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  • SIAM J. APPL. MATH. c 2009 Society for Industrial and Applied MathematicsVol. 69, No. 6, pp. 16621681



    Abstract. We develop a new variant of the factorization method that can be used to detectinclusions in electrical impedance tomography from either absolute current-to-voltage measurementsat a single, nonzero frequency or from frequency-difference measurements. This eliminates the needfor numerically simulated reference measurements at an inclusion-free body and thus greatly improvesthe methods robustness against forward modeling errors, e.g., in the assumed bodys shape.

    Key words. inverse problems, electrical impedance tomography, complex conductivity, frequency-difference measurements, factorization method

    AMS subject classifications. 35R30, 35Q60, 35J25, 35R05

    DOI. 10.1137/08072142X

    1. Introduction. In electrical impedance tomography (EIT), we inject a time-harmonic current of I mA at a fixed angular frequency into an imaging subjectusing a pair of surface electrodes attached to its boundary. Then the induced time-harmonic electrical potential is dictated by the complex conductivity distribution

    of the subject, the applied current, and the shape of the subject, where the real andimaginary part of the complex conductivity, () and ( ), are the real conduc-tivity and the permittivity at the angular frequency , respectively. In EIT, we usemeasured boundary voltages generated by multiple injection currents to reconstructan image of inside the subject. It is well known that these boundary measurementsare very insensitive and highly nonlinear to any local change of conductivity valuesaway from the measuring points. Hence, the reconstructed image quality in terms ofaccuracy would be affected sensitively by unavoidable errors including the modelingerrors and measurement noises.

    Understanding the limited capabilities of static EIT imaging under realistic en-vironments, numerous recent studies in EIT focus on the detection of conductivityanomalies instead of (e.g., cross-sectional) conductivity imaging; cf., e.g., [17, 21,27, 28, 2, 9, 1, 20, 8, 15], the references therein, and the works connected with thefactorization method cited further below.

    Let us briefly explain the anomaly detection problem in EIT. Let the imagingobject occupy a two- or three-dimensional region B with its smooth boundary B,and let anomalies occupy a region inside a background domain B of constant con-ductivity. We furthermore assume that the conductivity is isotropic. To distinguishthe conductivity of the anomaly and the surrounding homogeneous domain B \ ,we denote the conductivity distribution at = 0 by

    (x) = 0 + (x)(x),

    Received by the editors April 16, 2008; accepted for publication (in revised form) January 7,2009; published electronically March 27, 2009. name: Bastian Gebauer, Institut fur Mathematik, Johannes Gutenberg-Universitat, 55099

    Mainz, Germany ([email protected]). The work of this author was supported by the Ger-man Federal Ministry of Education and Research (BMBF) under grant 03HBPAM2, Regularizationtechniques for electrical impedance tomography in medical and geological sciences.

    Computational Science and Engineering, Yonsei University, Seoul, 120-749, Korea ([email protected]). The work of this author was supported by WCU program R31-2008-000-10049-0.



    where is the characteristic function of and is a positive and bounded functionin B. The inverse problem is to identify from several pairs of Neumann-to-Dirichletdata

    (gj ,(gj)) L2(B) L2(B), j = 1, . . . , L,where L2(B) = { L2(B) :

    B dx = 0}. Here, (g) = u|B and u is the

    H1(B)-solution for the Neumann boundary value problem:

    (u) =0 in B,


    |B = g,


    u dx = 0,

    where is the unit outward normal vector to the boundary B.One of the most successful EIT-methods for locating multiple anomalies would

    be the factorization method introduced by Kirsch [24] for inverse scattering problemsand generalized to EIT-problems by Bruhl and Hanke in [5, 4]; see also the recentbook of Kirsch and Grinberg [26], the work of Kirsch [25] on the complex conductivitycase, and [6, 14, 18, 11, 16, 19, 30, 13, 29] for further extensions of the method inthe context of EIT. The factorization method is based on a characterization using therange of the difference between the Neumann-to-Dirichlet (NtD) map in the presenceof anomalies and that in the absence of anomalies: z if and only if z|B is inthe range of the operator |0|1/2, where 0 is the NtD map corresponding to thereference homogeneous conductivity (x) = 0 and z(x) is the solution of

    xz(x) = d xz(x) in B,

    z | = 0, and


    z(x) dx = 0,

    where d is any unit vector and z is the Dirac delta function at z.For the practical application of the factorization method for static EIT systems,

    the requirement of the reference NtD data 0 is a drawback. While, in practice, arough approximation of the NtD map can be obtained from the current-to-voltagedata, the corresponding current-to-voltage data for the reference NtD map 0 in theabsence of anomalies is usually not available.

    Hence, one uses numerically simulated data corresponding to 0 by solving theforward problem (0u) = 0 in B with mimicked Neumann data representing theinjection current in the EIT system. Noting that the simulated Dirichlet data is mainlydepending on the geometry of B and the Neumann data, instead of the conductivity0 (which acts merely as a scaling factor), the requirement of the reference NtDdata 0 makes the factorization method very sensitive to forward modeling errorsincluding the boundary geometry error and electrodes position uncertainty (relatedto the mimicked Neumann data), since its image reconstruction problem is ill-posed.Hence, it is desirable to eliminate the requirement of the reference NtD data 0.

    In this work, we adopt the frequency-difference EIT system [31, 32] to obtaina subsidiary NtD data at a fixed angular frequency taken from the range of1kHz 2 500kHz. (g) is the Dirichlet data of the complex potential u whichsatisfies

    (u) = 0 in B, u|B = g, and


    u dx = 0.

    Our aim is to substitute for 0 in the conventional factorization method anduse an interrelation between and to locate the anomalies . However, due to


    {} = 0, the operator is not self-adjoint and is neither semipositive norseminegative.

    In this work, we show that, for an arbitrary fixed nonzero frequency , both, theimaginary part of 0 and the real part of the normalized difference 0 0 (or actually any other normalized difference of measurements taken at two differentfrequencies), provide a constructive way of locating , where 0 is the backgroundcomplex conductivity at the angular frequency . To our knowledge this is the firstcharacterization result that works without reference measurements. We numericallydemonstrate that the proposed new variant of the factorization method locates suc-cessfully the region with a reasonable accuracy in the presence of boundary geometryerrors and measurement noise. We also describe a heuristic approach to estimate anunknown background conductivity from the measured data and numerically test it ona homogeneous, as well as on a slightly inhomogeneous, background.

    In section 2 we formulate and prove our main results. In section 3 we test ourmethod numerically, compare its sensitivity to body shape errors with the conven-tional factorization method, and describe a heuristic approach to estimate an unknownbackground conductivity. Section 4 contains some concluding remarks.

    2. Characterization of an inclusion without reference data. Let B Rn,n 2, be a smoothly bounded domain describing the investigated body. Let > 0be an arbitrary fixed frequency and denote by the bodys complex conductivityat some fixed nonzero frequency > 0. We assume that () L+ (B; R) and() L(B; R), where () and () denote the real and imaginary part, thesubscript + indicates functions with positive (essential) infima, and throughout thiswork all function spaces consist of complex valued functions if not stated otherwise.

    A time-harmonic current with (complex) amplitude g L2(B) and frequency that is applied to the bodys surface gives rise to an electric potential u H1(B)that satisfies

    (2.1) (u) = 0 in B and u|B = g,where L2(B) is the subspace of L

    2(B)-functions with vanishing integral mean, isthe outer normal on B.

    It is a well-known consequence of the LaxMilgram theorem (cf., e.g., [7, Chap-ter VI, 3, Theorem 7]) that there exists a solution of (2.1) and that this solution isuniquely determined up to addition of a constant function. We denote the quotientspace of H1(B) modulo constant functions by H1 (B). The trace operator v v|Bcanonically extends to H1 (B) L2(B)/C, where we identify the latter space withL2(B) by appropriately fixing the ground level.

    The inverse problem of frequency-dependent EIT is the problem of determining(properties of) from measuring one or several pairs of Neumann and Dirichletboundary values (u|B, u|B). Mathematically, the knowledge of all such pairsis equivalent to knowing the Neumann-to-Dirichlet operator

    : L2(B) L2(B), g u|B,where u solves (2.1). It is easily checked that is linear and compact.

    2.1. The main results. In this work, we assume that the conductivity of thebody is constant outside one or several inclusions, i.e.,

    (x) = 0 + (x)(x),


    where is some open (pos