harri Hakula, Nuutti Hyvönen, Stratos Staboulis. Fine-tuning electrode information in electrical impedance tomography. Inverse Problems & Imaging, 2012, 6 (3) : 399-421. doi: 10.3934/ipi.2012.6.399

2017 Impact Factor: 1.465

American Institute of Mathematical Sciences