Identifying conductivity in electrical impedance tomography
with total variation regularization
Michael Hinze††Email: firstname.lastname@example.org, Barbara.Kaltenbacher@aau.at, email@example.com Barbara Kaltenbacher††M. Hinze gratefully acknowledges support of the Lothar Collatz Center for Computing in Science at the University of Hamburg††B. Kaltenbacher gratefully acknowledges support of the Austrian Wissenschaftsfonds through grant FWF I2271 entitled ”Solving inverse problems without forward operators”††T.N.T. Quyen gratefully acknowledges support of the Alexander von Humboldt-Foundation Tran Nhan Tam Quyen††Author to whom any correspondence should be addressed
University of Hamburg, Bundesstraße 55, D-20146 Hamburg, Germany
Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, A-9020 Klagenfurt, Austria
Abstract: In this paper we investigate the problem of identifying the conductivity in electrical impedance tomography from one boundary measurement. A variational method with total variation regularization is here proposed to tackle this problem. We discretize the PDE as well as the conductivity with piecewise linear, continuous finite elements. We prove the stability and convergence of this technique. For the numerical solution we propose a projected Armijo algorithm. Finally, a numerical experiment is presented to illustrate our theoretical results.
Key words and phrases: Conductivity identification, electrical impedance tomography, total variation regularization, finite element method, Neumann problem, Dirichlet problem, ill-posed problems.
AMS Subject Classifications: 65N21; 65N12; 35J25; 35R30
Let be an open bounded connected domain in with polygonal boundary and be given. We consider the following elliptic boundary value problem
where is the unit outward normal on .
are given, then there may be no satisfying this system. Here and are some given positive constants.
In this paper we assume that the system is consistent and our aim is to identify the conductivity and the electric potential in the system (1.1)–(1.3) from current and voltage i.e., Neumann and Dirichlet measurements at the boundary of the exact satisfying
Note that using the topology for the data is natural from the point of view of solution theory for elliptic PDEs but unrealistic with regard to practical measurements. We will comment in this issue in Remark 2.2 below.
For the purpose of conductivity identification — a problem which is very well known in literature and practice as electrical impedance tomography EIT, see below for some references — we simultaneously consider the Neumann problem
and the Dirichlet problem
and respectively denote by , the unique weak solutions of the problems (1.5), (1.6), which depend nonlinearly on , where is normalized with vanishing mean on the boundary. We adopt the variational approach of Kohn and Vogelius in [30, 31, 32] to the identification problem. In fact, for estimating the conductivity from the observation of the exact data , we use the functional
For simplicity of exposition we restrict ourselves to the case of just one Neumann-Dirichlet pair, while the approach described here can be easily extended to multiple measurements , see also Example 5.3 below. It is well-known that such a finite number of boundary data in general only allows to identify conductivities taking finitely many different values in the domain , see, e.g., .
Indeed, we are interested in estimating such piecewise constant conductivities and therefore use total variation regularization, i.e., we consider the minimization problem
where is the admissible set of the sought conductivities, is the space of all functions with bounded total variation (see §2.1 for its definition) and is the regularization parameter, and consider a minimizer of (1.7) as reconstruction.
For each let and be corresponding approximations of and in the finite dimensional space of piecewise linear, continuous finite elements and denote a minimizer of the discrete regularized problem corresponding to (1.7), i.e. of the following minimization problem
with and being a positive functional of the mesh size satisfying .
In Section 4 we will show the stability of approximations for fixed positive . Furthermore as and with an appropriate a priori regularization parameter choice , there exists a subsequence of converging in the -norm to a total variation-minimizing solution defined by
In particular, if is uniquely defined, then this convergence holds for the whole sequence . The corresponding state sequences and converge in the -norm to solving the system (1.1)-(1.3). Finally, for the numerical solution of the discrete regularized problem (1.8), in Section 5 we employ a projected Armijo algorithm. Numerical results show the efficiency of the proposed method and illustrate our theoretical findings.
We conclude this introduction with a selection of references from the vast literature on EIT, which has evolved to a highly relevant imaging and diagnostics tool in industrial and medical applications and has attracted great attention of many scientists in the last few decades.
To this end, for any fixed we define the Neumann-to-Dirichlet map , by
where is the Dirichlet trace operator. Calderón in 1980 posed the question whether an unknown conductivity distribution inside a domain can be determined from an infinite number of boundary observations, i.e. from the Neumann-to-Dirichlet map :
Calderón did not answer his question (1.9); however, in  he proved that the problem linearized at constant conductivities has a unique solution. In dimensions three and higher Sylvester and Uhlmann  proved the unique identifiability of a -smooth conductivity. Päivärinta el al.  and Brown and Torres  established uniqueness in the inverse conductivity problem for -smooth conductivities with and , respectively. In the two dimensional setting, Nachman  and Brown and Uhlmann  proved uniqueness results for conductivities which are in with and with , respectively. Finally, in 2006 the question (1.9) has been answered to be positive by Astala and Päivärinta  in dimension two. For surveys on the subject, we refer the reader to [10, 17, 20, 33, 43] and the references therein.
Although there exists a large number of papers on the numerical solution of the inverse problems of EIT, among these also papers considering the Kohn-Vogelius functional (see, e.g., [28, 29]) and total variation regularization (see, e.g., [21, 36]), we have not yet found investigations on the discretization error in a combination of both functionals for the fully nonlinear setting, a fact which motivated the research presented in this paper.
Throughout the paper we use the standard notion of Sobolev spaces , , , etc from, for example, . If not stated otherwise we write instead of .
2 Problem setting and preliminaries
Let us denote by
the continuous Dirichlet trace operator while
is the continuous right inverse operator of , i.e. for all . With (with a slight abuse of notation) in (1.1) being given, let us denote
where the expression denotes the value of the functional at . We also denote
where the notation stands for the value of the functional at . Similarly, we denote
while is the closed subspace of consisting of all functions with zero mean on the boundary, i.e.
Let us denote by the positive constant appearing in the Poincaré-Friedrichs inequality (see, for example, )
Then for all defined by (1.4), the coercivity condition
holds for all . Furthermore, since , the inequality (2.2) remains valid for all .
Finally, for the sake of completeness we briefly introduce the space of functions with bounded total variation; for more details one may consult [4, 24]. A scalar function is said to be of bounded total variation if
Here denotes the -norm on defined by and the space of continuously differentiable functions with compact support in . The space of all functions in with bounded total variation is denoted by
which is a Banach space with the norm
Furthermore, if is an open bounded set with Lipschitz boundary, then .
2.2 Neumann operator, Dirichlet operator and Neumann-to-Dirichlet map
2.2.1 Neumann operator
We consider the following Neumann problem
By the coercivity condition (2.2) and the Riesz representation theorem, we conclude that for each and there exists a unique weak solution of the problem (2.3) in the sense that and satisfies the identity
for all . By the imposed compatibility condition , i.e.
and the fact that , equation (2.4) is satisfied for all . Furthermore, this solution satisfies the following estimate
Then for any fixed we can define the Neumann operator
which maps each to the unique weak solution of the problem (2.3).
We note that the restriction instead of preserves the compatibility condition (2.5) for the pure Neumann problem. In case this condition fails, the strong form of the problem (2.3) has no solution. This is the reason why we require . However, its weak form, i.e. the variational equation (2.4), attains a unique solution independently of the compatibility condition. By working with the weak form only, all results in the present paper remain valid for .
2.2.2 Dirichlet operator
We now consider the following Dirichlet problem
for all . We can rewrite
where and is the unique solution to the following variational problem
for all . Since
we thus obtain the priori estimate
The Dirichlet operator is for any fixed defined as
which maps each to the unique weak solution of the problem (2.7).
2.2.3 Neumann-to-Dirichlet map
For any fixed we can define the Neumann-to-Dirichlet map
for all , in view of (2.8) we conclude that
2.3 Identification problem
2.4 Total variation regularization
Assume that is the measured data of the exact boundary values with
for some measurement error . Our problem is now to reconstruct the conductivity from this perturbed data . For this purpose we consider the cost functional
where and is the unique weak solutions of the problems (2.3) and (2.7), respectively, with in (2.3) and in (2.7) being replaced by and . Furthermore, to estimate the possibly discontinuous conductivity, we here use the total variation regularization (cf., e.g., [14, 21, 22]), i.e., we consider the minimization problem
is the admissible set of the sought conductivities.
The noise model (2.11) is to some extent an idealized one, since in practice, measurement precision might be different for the current and the voltage , and, more importantly, it will be first of all be given with respect to some norm (e.g., corresponding to normally and to uniformly distributed noise) rather than in . While the Neumann data part is unproblematic, by continuity of the embedding of in for , we can obtain an version of the originally Dirichlet data e.g. by Tikhonov regularization (cf.  and the references therein) as follows. For simplicity, we restrict ourselves to the Hilbert space case and assume that we have measurements such that
Tikhonov regularization applied to the embedding operator amounts to finding a minimizer of
where we use
as a norm on . The first order optimality conditions for this quadratic minimization problem yield
which is equivalent to
for , i.e., the weak form of the Robin problem
Thus, according to well-known results from regularization theory (cf., e.g. ), the smoothed version (where weakly solves (2.13)) of converges to as tends to zero, provided the regularization parameter is chosen appropriately. The latter can, e.g., be done by the discrepancy principle, where is chosen such that
We also wish to mention the complete electrode model cf., e.g., , which fully takes into account the fact that current and voltage are typically not measured pointwise on the whole boundary, but via a set of finitely many electrodes with finite geometric extensions as well as contact impedances.
2.5 Auxiliary results
Now we summarize some useful properties of the Neumann and Dirichlet operators. The proof of the following result is based on standard arguments and therefore omitted.
Let be fixed.
(i) The Neumann operator is continuously Fréchet differentiable on the set . For each the action of the Fréchet derivative in direction denoted by is the unique weak solution in to the Neumann problem
in the sense that the identity
holds for all . Furthermore, the following estimate is fulfilled
(ii) The Dirichlet operator is continuously Fréchet differentiable on the set . For each the action of the Fréchet derivative in direction denoted by is the unique weak solution in to the Dirichlet problem
in the sense that it satisfies the equation
for all . Furthermore, the following estimate is fulfilled
If the sequence converges to in the -norm, then and for any fixed the sequence converges to in the -norm. Furthermore, there holds
where the functional is defined in (2.12).
Since converges to in the -norm, up to a subsequence we assume that it converges to a.e. in , which implies that . For all we infer from (2.4) that
and so that
Taking , by (2.2), we get
and so that
Hence, by the Lebesgue dominated convergence theorem, we deduce from the last inequality that
Similarly to (2.16), we also get
for all . Since , taking in the last equation, we also obtain the limit
Next, we rewrite the functional as follows
as tends to . We now consider the difference
and note that
as goes to , by the Lebesgue dominated convergence theorem. Furthermore, then applying the Cauchy-Schwarz inequality, we also get that
which finishes the proof. ∎
Lemma 2.5 ().
(i) Let be a bounded sequence in the -norm. Then a subsequence which is denoted by the same symbol and an element exist such that converges to in the -norm.
(ii) Let be a sequence in converging to in the -norm. Then and
We mention that equality need not be achieved in (2.22