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 : https://doi.org/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},
zbl = {1259.49056},
mrnumber = {3019471},
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
DA  - 2012///
SP  - 1027
EP  - 1048
VL  - 18
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2011193/
UR  - https://zbmath.org/?q=an%3A1259.49056
UR  - https://www.ams.org/mathscinet-getitem?mr=3019471
UR  - https://doi.org/10.1051/cocv/2011193
DO  - 10.1051/cocv/2011193
LA  - en
ID  - COCV_2012__18_4_1027_0
ER  - 
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 2367859 | Zbl 1221.35443

[2] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane. Ann. of Math. (2) 163 (2006) 265-299. | MR 2195135 | Zbl 1111.35004

[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 2413671 | Zbl 1164.30014

[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 2393115

[6] K. Bredies and D.A. Lorenz, Regularization with non-convex separable constraints. Inverse Problems 25 (2009) 085011. | MR 2529201 | Zbl 1180.65068

[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 215617 | Zbl 0186.23807

[8] M. Burger and S. Osher, Convergence rates of convex variational regularization. Inverse Problems 20 (2004) 1411-1420. | MR 2109126 | Zbl 1068.65085

[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 590275 | Zbl 1182.35230

[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 1669729 | Zbl 0927.35130

[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 2132313 | Zbl 1072.65143

[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 2077704 | Zbl 1077.65055

[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 1918308 | Zbl 1010.35115

[16] D. Dobson, Convergence of a reconstruction method for the inverse conductivity problem. SIAM J. Appl. Math. 52 (1992) 442-458. | MR 1154782 | Zbl 0747.35051

[17] D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theor. 52 (2006) 1289-1306. | MR 2241189 | Zbl 1288.94016

[18] H. Egger and M. Schlottbom, Analysis and regularization of problems in diffuse optical tomography. SIAM J. Math. Anal. 42 (2010) 1934-1948. | MR 2684305 | Zbl 1219.35355

[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 1009037 | Zbl 0695.65037

[20] H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer Academic, Dordrecht (1996). | MR 1408680 | Zbl 0859.65054

[21] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992). | MR 1158660 | Zbl 0804.28001

[22] T. Gallouet and A. Monier, On the regularity of solutions to elliptic equations. Rend. Mat. Appl. (7) 19 (1999) 471-488. | MR 1789483 | Zbl 0961.35036

[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 2876676 | Zbl 1251.78008

[24] M. Grasmair, M. Haltmeier and O. Scherzer, Sparse regularization with lq penalty term. Inverse Problems 24 (2008) 055020. | MR 2438955 | Zbl 1157.65033

[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 990595 | Zbl 0646.35024

[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 2679585 | Zbl 1215.35167

[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 2683677 | Zbl 1207.47065

[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 2329928 | Zbl 1131.65046

[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 2068447 | Zbl 1059.35168

[30] M. Ikehata and S. Siltanen, Electrical impedance tomography and Mittag-Leffler's function. Inverse Problems 20 (2004) 1325-1348. | MR 2087994 | Zbl 1074.35087

[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 2629983 | Zbl 1201.35183

[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 1862188 | Zbl 0986.35130

[34] K. Ito, B. Jin and T. Takeuchi, A regularization parameter for nonsmooth Tikhonov regularization. SIAM J. Sci. Comput. 33 (2011) 1415-1438. | MR 2813246 | Zbl 1235.65054

[35] B. Jin, Y. Zhao and J. Zou, Iterative parameter choice by discrepancy principle. IMA J. Numer. Anal. (2011), in press. | MR 2991843 | Zbl 1261.65052

[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 1242.78016

[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 1800606 | Zbl 1044.78513

[38] B. Kaltenbacher and B. Hofmann, Convergence rates for the iteratively regularized Gauss-Newton method in Banach spaces. Inverse Problems 26 (2010) 035007. | MR 2594377 | Zbl 1204.65060

[39] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems. Oxford University Press, Oxford (2008). | MR 2378253 | Zbl 1222.35001

[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 2557921 | Zbl 1184.35314

[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 1872930 | Zbl 0991.35109

[42] A. Lechleiter, A regularization technique for the factorization method. Inverse Problems 22 (2006) 1605-1625. | MR 2261257 | Zbl 1106.35136

[43] A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography : a numerical study. Inverse Problems 22 (2006) 1967-1987. | MR 2277524 | Zbl 1109.65100

[44] A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography : convergence by local injectivity. Inverse Problems, 24 (2008) 065009. | MR 2456956 | Zbl 1152.35516

[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 2442066 | Zbl 1161.65041

[47] M. Lukaschewitsch, P. Maass and M. Pidcock, Tikhonov regularization for electrical impedance tomography on unbounded domains. Inverse Problems 19 (2003) 585-610. | MR 1984879 | Zbl 1034.65043

[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 159110 | Zbl 0127.31904

[49] A. Neubauer, When do Sobolev spaces form a Hilbert scale? Proc. Amer. Math. Soc. 103 (1988) 557-562. | MR 943084 | Zbl 0665.46029

[50] E. Resmerita, Regularization of ill-posed problems in Banach spaces : convergence rates. Inverse Problems 21 (2005) 1303-1314. | MR 2158110 | Zbl 1082.65055

[51] L. Rondi, On the regularization of the inverse conductivity problem with discontinuous conductivities. IPI 2 (2008) 397-409. | MR 2424823 | Zbl 1180.35572

[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 1849414 | Zbl 0989.35136

[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 1174044 | Zbl 0759.35055

[54] A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-Posed Problems. John Wiley, New York (1977). | MR 455365 | Zbl 0354.65028

[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 2435340 | Zbl 1140.01021

[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 :