American Institute of Mathematical Sciences

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

Localized potentials in electrical impedance tomography

Bastian Gebauer 1,

1. 

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

Received  January 2008 Revised  March 2008 Published  April 2008

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), 1425-1448. doi: 10.1090/S0025-5718-07-01946-1.  Google Scholar

[2]

T. Arens, Why linear sampling works, Inverse Problems, 20 (2004), 163-173. 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), 265-299. 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, vol. 2007, Article ID 83016, 7 pages, 2007. Google Scholar

[5]

N. Bourbaki, "Elements of Mathematics, Topological Vector Spaces, Chapters 1-5," Springer-Verlag, Berlin, 2003. Google Scholar

[6]

M. Brühl, Explicit characterization of inclusions in electrical impedance tomography, SIAM J. Math. Anal., 32 (2001), 1327-1341. 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), 1029-1042. 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), 653-668. doi: 10.1081/PDE-120002868.  Google Scholar

[9]

A. P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Application to Continuum Physics" (eds. W. H. Meyer and M. A. Raupp), Math. Soc., Rio de Janeiro, Brasil, (1980), 65-73.  Google Scholar

[10]

A. P. Calderón, On an inverse boundary value problem, Comput. Appl. Math., 25 (2006), 133-138.  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, Berlin, 2000. Google Scholar

[12]

V. Druskin, On the uniqueness of inverse problems from incomplete boundary data, SIAM J. Appl. Math., 58 (1998), 1591-1603. 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, Kluwer Academic Publishers, Dordrecht/Boston/London, 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), 810-836. doi: 10.1137/050641545.  Google Scholar

[15]

B. Gebauer, The factorization method for real elliptic problems, Z. Anal. Anwend., 25 (2006), 81-102. 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), 2159-2170. 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) S65-S90.  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), 907-924. Google Scholar

[19]

N. Hyvönen, Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions, SIAM J. Appl. Math., 64 (2004), 902-931. 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), 299-317.  Google Scholar

[21]

V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math., 41 (1988), 865-877. 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), 95-105.  Google Scholar

[23]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The calderón problem with partial data, Ann. of Math., 165 (2007), 567-591. 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), 1489-1512. 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), 258-277. 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), 333-358. doi: 10.1216/jiea/1190905490.  Google Scholar

[27]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), 289-298. 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), 643-667. 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), 1097-1121. doi: 10.1137/070683295.  Google Scholar

[30]

C. Miranda, "Partial Differential Equations of Elliptic Type," Springer-Verlag, Berlin, 1970. Google Scholar

[31]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. doi: 10.2307/2118653.  Google Scholar

[32]

W. Rudin, "Functional Analysis," 2nd ed., McGraw-Hill, New-York, 1991. Google Scholar

[33]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. 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), 1425-1448. doi: 10.1090/S0025-5718-07-01946-1.  Google Scholar

[2]

T. Arens, Why linear sampling works, Inverse Problems, 20 (2004), 163-173. 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), 265-299. 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, vol. 2007, Article ID 83016, 7 pages, 2007. Google Scholar

[5]

N. Bourbaki, "Elements of Mathematics, Topological Vector Spaces, Chapters 1-5," Springer-Verlag, Berlin, 2003. Google Scholar

[6]

M. Brühl, Explicit characterization of inclusions in electrical impedance tomography, SIAM J. Math. Anal., 32 (2001), 1327-1341. 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), 1029-1042. 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), 653-668. doi: 10.1081/PDE-120002868.  Google Scholar

[9]

A. P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Application to Continuum Physics" (eds. W. H. Meyer and M. A. Raupp), Math. Soc., Rio de Janeiro, Brasil, (1980), 65-73.  Google Scholar

[10]

A. P. Calderón, On an inverse boundary value problem, Comput. Appl. Math., 25 (2006), 133-138.  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, Berlin, 2000. Google Scholar

[12]

V. Druskin, On the uniqueness of inverse problems from incomplete boundary data, SIAM J. Appl. Math., 58 (1998), 1591-1603. 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, Kluwer Academic Publishers, Dordrecht/Boston/London, 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), 810-836. doi: 10.1137/050641545.  Google Scholar

[15]

B. Gebauer, The factorization method for real elliptic problems, Z. Anal. Anwend., 25 (2006), 81-102. 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), 2159-2170. 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) S65-S90.  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), 907-924. Google Scholar

[19]

N. Hyvönen, Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions, SIAM J. Appl. Math., 64 (2004), 902-931. 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), 299-317.  Google Scholar

[21]

V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math., 41 (1988), 865-877. 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), 95-105.  Google Scholar

[23]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The calderón problem with partial data, Ann. of Math., 165 (2007), 567-591. 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), 1489-1512. 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), 258-277. 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), 333-358. doi: 10.1216/jiea/1190905490.  Google Scholar

[27]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), 289-298. 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), 643-667. 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), 1097-1121. doi: 10.1137/070683295.  Google Scholar

[30]

C. Miranda, "Partial Differential Equations of Elliptic Type," Springer-Verlag, Berlin, 1970. Google Scholar

[31]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. doi: 10.2307/2118653.  Google Scholar

[32]

W. Rudin, "Functional Analysis," 2nd ed., McGraw-Hill, New-York, 1991. Google Scholar

[33]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar

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