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Identifiability and reconstruction of shapes from integral invariants
Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem
1. | Institut für Mathematik, Johannes Gutenberg-Universität Maint, 55099 Mainz |
2. | Institute of Mathematics, Helsinki University of Technology, FI-02015 HUT |
References:
[1] |
S. R. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93.
doi: 10.1088/0266-5611/15/2/022. |
[2] |
S. R. Arridge and W. R. B. Lionheart, Nonuniqueness in diffusion-based optical tomography, Opt. Lett., 23 (1998), 882-884.
doi: 10.1364/OL.23.000882. |
[3] |
K. Astala and L. Päivärinta, Calderon's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299.
doi: 10.4007/annals.2006.163.265. |
[4] |
G. Bal, Reconstructions in impedance and optical tomography with singular interfaces, Inverse Problems, 21 (2005), 113-131.
doi: 10.1088/0266-5611/21/1/008. |
[5] |
M. Brühl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Problems, 16 (2000), 1029-1042.
doi: 10.1088/0266-5611/16/4/310. |
[6] |
M. Brühl, Explicit characterization of inclusions in electrical impedance tomography, SIAM J. Math. Anal., 32 (2001), 1327-1341.
doi: 10.1137/S003614100036656X. |
[7] |
J. Diestel and J. J. Uhl, "Vector Measures,'' vol. 15 of Mathematical Surveys, American Mathematical Society, Providence, Rhode Island, 1977. |
[8] |
H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,'' vol. 375 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 1996. |
[9] |
F. Frühauf, B. Gebauer and O. Scherzer, Detecting interfaces in a parabolic-elliptic problem from surface measurements, SIAM J. Numer. Anal., 45 (2007), 810-836.
doi: 10.1137/050641545. |
[10] |
B. Gebauer, M. Hanke, A. Kirsch, W. Muniz and C. Schneider, A sampling method for detecting buried objects using electromagnetic scattering, Inverse Problems, 21 (2005), 2035-2050.
doi: 10.1088/0266-5611/21/6/015. |
[11] |
B. Gebauer, The factorization method for real elliptic problems, Z. Anal. Anwend., 25 (2006), 81-102.
doi: 10.4171/ZAA/1279. |
[12] |
B. Gebauer and N. Hyvönen, Factorization method and irregular inclusions in electrical impedance tomography, Inverse Problems, 23 (2007), 2159-2170.
doi: 10.1088/0266-5611/23/5/020. |
[13] |
M. Hanke and M. Brühl, Recent progress in electrical impedance tomography, Inverse Problems, 19 (2003), S65-S90.
doi: 10.1088/0266-5611/19/6/055. |
[14] |
J. Heino and E. Somersalo, Estimation of optical absorption in anisotropic background, Inverse Problems, 18 (2002), 559-573.
doi: 10.1088/0266-5611/18/3/304. |
[15] |
N. Hyvönen, Characterizing inclusions in optical tomography, Inverse Problems, 20 (2004), 737-751.
doi: 10.1088/0266-5611/20/3/006. |
[16] |
N. Hyvönen, Application of a weaker formulation of the factorization method to the characterization of absorbing inclusions in optical tomography, Inverse Problems, 21 (2005), 1331-1343.
doi: 10.1088/0266-5611/21/4/009. |
[17] |
N. Hyvönen, Application of the factorization method to the characterization of weak inclusions in electrical impedance tomography, Adv. in Appl. Math., 39 (2007), 197-221.
doi: 10.1016/j.aam.2006.12.004. |
[18] |
N. Hyvönen, Locating transparent regions in optical absorption and scattering tomography, SIAM J. Appl. Math., 67 (2007), 1101-1123.
doi: 10.1137/06066299X. |
[19] |
A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512.
doi: 10.1088/0266-5611/14/6/009. |
[20] |
A. Kirsch, The factorization method for a class of inverse elliptic problems, Math. Nachr., 278 (2005), 258-277.
doi: 10.1002/mana.200310239. |
[21] |
S. Kusiak and J. Sylvester, The scattering support, Comm. Pure Appl. Math., 56 (2003), 1525-1548.
doi: 10.1002/cpa.3038. |
[22] |
A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96.
doi: 10.2307/2118653. |
[23] |
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. |
show all references
References:
[1] |
S. R. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93.
doi: 10.1088/0266-5611/15/2/022. |
[2] |
S. R. Arridge and W. R. B. Lionheart, Nonuniqueness in diffusion-based optical tomography, Opt. Lett., 23 (1998), 882-884.
doi: 10.1364/OL.23.000882. |
[3] |
K. Astala and L. Päivärinta, Calderon's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299.
doi: 10.4007/annals.2006.163.265. |
[4] |
G. Bal, Reconstructions in impedance and optical tomography with singular interfaces, Inverse Problems, 21 (2005), 113-131.
doi: 10.1088/0266-5611/21/1/008. |
[5] |
M. Brühl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Problems, 16 (2000), 1029-1042.
doi: 10.1088/0266-5611/16/4/310. |
[6] |
M. Brühl, Explicit characterization of inclusions in electrical impedance tomography, SIAM J. Math. Anal., 32 (2001), 1327-1341.
doi: 10.1137/S003614100036656X. |
[7] |
J. Diestel and J. J. Uhl, "Vector Measures,'' vol. 15 of Mathematical Surveys, American Mathematical Society, Providence, Rhode Island, 1977. |
[8] |
H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,'' vol. 375 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 1996. |
[9] |
F. Frühauf, B. Gebauer and O. Scherzer, Detecting interfaces in a parabolic-elliptic problem from surface measurements, SIAM J. Numer. Anal., 45 (2007), 810-836.
doi: 10.1137/050641545. |
[10] |
B. Gebauer, M. Hanke, A. Kirsch, W. Muniz and C. Schneider, A sampling method for detecting buried objects using electromagnetic scattering, Inverse Problems, 21 (2005), 2035-2050.
doi: 10.1088/0266-5611/21/6/015. |
[11] |
B. Gebauer, The factorization method for real elliptic problems, Z. Anal. Anwend., 25 (2006), 81-102.
doi: 10.4171/ZAA/1279. |
[12] |
B. Gebauer and N. Hyvönen, Factorization method and irregular inclusions in electrical impedance tomography, Inverse Problems, 23 (2007), 2159-2170.
doi: 10.1088/0266-5611/23/5/020. |
[13] |
M. Hanke and M. Brühl, Recent progress in electrical impedance tomography, Inverse Problems, 19 (2003), S65-S90.
doi: 10.1088/0266-5611/19/6/055. |
[14] |
J. Heino and E. Somersalo, Estimation of optical absorption in anisotropic background, Inverse Problems, 18 (2002), 559-573.
doi: 10.1088/0266-5611/18/3/304. |
[15] |
N. Hyvönen, Characterizing inclusions in optical tomography, Inverse Problems, 20 (2004), 737-751.
doi: 10.1088/0266-5611/20/3/006. |
[16] |
N. Hyvönen, Application of a weaker formulation of the factorization method to the characterization of absorbing inclusions in optical tomography, Inverse Problems, 21 (2005), 1331-1343.
doi: 10.1088/0266-5611/21/4/009. |
[17] |
N. Hyvönen, Application of the factorization method to the characterization of weak inclusions in electrical impedance tomography, Adv. in Appl. Math., 39 (2007), 197-221.
doi: 10.1016/j.aam.2006.12.004. |
[18] |
N. Hyvönen, Locating transparent regions in optical absorption and scattering tomography, SIAM J. Appl. Math., 67 (2007), 1101-1123.
doi: 10.1137/06066299X. |
[19] |
A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512.
doi: 10.1088/0266-5611/14/6/009. |
[20] |
A. Kirsch, The factorization method for a class of inverse elliptic problems, Math. Nachr., 278 (2005), 258-277.
doi: 10.1002/mana.200310239. |
[21] |
S. Kusiak and J. Sylvester, The scattering support, Comm. Pure Appl. Math., 56 (2003), 1525-1548.
doi: 10.1002/cpa.3038. |
[22] |
A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96.
doi: 10.2307/2118653. |
[23] |
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. |
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