An analysis of electrical impedance tomography with applications to Tikhonov regularization
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 1027-1048.

This paper analyzes the continuum model/complete electrode model in the electrical impedance tomography inverse problem of determining the conductivity parameter from boundary measurements. The continuity and differentiability of the forward operator with respect to the conductivity parameter in Lp-norms are proved. These analytical results are applied to several popular regularization formulations, which incorporate a priori information of smoothness/sparsity on the inhomogeneity through Tikhonov regularization, for both linearized and nonlinear models. Some important properties, e.g., existence, stability, consistency and convergence rates, are established. This provides some theoretical justifications of their practical usage.

DOI : 10.1051/cocv/2011193
Classification : 49N45, 65N21
Mots-clés : electrical impedance tomography, Tikhonov regularization, convergence rate 
@article{COCV_2012__18_4_1027_0,
     author = {Jin, Bangti and Maass, Peter},
     title = {An analysis of electrical impedance tomography with applications to {Tikhonov} regularization},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1027--1048},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {4},
     year = {2012},
     doi = {10.1051/cocv/2011193},
     mrnumber = {3019471},
     zbl = {1259.49056},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2011193/}
}
TY  - JOUR
AU  - Jin, Bangti
AU  - Maass, Peter
TI  - An analysis of electrical impedance tomography with applications to Tikhonov regularization
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2012
SP  - 1027
EP  - 1048
VL  - 18
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2011193/
DO  - 10.1051/cocv/2011193
LA  - en
ID  - COCV_2012__18_4_1027_0
ER  - 
%0 Journal Article
%A Jin, Bangti
%A Maass, Peter
%T An analysis of electrical impedance tomography with applications to Tikhonov regularization
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2012
%P 1027-1048
%V 18
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2011193/
%R 10.1051/cocv/2011193
%G en
%F COCV_2012__18_4_1027_0
Jin, Bangti; Maass, Peter. An analysis of electrical impedance tomography with applications to Tikhonov regularization. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 1027-1048. doi : 10.1051/cocv/2011193. http://www.numdam.org/articles/10.1051/cocv/2011193/

[1] G. Alessandrini, Open issues of stability for the inverse conductivity problem. Journal Inverse Ill-Posed Problems 15 (2007) 451-460. | MR | Zbl

[2] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane. Ann. of Math. (2) 163 (2006) 265-299. | MR | Zbl

[3] K. Astala, D. Faraco, and L. Székelyhidi Jr., Convex integration and the Lp theory of elliptic equations. Ann. Scuola Norm. Super. Pisa Cl. Sci. (5) 7 (2008) 1-50. | Numdam | MR | Zbl

[4] R.H. Bayford, Bioimpedance tomography (electrical impedance tomography). Ann. Rev. Biomed. Eng. 8 (2006) 63-91.

[5] T. Bonesky, K.S. Kazimierski, P. Maass, F. Schöpfer and T. Schuster, Minimization of Tikhonov functionals in Banach spaces. Abstr. Appl. Anal. (2008) 19 pages. | MR

[6] K. Bredies and D.A. Lorenz, Regularization with non-convex separable constraints. Inverse Problems 25 (2009) 085011. | MR | Zbl

[7] L.M. Bregman, The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7 (1967) 200-217. | MR | Zbl

[8] M. Burger and S. Osher, Convergence rates of convex variational regularization. Inverse Problems 20 (2004) 1411-1420. | MR | Zbl

[9] A.-P. Calderón, On an inverse boundary value problem. In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980). Soc. Brasil. Mat., Rio de Janeiro (1980) 65-73. | MR | Zbl

[10] M. Cheney, D. Isaacson, J.C. Newell, S. Simske and J. Goble, NOSER : An algorithm for solving the inverse conductivity problem. Int. J. Imag. Syst. Tech. 2 (1990) 66-75.

[11] M. Cheney, D. Isaacson and J.C. Newell, Electrical impedance tomography. SIAM Rev. 41 (1999) 85-101. | MR | Zbl

[12] K.-S. Cheng, D. Isaacson, J.C. Newell and D.G. Gisser, Electrode models for electric current computed tomography. IEEE Trans. Biomed. Eng. 36 (1989) 918-924.

[13] E.T. Chung, T.F. Chan and X.-C. Tai, Electrical impedance tomography using level set representation and total variational regularization. J. Comput. Phys. 205 (2005) 357-372. | MR | Zbl

[14] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57 (2004) 1413-1457. | MR | Zbl

[15] T. Dierkes, O. Dorn, F. Natterer, V. Palamodov and H. Sielschott, Fréchet derivatives for some bilinear inverse problems. SIAM J. Appl. Math. 62 (2002) 2092-2113. | MR | Zbl

[16] D. Dobson, Convergence of a reconstruction method for the inverse conductivity problem. SIAM J. Appl. Math. 52 (1992) 442-458. | MR | Zbl

[17] D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theor. 52 (2006) 1289-1306. | MR | Zbl

[18] H. Egger and M. Schlottbom, Analysis and regularization of problems in diffuse optical tomography. SIAM J. Math. Anal. 42 (2010) 1934-1948. | MR | Zbl

[19] H.W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularisation of nonlinear ill-posed problems. Inverse Problems 5 (1989) 523-540. | MR | Zbl

[20] H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer Academic, Dordrecht (1996). | MR | Zbl

[21] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992). | MR | Zbl

[22] T. Gallouet and A. Monier, On the regularity of solutions to elliptic equations. Rend. Mat. Appl. (7) 19 (1999) 471-488. | MR | Zbl

[23] M. Gehre, T. Kluth, A. Lipponen, B. Jin, A. Seppänen, J. Kaipio and P. Maass, Sparsity reconstruction in electrical impedance tomography : an experimental evaluation. J. Comput. Appl. Math. (2011), in press, DOI : 10.1016/j.cam.2011.09.035. | MR | Zbl

[24] M. Grasmair, M. Haltmeier and O. Scherzer, Sparse regularization with lq penalty term. Inverse Problems 24 (2008) 055020. | MR | Zbl

[25] K. Gröger, A W1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283 (1989) 679-687. | MR | Zbl

[26] B. Harrach and J.K. Seo, Exact shape-reconstruction by one-step linearization in electrical impedance tomography. SIAM J. Math. Anal. 42 (2010) 1505-1518. | MR | Zbl

[27] B. Hofmann and M. Yamamoto, On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems. Appl. Anal. 89 (2010) 1705-1727. | MR | Zbl

[28] B. Hofmann, B. Kaltenbacher, C. Poeschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Problems 23 (2007) 987-1010. | MR | Zbl

[29] N. Hyvönen, Complete electrode model of electrical impedance tomography : approximation properties and characterization of inclusions. SIAM J. Appl. Math. 64 (2004) 902-931. | MR | Zbl

[30] M. Ikehata and S. Siltanen, Electrical impedance tomography and Mittag-Leffler's function. Inverse Problems 20 (2004) 1325-1348. | MR | Zbl

[31] O.Y. Imanuvilov, G. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions. J. Amer. Math. Soc. 23 (2010) 655-691. | MR | Zbl

[32] D. Isaacson, J.L. Mueller, J.C. Newell and S. Siltanen, Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography. IEEE Trans. Med. Imag. 23 (2004) 821-828.

[33] K. Ito, K. Kunisch and Z. Li, Level-set function approach to an inverse interface problem. Inverse Problems 17 (2001) 1225-1242. | MR | Zbl

[34] K. Ito, B. Jin and T. Takeuchi, A regularization parameter for nonsmooth Tikhonov regularization. SIAM J. Sci. Comput. 33 (2011) 1415-1438. | MR | Zbl

[35] B. Jin, Y. Zhao and J. Zou, Iterative parameter choice by discrepancy principle. IMA J. Numer. Anal. (2011), in press. | MR | Zbl

[36] B. Jin, Y. Zhao and P. Maass, A reconstruction algorithm for electrical impedance tomography based on sparsity regularization. Internat. J. Numer. Methods Engrg. (2011), DOI : 10.2002/nme.3247. | Zbl

[37] J.P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography. Inverse Problems 16 (2000) 1487-1522. | MR | Zbl

[38] B. Kaltenbacher and B. Hofmann, Convergence rates for the iteratively regularized Gauss-Newton method in Banach spaces. Inverse Problems 26 (2010) 035007. | MR | Zbl

[39] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems. Oxford University Press, Oxford (2008). | MR | Zbl

[40] K. Knudsen, M. Lassas, J.L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem. IPI 3 (2009) 599-624. | MR | Zbl

[41] V. Kolehmain, A. Voutilainen and J.P. Kaipio, Estimation of nonstionary region boundaries in EIT-state estimation approach. Inverse Problems 17 (2001) 1937-1956. | MR | Zbl

[42] A. Lechleiter, A regularization technique for the factorization method. Inverse Problems 22 (2006) 1605-1625. | MR | Zbl

[43] A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography : a numerical study. Inverse Problems 22 (2006) 1967-1987. | MR | Zbl

[44] A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography : convergence by local injectivity. Inverse Problems, 24 (2008) 065009. | MR | Zbl

[45] W.R.B. Lionheart, EIT reconstruction algorithms : pitfalls, challenges and recent developments. Physiol. Meas. 25 (2004) 125-142.

[46] D.A. Lorenz, Convergence rates and source conditions for Tikhonov regularization with sparsity constraints. Journal Inverse Ill-Posed Problems 16 (2008) 463-478. | MR | Zbl

[47] M. Lukaschewitsch, P. Maass and M. Pidcock, Tikhonov regularization for electrical impedance tomography on unbounded domains. Inverse Problems 19 (2003) 585-610. | MR | Zbl

[48] N.G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa (3) 17 (1963) 189-206. | Numdam | MR | Zbl

[49] A. Neubauer, When do Sobolev spaces form a Hilbert scale? Proc. Amer. Math. Soc. 103 (1988) 557-562. | MR | Zbl

[50] E. Resmerita, Regularization of ill-posed problems in Banach spaces : convergence rates. Inverse Problems 21 (2005) 1303-1314. | MR | Zbl

[51] L. Rondi, On the regularization of the inverse conductivity problem with discontinuous conductivities. IPI 2 (2008) 397-409. | MR | Zbl

[52] L. Rondi and F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional. ESAIM Control Optim. Calc. Var. 6 (2001) 517-538. | Numdam | MR | Zbl

[53] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography. SIAM J. Appl. Math. 52 (1992) 1023-1040. | MR | Zbl

[54] A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-Posed Problems. John Wiley, New York (1977). | MR | Zbl

[55] G. Uhlmann, Commentary on Calderón's paper (29), on an inverse boundary value problem, in Selected papers of Alberto P. Calderón. Amer. Math. Soc., Providence, RI (2008) 623-636. | MR | Zbl

[56] A. Wexler, B. Fry and M.R. Neuman, Impedance-computed tomography algorithm and system. Appl. Opt. 24 (1985) 3985-3992.

[57] T.J. Yorkey, J.G. Webster and W.J. Tompkins, Comparing reconstruction algorithms for electrical impedance tomography. IEEE Trans. Biomed. Eng. 34 (1987) 843-852.

Cité par Sources :