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A real-time D-bar algorithm for 2-D electrical impedance tomography data
Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields
1. | Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027 |
2. | Department of Applied Physics and Applied Mathematics, Columbia University, 200 S. W. Mudd Building, MC 4701, 500 W. 120th Street, New York, NY 10027 |
References:
[1] |
G. Alessandrini, An identification problem for an elliptic equation in two variables, Ann. Mat. Pura Appl., 145 (1986), 265-295.
doi: 10.1007/BF01790543. |
[2] |
H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter and M. Fink, Electrical Impedance Tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573.
doi: 10.1137/070686408. |
[3] |
G. Bal, Inside Out, Cambridge University Press, 2012, ch. Hybrid inverse problems and internal functionals. |
[4] |
G. Bal, C. Guo and F. Monard, Linearized internal functionals for anisotropic conductivities, Inv. Probl. and Imaging, 8 (2014), 1-22.
doi: 10.3934/ipi.2014.8.1. |
[5] |
_______, Inverse anisotropic conductivity from internal current densities, Inverse Problems, 30 (2014), 025001. |
[6] |
_______, Imaging of anisotropic conductivities from current densities in two dimensions, submitted, (2014)., , ().
|
[7] |
G. Bal and J. C. Schotland, Inverse scattering and acousto-optic imaging, Phys. Rev. Letters, 104 (2010), p. 043902.
doi: 10.1103/PhysRevLett.104.043902. |
[8] |
G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics, Inverse Problems, 26 (2010), 085010, 20pp.
doi: 10.1088/0266-5611/26/8/085010. |
[9] |
_______, Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions, Communications on Pure and Applied Mathematics, 66 (2013), 1629-1652. |
[10] |
A. Calderón, Uniqueness in the cauchy problem for partial differential equations, Amer.J.Math., 80 (1958), 16-36.
doi: 10.2307/2372819. |
[11] |
P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data, Comm.PDE., 34 (2009), 1425-1464.
doi: 10.1080/03605300903296272. |
[12] |
J. Chen and Y. Yang, Inverse problem of electro-seismic conversion, Inverse Problems, 29 (2013), 115006, 15pp.
doi: 10.1088/0266-5611/29/11/115006. |
[13] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, Vol. 93. Springer, 2012.
doi: 10.1007/978-1-4614-4942-3. |
[14] |
D. Colton and L. Päivärinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves, Arch.Rational Mech.Anal., 119 (1992), 59-70.
doi: 10.1007/BF00376010. |
[15] |
M. Eller and M.Yamamoto, A Carleman inequality for the stationary anisotropic Maxwell system, J.Math.Pures Appl., 86 (2006), 449-462.
doi: 10.1016/j.matpur.2006.10.004. |
[16] |
L. Hormander, The Analysis of Linear Partial Differential Operators:Pseudo-Differential Operators I-IV, Springer Verlag, 1983. |
[17] |
R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1990. |
[18] |
Y. Ider and L. Muftuler, Measurement of AC magnetic field distribution using magnetic resonance imaging, IEEE Transactions on Medical Imaging, 16 (1997), 617-622.
doi: 10.1109/42.640752. |
[19] |
Y. Ider and Özlem Birgül, Use of the magnetic field generated by the internal distribution of injected currents for electrical impedance tomography (MR-EIT), Elektrik, 6 (1998), 215-225. |
[20] |
C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations, Duke Math.J., 157 (2011), 369-419.
doi: 10.1215/00127094-1272903. |
[21] |
R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Commun. Pure Appl. Math., 37 (1984), 289-298.
doi: 10.1002/cpa.3160370302. |
[22] |
P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 055013, 21pp.
doi: 10.1088/0266-5611/27/5/055013. |
[23] |
P. Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data, Inverse Problems, 28 (2012), 084007, 20pp.
doi: 10.1088/0266-5611/28/8/084007. |
[24] |
O. Kwon, E. Woo, J. Yoon and J. Seo, Magnetic resonance electrical impedance tomography (MREIT): Simulation study of J-substitution algorithm, IEEE Trans. Biomed. Eng., 49 (2002), 160-167. |
[25] |
P. Lax, A stability theorem for solutions of abstract differential equations, and tis application to the study of the local behavior of solutions to elliptic equations, Comm.Pure Applied Math., 9 (1956), 747-766.
doi: 10.1002/cpa.3160090407. |
[26] |
F. Monard and G. Bal, Inverse anisotropic conductivity from power densities in dimension $n \ge 3$, Comm. Partial Differential Equations, 38 (2013), 1183-1207.
doi: 10.1080/03605302.2013.787089. |
[27] |
A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014, 16pp.
doi: 10.1088/0266-5611/25/3/035014. |
[28] |
________, Reconstruction of planar conductivities in subdomains from incomplete data, SIAM J. Appl. Math., 70 (2010), 3342-3362. |
[29] |
G. Nakamura, G. Uhlmann and J.N. Wang, Oscillating-decaying solutions, Runge approximation property for the anisotropic elasticity system and their applications to inverse problems, J.Math.Pures Appl., 84 (2005), 21-54.
doi: 10.1016/j.matpur.2004.09.002. |
[30] |
L. Nirenberg, Lectures on Linear Partial Differential Equations, Amer. Math. Soc., Providencem R.I., 1973. |
[31] |
P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electromagnetics, Duke Math. J., 70 (1993), 617-653.
doi: 10.1215/S0012-7094-93-07014-7. |
[32] |
P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized Sommerfeld potentials, SIAM J.Appl.Math., 56 (1996), 1129-1145.
doi: 10.1137/S0036139995283948. |
[33] |
J. K. Seo, D.-H. Kim, J. Lee, O. I. Kwon, S. Z. K. Sajib and E. J. Woo, Electrical tissue property imaging using MRI at dc and larmor frequency, Inverse Problems, 28 (2012), 084002, 26pp.
doi: 10.1088/0266-5611/28/8/084002. |
[34] |
E. Somersalo, D. Isaacson and M. Cheney, A linearized inverse boundary value problem for Maxwell's equations, J.Comp.Appl.Math, 42 (1992), 123-136.
doi: 10.1016/0377-0427(92)90167-V. |
[35] |
Z. Sun and G. Uhlmann, An inverse boundary value problem for Maxwell's equations, Arch.Rational Mech.Anal., 119 (1992), 71-93.
doi: 10.1007/BF00376011. |
[36] |
J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
[37] |
G. Uhlmann, Calderón's problem and electrical impedance tomography, Inverse Problems, 25 (2009), 123011.
doi: 10.1088/0266-5611/25/12/123011. |
[38] |
C. Weber, Regularity theorems for Maxwell's equations, Mathematical Methods in the Applied Sciences, 3 (1981), 523-536.
doi: 10.1002/mma.1670030137. |
show all references
References:
[1] |
G. Alessandrini, An identification problem for an elliptic equation in two variables, Ann. Mat. Pura Appl., 145 (1986), 265-295.
doi: 10.1007/BF01790543. |
[2] |
H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter and M. Fink, Electrical Impedance Tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573.
doi: 10.1137/070686408. |
[3] |
G. Bal, Inside Out, Cambridge University Press, 2012, ch. Hybrid inverse problems and internal functionals. |
[4] |
G. Bal, C. Guo and F. Monard, Linearized internal functionals for anisotropic conductivities, Inv. Probl. and Imaging, 8 (2014), 1-22.
doi: 10.3934/ipi.2014.8.1. |
[5] |
_______, Inverse anisotropic conductivity from internal current densities, Inverse Problems, 30 (2014), 025001. |
[6] |
_______, Imaging of anisotropic conductivities from current densities in two dimensions, submitted, (2014)., , ().
|
[7] |
G. Bal and J. C. Schotland, Inverse scattering and acousto-optic imaging, Phys. Rev. Letters, 104 (2010), p. 043902.
doi: 10.1103/PhysRevLett.104.043902. |
[8] |
G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics, Inverse Problems, 26 (2010), 085010, 20pp.
doi: 10.1088/0266-5611/26/8/085010. |
[9] |
_______, Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions, Communications on Pure and Applied Mathematics, 66 (2013), 1629-1652. |
[10] |
A. Calderón, Uniqueness in the cauchy problem for partial differential equations, Amer.J.Math., 80 (1958), 16-36.
doi: 10.2307/2372819. |
[11] |
P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data, Comm.PDE., 34 (2009), 1425-1464.
doi: 10.1080/03605300903296272. |
[12] |
J. Chen and Y. Yang, Inverse problem of electro-seismic conversion, Inverse Problems, 29 (2013), 115006, 15pp.
doi: 10.1088/0266-5611/29/11/115006. |
[13] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, Vol. 93. Springer, 2012.
doi: 10.1007/978-1-4614-4942-3. |
[14] |
D. Colton and L. Päivärinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves, Arch.Rational Mech.Anal., 119 (1992), 59-70.
doi: 10.1007/BF00376010. |
[15] |
M. Eller and M.Yamamoto, A Carleman inequality for the stationary anisotropic Maxwell system, J.Math.Pures Appl., 86 (2006), 449-462.
doi: 10.1016/j.matpur.2006.10.004. |
[16] |
L. Hormander, The Analysis of Linear Partial Differential Operators:Pseudo-Differential Operators I-IV, Springer Verlag, 1983. |
[17] |
R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1990. |
[18] |
Y. Ider and L. Muftuler, Measurement of AC magnetic field distribution using magnetic resonance imaging, IEEE Transactions on Medical Imaging, 16 (1997), 617-622.
doi: 10.1109/42.640752. |
[19] |
Y. Ider and Özlem Birgül, Use of the magnetic field generated by the internal distribution of injected currents for electrical impedance tomography (MR-EIT), Elektrik, 6 (1998), 215-225. |
[20] |
C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations, Duke Math.J., 157 (2011), 369-419.
doi: 10.1215/00127094-1272903. |
[21] |
R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Commun. Pure Appl. Math., 37 (1984), 289-298.
doi: 10.1002/cpa.3160370302. |
[22] |
P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 055013, 21pp.
doi: 10.1088/0266-5611/27/5/055013. |
[23] |
P. Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data, Inverse Problems, 28 (2012), 084007, 20pp.
doi: 10.1088/0266-5611/28/8/084007. |
[24] |
O. Kwon, E. Woo, J. Yoon and J. Seo, Magnetic resonance electrical impedance tomography (MREIT): Simulation study of J-substitution algorithm, IEEE Trans. Biomed. Eng., 49 (2002), 160-167. |
[25] |
P. Lax, A stability theorem for solutions of abstract differential equations, and tis application to the study of the local behavior of solutions to elliptic equations, Comm.Pure Applied Math., 9 (1956), 747-766.
doi: 10.1002/cpa.3160090407. |
[26] |
F. Monard and G. Bal, Inverse anisotropic conductivity from power densities in dimension $n \ge 3$, Comm. Partial Differential Equations, 38 (2013), 1183-1207.
doi: 10.1080/03605302.2013.787089. |
[27] |
A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014, 16pp.
doi: 10.1088/0266-5611/25/3/035014. |
[28] |
________, Reconstruction of planar conductivities in subdomains from incomplete data, SIAM J. Appl. Math., 70 (2010), 3342-3362. |
[29] |
G. Nakamura, G. Uhlmann and J.N. Wang, Oscillating-decaying solutions, Runge approximation property for the anisotropic elasticity system and their applications to inverse problems, J.Math.Pures Appl., 84 (2005), 21-54.
doi: 10.1016/j.matpur.2004.09.002. |
[30] |
L. Nirenberg, Lectures on Linear Partial Differential Equations, Amer. Math. Soc., Providencem R.I., 1973. |
[31] |
P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electromagnetics, Duke Math. J., 70 (1993), 617-653.
doi: 10.1215/S0012-7094-93-07014-7. |
[32] |
P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized Sommerfeld potentials, SIAM J.Appl.Math., 56 (1996), 1129-1145.
doi: 10.1137/S0036139995283948. |
[33] |
J. K. Seo, D.-H. Kim, J. Lee, O. I. Kwon, S. Z. K. Sajib and E. J. Woo, Electrical tissue property imaging using MRI at dc and larmor frequency, Inverse Problems, 28 (2012), 084002, 26pp.
doi: 10.1088/0266-5611/28/8/084002. |
[34] |
E. Somersalo, D. Isaacson and M. Cheney, A linearized inverse boundary value problem for Maxwell's equations, J.Comp.Appl.Math, 42 (1992), 123-136.
doi: 10.1016/0377-0427(92)90167-V. |
[35] |
Z. Sun and G. Uhlmann, An inverse boundary value problem for Maxwell's equations, Arch.Rational Mech.Anal., 119 (1992), 71-93.
doi: 10.1007/BF00376011. |
[36] |
J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
[37] |
G. Uhlmann, Calderón's problem and electrical impedance tomography, Inverse Problems, 25 (2009), 123011.
doi: 10.1088/0266-5611/25/12/123011. |
[38] |
C. Weber, Regularity theorems for Maxwell's equations, Mathematical Methods in the Applied Sciences, 3 (1981), 523-536.
doi: 10.1002/mma.1670030137. |
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