Localized potentials in electrical impedance tomography

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May 2008, 2(2): 251-269. doi: 10.3934/ipi.2008.2.251

Localized potentials in electrical impedance tomography

1.	Institut für Mathematik, Johannes Gutenberg-Universität Maint, 55099 Mainz, Germany

Received January 2008 Revised March 2008 Published April 2008

Abstract
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In this work we study localized electric potentials that have an arbitrarily high energy on some given subset of a domain and low energy on another. We show that such potentials exist for general $L^\infty_+$-conductivities in almost arbitrarily shaped subregions of a domain, as long as these regions are connected to the boundary and a unique continuation principle is satisfied. From this we deduce a simple, but new, theoretical identifiability result for the famous Calderón problem with partial data. We also show how to construct such potentials numerically and use a connection with the factorization method to derive a new non-iterative algorithm for the detection of inclusions in electrical impedance tomography.

Keywords: Electrical impedance tomography, factorization method., Calderon problem.

Mathematics Subject Classification: 35J25, 35R05, 35R3.

Citation: Bastian Gebauer. Localized potentials in electrical impedance tomography. Inverse Problems & Imaging, 2008, 2 (2) : 251-269. doi: 10.3934/ipi.2008.2.251

References:

[1]	H. Ammari, R. Griesmaier and M. Hanke, Identification of small inhomogeneities: asymptotic factorization,, Math. Comp., 76 (2007), 1425. doi: 10.1090/S0025-5718-07-01946-1. Google Scholar
[2]	T. Arens, Why linear sampling works,, Inverse Problems, 20 (2004), 163. doi: 10.1088/0266-5611/20/1/010. Google Scholar
[3]	K. Astala and L. Päivärinta, Calderon's inverse conductivity problem in the plane,, Ann. of Math., 163 (2006), 265. doi: 10.4007/annals.2006.163.265. Google Scholar
[4]	M. Azzouz, M. Hanke, C. Oesterlein and K. Schilcher, The factorization method for electrical impedance tomography data from a new planar device,, International Journal of Biomedical Imaging, (2007). Google Scholar
[5]	N. Bourbaki, "Elements of Mathematics, Topological Vector Spaces, Chapters 1-5,", Springer-Verlag, (2003). Google Scholar
[6]	M. Brühl, Explicit characterization of inclusions in electrical impedance tomography,, SIAM J. Math. Anal., 32 (2001), 1327. doi: 10.1137/S003614100036656X. Google Scholar
[7]	M. Brühl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography,, Inverse Problems, 16 (2000), 1029. doi: 10.1088/0266-5611/16/4/310. Google Scholar
[8]	A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial cauchy data,, Comm. Partial Differential Equations, 27 (2002), 653. doi: 10.1081/PDE-120002868. Google Scholar
[9]	A. P. Calderón, On an inverse boundary value problem,, in, (1980), 65. Google Scholar
[10]	A. P. Calderón, On an inverse boundary value problem,, Comput. Appl. Math., 25 (2006), 133. Google Scholar
[11]	R. Dautray and J. L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology - Volume 2: Functional and Variational Methods,", Springer-Verlag, (2000). Google Scholar
[12]	V. Druskin, On the uniqueness of inverse problems from incomplete boundary data,, SIAM J. Appl. Math., 58 (1998), 1591. doi: 10.1137/S0036139996298292. Google Scholar
[13]	H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,", volume 375 of Mathematics and Its Applications, (2000). Google Scholar
[14]	F. Frühauf, B. Gebauer and O. Scherzer, Detecting interfaces in a parabolic-elliptic problem from surface measurements,, SIAM J. Numer. Anal., 45 (2007), 810. doi: 10.1137/050641545. Google Scholar
[15]	B. Gebauer, The factorization method for real elliptic problems,, Z. Anal. Anwend., 25 (2006), 81. doi: 10.4171/ZAA/1279. Google Scholar
[16]	B. Gebauer and N. Hyvönen, Factorization method and irregular inclusions in electrical impedance tomography,, Inverse Problems, 23 (2007), 2159. doi: 10.1088/0266-5611/23/5/020. Google Scholar
[17]	M. Hanke and M. Brühl, Recent progress in electrical impedance tomography,, Inverse Problems, 19 (2003). Google Scholar
[18]	M. Hanke and B. Schappel, The factorization method for electrical impedance tomography in the half space,, SIAM J. Appl. Math., 68 (2008), 907. Google Scholar
[19]	N. Hyvönen, Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions,, SIAM J. Appl. Math., 64 (2004), 902. doi: 10.1137/S0036139903423303. Google Scholar
[20]	N. Hyvönen, H. Hakula and S. Pursiainen, Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography,, Inverse Probl. Imaging, 1 (2007), 299. Google Scholar
[21]	V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient,, Comm. Pure Appl. Math., 41 (1988), 865. doi: 10.1002/cpa.3160410702. Google Scholar
[22]	V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Probl. Imaging, 1 (2007), 95. Google Scholar
[23]	C. E. Kenig, J. Sjöstrand and G. Uhlmann, The calderón problem with partial data,, Ann. of Math., 165 (2007), 567. doi: 10.4007/annals.2007.165.567. Google Scholar
[24]	A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator,, Inverse Problems, 14 (1998), 1489. doi: 10.1088/0266-5611/14/6/009. Google Scholar
[25]	A. Kirsch, The factorization method for a class of inverse elliptic problems,, Math. Nachr., 278 (2005), 258. doi: 10.1002/mana.200310239. Google Scholar
[26]	A. Kirsch, An integral equation for maxwell's equations in a layered medium with an application to the factorization method,, J. Integral Equations Appl., 19 (2007), 333. doi: 10.1216/jiea/1190905490. Google Scholar
[27]	R. Kohn and M. Vogelius, Determining conductivity by boundary measurements,, Comm. Pure Appl. Math., 37 (1984), 289. doi: 10.1002/cpa.3160370302. Google Scholar
[28]	R. Kohn and M. Vogelius, Determining conductivity by boundary measurements II. interior results,, Comm. Pure Appl. Math., 38 (1985), 643. doi: 10.1002/cpa.3160380513. Google Scholar
[29]	A. Lechleiter, N. Hyvönen and H. Hakula, The factorization method applied to the complete electrode model of impedance tomography,, SIAM J. Appl. Math., 68 (2008), 1097. doi: 10.1137/070683295. Google Scholar
[30]	C. Miranda, "Partial Differential Equations of Elliptic Type,", Springer-Verlag, (1970). Google Scholar
[31]	A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math., 143 (1996), 71. doi: 10.2307/2118653. Google Scholar
[32]	W. Rudin, "Functional Analysis," 2nd ed.,, McGraw-Hill, (1991). Google Scholar
[33]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153. doi: 10.2307/1971291. Google Scholar

show all references

References:

[1]	H. Ammari, R. Griesmaier and M. Hanke, Identification of small inhomogeneities: asymptotic factorization,, Math. Comp., 76 (2007), 1425. doi: 10.1090/S0025-5718-07-01946-1. Google Scholar
[2]	T. Arens, Why linear sampling works,, Inverse Problems, 20 (2004), 163. doi: 10.1088/0266-5611/20/1/010. Google Scholar
[3]	K. Astala and L. Päivärinta, Calderon's inverse conductivity problem in the plane,, Ann. of Math., 163 (2006), 265. doi: 10.4007/annals.2006.163.265. Google Scholar
[4]	M. Azzouz, M. Hanke, C. Oesterlein and K. Schilcher, The factorization method for electrical impedance tomography data from a new planar device,, International Journal of Biomedical Imaging, (2007). Google Scholar
[5]	N. Bourbaki, "Elements of Mathematics, Topological Vector Spaces, Chapters 1-5,", Springer-Verlag, (2003). Google Scholar
[6]	M. Brühl, Explicit characterization of inclusions in electrical impedance tomography,, SIAM J. Math. Anal., 32 (2001), 1327. doi: 10.1137/S003614100036656X. Google Scholar
[7]	M. Brühl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography,, Inverse Problems, 16 (2000), 1029. doi: 10.1088/0266-5611/16/4/310. Google Scholar
[8]	A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial cauchy data,, Comm. Partial Differential Equations, 27 (2002), 653. doi: 10.1081/PDE-120002868. Google Scholar
[9]	A. P. Calderón, On an inverse boundary value problem,, in, (1980), 65. Google Scholar
[10]	A. P. Calderón, On an inverse boundary value problem,, Comput. Appl. Math., 25 (2006), 133. Google Scholar
[11]	R. Dautray and J. L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology - Volume 2: Functional and Variational Methods,", Springer-Verlag, (2000). Google Scholar
[12]	V. Druskin, On the uniqueness of inverse problems from incomplete boundary data,, SIAM J. Appl. Math., 58 (1998), 1591. doi: 10.1137/S0036139996298292. Google Scholar
[13]	H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,", volume 375 of Mathematics and Its Applications, (2000). Google Scholar
[14]	F. Frühauf, B. Gebauer and O. Scherzer, Detecting interfaces in a parabolic-elliptic problem from surface measurements,, SIAM J. Numer. Anal., 45 (2007), 810. doi: 10.1137/050641545. Google Scholar
[15]	B. Gebauer, The factorization method for real elliptic problems,, Z. Anal. Anwend., 25 (2006), 81. doi: 10.4171/ZAA/1279. Google Scholar
[16]	B. Gebauer and N. Hyvönen, Factorization method and irregular inclusions in electrical impedance tomography,, Inverse Problems, 23 (2007), 2159. doi: 10.1088/0266-5611/23/5/020. Google Scholar
[17]	M. Hanke and M. Brühl, Recent progress in electrical impedance tomography,, Inverse Problems, 19 (2003). Google Scholar
[18]	M. Hanke and B. Schappel, The factorization method for electrical impedance tomography in the half space,, SIAM J. Appl. Math., 68 (2008), 907. Google Scholar
[19]	N. Hyvönen, Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions,, SIAM J. Appl. Math., 64 (2004), 902. doi: 10.1137/S0036139903423303. Google Scholar
[20]	N. Hyvönen, H. Hakula and S. Pursiainen, Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography,, Inverse Probl. Imaging, 1 (2007), 299. Google Scholar
[21]	V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient,, Comm. Pure Appl. Math., 41 (1988), 865. doi: 10.1002/cpa.3160410702. Google Scholar
[22]	V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Probl. Imaging, 1 (2007), 95. Google Scholar
[23]	C. E. Kenig, J. Sjöstrand and G. Uhlmann, The calderón problem with partial data,, Ann. of Math., 165 (2007), 567. doi: 10.4007/annals.2007.165.567. Google Scholar
[24]	A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator,, Inverse Problems, 14 (1998), 1489. doi: 10.1088/0266-5611/14/6/009. Google Scholar
[25]	A. Kirsch, The factorization method for a class of inverse elliptic problems,, Math. Nachr., 278 (2005), 258. doi: 10.1002/mana.200310239. Google Scholar
[26]	A. Kirsch, An integral equation for maxwell's equations in a layered medium with an application to the factorization method,, J. Integral Equations Appl., 19 (2007), 333. doi: 10.1216/jiea/1190905490. Google Scholar
[27]	R. Kohn and M. Vogelius, Determining conductivity by boundary measurements,, Comm. Pure Appl. Math., 37 (1984), 289. doi: 10.1002/cpa.3160370302. Google Scholar
[28]	R. Kohn and M. Vogelius, Determining conductivity by boundary measurements II. interior results,, Comm. Pure Appl. Math., 38 (1985), 643. doi: 10.1002/cpa.3160380513. Google Scholar
[29]	A. Lechleiter, N. Hyvönen and H. Hakula, The factorization method applied to the complete electrode model of impedance tomography,, SIAM J. Appl. Math., 68 (2008), 1097. doi: 10.1137/070683295. Google Scholar
[30]	C. Miranda, "Partial Differential Equations of Elliptic Type,", Springer-Verlag, (1970). Google Scholar
[31]	A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math., 143 (1996), 71. doi: 10.2307/2118653. Google Scholar
[32]	W. Rudin, "Functional Analysis," 2nd ed.,, McGraw-Hill, (1991). Google Scholar
[33]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153. doi: 10.2307/1971291. Google Scholar

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