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Electrical impedance tomography. (English) Zbl 1031.35147

Updated (up to 2002, 153 references) theoretical and numerical studies of the inverse problem for the continuum model of EIT which seeks the admittivity $\left(\gamma \left(x\right)=\sigma \left(x\right)+i\omega \epsilon \left(x\right)\right)$ inside a body ${\Omega }$ from the knowledge of the Dirichlet to Neumann (DtN) (or the Neumann to Dirichlet (NtD)) map at the boundary $\partial {\Omega }$, are reviewed. A similar review has been performed in 1999 by M. Cheney, D. Isaacson and J. C. Newell [SIAM Rev. 41, 85-101 (1999; Zbl 0927.35130)]. It is very useful because it discusses the performances and limitations of some of the previously proposed methods for solving the inverse problem.

The DtN and the NtD maps are characterized through the Dirichlet and Thompson variational principles, respectively. For Lipschitz bounded domains ${\Omega }$ in ${ℝ}^{d}$, the DtN determines uniquely a positive isotropic conductivity in ${W}^{2,p}\left({\Omega }\right)$, for $p>1$, if $d=2$, see A. I. Nachman [Ann. Math. (2) 143, 71-96 (1996; Zbl 0857.35135)], and a positive isotropic Lipschitz conductivity, if $d\ge 3$, see L. Päivärinta, A. Panchenko and G. Uhlmann [Rev. Mat. Iberoam. 19, 57-72 (2003; Zbl 1055.35144)]. For a positive anisotropi conductivity $\sigma$ the DtN map determines uniquely $\sigma$ up to a diffeomorphism, if $\sigma \in {C}^{2,\alpha }\left(\overline{{\Omega }}\right)$, $\alpha \in \left(0,1\right)$ and $\partial {\Omega }\in {C}^{3,\alpha }$, see J. Sylvester [Commun. Pure Appl. Math. 43, 201-232 (1990; Zbl 0709.35102)] if $\lambda =2$, and if $\sigma$ is analytic, if $d=3$, see J. M. Lee and G. Uhlmann [Commun. Pure Appl. Math. 42, 1087-1112 (1989; Zbl 0702.35036)]. An interesting analogy between EIT and electrical networks and magnetotellurics is made, giving new areas for research in electrical engineering geophysics. The stability of the inverse problem relies on the logarithmic estimates for $\sigma \in {H}^{2+s}\left({\Omega }\right)$, $s>d/2$ given by G. Alessandrini [Appl. Anal. 27, 153-172 (1988; Zbl 0616.35082)].

The review continues on how to stabilize the inverse problem by some regularization approach which ensures convergence of reconstruction algorithms, by restricting the admittivity $\gamma$ to a compact subset of ${L}^{\infty }\left({\Omega }\right)$. It is noted that basically all the known regularization methods make use of some ‘a priori’ information about the unknown $\sigma$ or $\gamma$ and, as a result, they may produce artifacts in the images. It also stresses the need for criteria for comparing regularization methods. Various imaging methods are described. First, the linearized EIT problem $\sigma =1+\delta \sigma$ is reviewed and it is concluded that there is no known exact (or fully satisfactory) reconstruction of $\delta \sigma$ inside ${\Omega }$. For the nonliner EIT problem the layer stripping algorithm is not stable, but for the inverse conductivity problem there are other methods such as the signal processing method, see M. Brühl and M. Hanke [Inverse Probl. 16, 1029-1042 (2000; Zbl 0955.35076)] and the level set method, see F. Santosa [ESAIM Control Optim. Calc. Var. 1, 17-33 (1996; Zbl 0870.49016)]. Iterative algorithms are also reviewed, unfortunately the exposition is rather restrictive since $\gamma$ is assumed known at the boundary.

The output least-squares method with regularization is described and two important questions that affect the quality of the final image, namely: (1) How to discretize the unknown $\gamma$?; (2) What current flux excitations to apply?, are addressed. For the first question one can use multigrid methods, or optimal finite-difference grids, but one may as well employ the finite element method. For the second question, if one is interested in distinguishing $\sigma$ from a given ${\sigma }^{0}$, then one should apply the current flux given by the leading eigenvectors (or right singular vectors) of the difference between the inverse DtN operator for $\sigma$ and the inverse DtN operator for ${\sigma }^{0}$.

Alternatively, one may use variational algorithms. It is highlighted that there exists no successful imaging algorithm which uses Kohn and Vogelius relaxed variational formulation of EIT. More powerful seem to be variational feasibility constraints. Some interesting remarks are made on the regularity of the forward map, see D. C. Dobson [SIAM J. Appl. Math. 52, 442-458 (1992; Zbl 0747.35051)], E. Bonnetier and M. Vogelius [SIAM J. Math. Anal. 93, 651-677 (2000; Zbl 0947.35044)].

Finally, several open questions are addressed, such as: (i) the injectivity of the DtN at the boundary for the complete electrode model; (ii) better parametrization of the unknown $\sigma$ or $\gamma$; (iii) anisotropy.

Overall, this work is an excellent topical review paper on EIT for the continuum model.

An Addendum is given in ibid. 19, 997-998 (2003).

Reviewer: D.Lesnic (Leeds)
##### MSC:
 35R30 Inverse problems for PDE 35-02 Research monographs (partial differential equations) 35Q60 PDEs in connection with optics and electromagnetic theory 92C55 Biomedical imaging and signal processing, tomography 78A55 Technical applications of optics and electromagnetic theory