Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-20T05:14:01.808Z Has data issue: false hasContentIssue false
  • English
  • Français

An analysis of electrical impedance tomography with applications to Tikhonov regularization

Published online by Cambridge University Press:  16 January 2012

Bangti Jin  and
Peter Maass
Bangti Jin
Affiliation:
Department of Mathematics and Institute for Applied Mathematics and Computational Sciences, Texas A&M University, College Station, 77843-3368 TX, USA. btjin@math.tamu.edu
Peter Maass
Affiliation:
Center for Industrial Mathematics, University of Bremen, 28334 Bremen, Germany; pmaass@math.uni-bremen.de
Get access

Abstract

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.

Keywords

Electrical impedance tomographyTikhonov regularizationconvergence rate
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 4 , October 2012 , pp. 1027 - 1048
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alessandrini, G., Open issues of stability for the inverse conductivity problem. Journal Inverse Ill-Posed Problems 15 (2007) 451–460. Google Scholar
Astala, K. and Päivärinta, L., Calderón’s inverse conductivity problem in the plane. Ann. of Math. (2) 163 (2006) 265–299. Google Scholar
Astala, K., Faraco, D., and Székelyhidi, L. Jr., Convex integration and the L p theory of elliptic equations. Ann. Scuola Norm. Super. Pisa Cl. Sci. (5) 7 (2008) 1–50. Google Scholar
Bayford, R.H., Bioimpedance tomography (electrical impedance tomography). Ann. Rev. Biomed. Eng. 8 (2006) 63–91. Google Scholar
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.
Bredies, K. and Lorenz, D.A., Regularization with non-convex separable constraints. Inverse Problems 25 (2009) 085011. Google Scholar
Bregman, L.M., 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. Google Scholar
Burger, M. and Osher, S., Convergence rates of convex variational regularization. Inverse Problems 20 (2004) 1411–1420. Google Scholar
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.
Cheney, M., Isaacson, D., Newell, J.C., Simske, S. and Goble, J., NOSER : An algorithm for solving the inverse conductivity problem. Int. J. Imag. Syst. Tech. 2 (1990) 66–75. Google Scholar
Cheney, M., Isaacson, D. and Newell, J.C., Electrical impedance tomography. SIAM Rev. 41 (1999) 85–101. Google Scholar
Cheng, K.-S., Isaacson, D., Newell, J.C. and Gisser, D.G., Electrode models for electric current computed tomography. IEEE Trans. Biomed. Eng. 36 (1989) 918–924. Google ScholarPubMed
Chung, E.T., Chan, T.F. and Tai, X.-C., Electrical impedance tomography using level set representation and total variational regularization. J. Comput. Phys. 205 (2005) 357–372. Google Scholar
Daubechies, I., Defrise, M. and De Mol, C., An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57 (2004) 1413–1457. Google Scholar
Dierkes, T., Dorn, O., Natterer, F., Palamodov, V. and Sielschott, H., Fréchet derivatives for some bilinear inverse problems. SIAM J. Appl. Math. 62 (2002) 2092–2113. Google Scholar
Dobson, D., Convergence of a reconstruction method for the inverse conductivity problem. SIAM J. Appl. Math. 52 (1992) 442–458. Google Scholar
Donoho, D.L., Compressed sensing. IEEE Trans. Inf. Theor. 52 (2006) 1289–1306. Google Scholar
Egger, H. and Schlottbom, M., Analysis and regularization of problems in diffuse optical tomography. SIAM J. Math. Anal. 42 (2010) 1934–1948. Google Scholar
Engl, H.W., Kunisch, K. and Neubauer, A., Convergence rates for Tikhonov regularisation of nonlinear ill-posed problems. Inverse Problems 5 (1989) 523–540. Google Scholar
H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer Academic, Dordrecht (1996).
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992).
Gallouet, T. and Monier, A., On the regularity of solutions to elliptic equations. Rend. Mat. Appl. (7) 19 (1999) 471–488. Google Scholar
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.
Grasmair, M., Haltmeier, M. and Scherzer, O., Sparse regularization with l q penalty term. Inverse Problems 24 (2008) 055020. Google Scholar
Gröger, K., A W 1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283 (1989) 679–687. Google Scholar
Harrach, B. and Seo, J.K., Exact shape-reconstruction by one-step linearization in electrical impedance tomography. SIAM J. Math. Anal. 42 (2010) 1505–1518. Google Scholar
Hofmann, B. and Yamamoto, M., On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems. Appl. Anal. 89 (2010) 1705–1727. Google Scholar
Hofmann, B., Kaltenbacher, B., Poeschl, C. and Scherzer, O., A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Problems 23 (2007) 987–1010. Google Scholar
Hyvönen, N., Complete electrode model of electrical impedance tomography : approximation properties and characterization of inclusions. SIAM J. Appl. Math. 64 (2004) 902–931. Google Scholar
Ikehata, M. and Siltanen, S., Electrical impedance tomography and Mittag-Leffler’s function. Inverse Problems 20 (2004) 1325–1348. Google Scholar
Imanuvilov, O.Y., Uhlmann, G. and Yamamoto, M., The Calderón problem with partial data in two dimensions. J. Amer. Math. Soc. 23 (2010) 655–691. Google Scholar
Isaacson, D., Mueller, J.L., Newell, J.C. and Siltanen, S., Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography. IEEE Trans. Med. Imag. 23 (2004) 821–828. Google ScholarPubMed
Ito, K., Kunisch, K. and Li, Z., Level-set function approach to an inverse interface problem. Inverse Problems 17 (2001) 1225–1242. Google Scholar
Ito, K., Jin, B. and Takeuchi, T., A regularization parameter for nonsmooth Tikhonov regularization. SIAM J. Sci. Comput. 33 (2011) 1415–1438. Google Scholar
B. Jin, Y. Zhao and J. Zou, Iterative parameter choice by discrepancy principle. IMA J. Numer. Anal. (2011), in press.
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.
Kaipio, J.P., Kolehmainen, V., Somersalo, E. and Vauhkonen, M., Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography. Inverse Problems 16 (2000) 1487–1522. Google Scholar
Kaltenbacher, B. and Hofmann, B., Convergence rates for the iteratively regularized Gauss-Newton method in Banach spaces. Inverse Problems 26 (2010) 035007. Google Scholar
A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems. Oxford University Press, Oxford (2008).
Knudsen, K., Lassas, M., Mueller, J.L. and Siltanen, S., Regularized D-bar method for the inverse conductivity problem. IPI 3 (2009) 599–624. Google Scholar
Kolehmain, V., Voutilainen, A. and Kaipio, J.P., Estimation of nonstionary region boundaries in EIT-state estimation approach. Inverse Problems 17 (2001) 1937–1956. Google Scholar
Lechleiter, A., A regularization technique for the factorization method. Inverse Problems 22 (2006) 1605–1625. Google Scholar
Lechleiter, A. and Rieder, A., Newton regularizations for impedance tomography : a numerical study. Inverse Problems 22 (2006) 1967–1987. Google Scholar
Lechleiter, A. and Rieder, A., Newton regularizations for impedance tomography : convergence by local injectivity. Inverse Problems, 24 (2008) 065009. Google Scholar
Lionheart, W.R.B., EIT reconstruction algorithms : pitfalls, challenges and recent developments. Physiol. Meas. 25 (2004) 125–142. Google ScholarPubMed
Lorenz, D.A., Convergence rates and source conditions for Tikhonov regularization with sparsity constraints. Journal Inverse Ill-Posed Problems 16 (2008) 463–478. Google Scholar
Lukaschewitsch, M., Maass, P. and Pidcock, M., Tikhonov regularization for electrical impedance tomography on unbounded domains. Inverse Problems 19 (2003) 585–610. Google Scholar
Meyers, N.G., An L p-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa (3) 17 (1963) 189–206. Google Scholar
Neubauer, A., When do Sobolev spaces form a Hilbert scale? Proc. Amer. Math. Soc. 103 (1988) 557–562. Google Scholar
Resmerita, E., Regularization of ill-posed problems in Banach spaces : convergence rates. Inverse Problems 21 (2005) 1303–1314. Google Scholar
Rondi, L., On the regularization of the inverse conductivity problem with discontinuous conductivities. IPI 2 (2008) 397–409. Google Scholar
Rondi, L. and Santosa, F., Enhanced electrical impedance tomography via the Mumford-Shah functional. ESAIM Control Optim. Calc. Var. 6 (2001) 517–538. Google Scholar
Somersalo, E., Cheney, M. and Isaacson, D., Existence and uniqueness for electrode models for electric current computed tomography. SIAM J. Appl. Math. 52 (1992) 1023–1040. Google Scholar
A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-Posed Problems. John Wiley, New York (1977).
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.
Wexler, A., Fry, B. and Neuman, M.R., Impedance-computed tomography algorithm and system. Appl. Opt. 24 (1985) 3985–3992. Google ScholarPubMed
Yorkey, T.J., Webster, J.G. and Tompkins, W.J., Comparing reconstruction algorithms for electrical impedance tomography. IEEE Trans. Biomed. Eng. 34 (1987) 843–852. Google ScholarPubMed