Identification of an inclusion in multifrequency electric impedance tomography

The multifrequency electrical impedance tomography is considered in order to image a conductivity inclusion inside a homogeneous background medium by injecting one current. An original spectral decomposition of the solution of the forward conductivity problem is used to retrieve the Cauchy data corresponding to the extreme case of perfect conductor. Using results based on the unique continuation we then prove the uniqueness of multifrequency electrical impedance tomography and obtain rigorous stability estimates. Our results in this paper are quite surprising in inverse conductivity problem since in general infinitely many input currents are needed in order to obtain the uniqueness in the determination of the conductivity. 1. The Mathematical Model and main results In this section we introduce the mathematical model of the multifrequency electrical impedance tomography (mfEIT). Let Ω be the open bounded smooth domain in R, d = 2, 3, occupied by the sample under investigation and denote by ∂Ω its boundary. The mfEIT forward problem is to determine the potential u(·, ω) ∈ H(Ω) := {v ∈ L(Ω) : ∇v ∈ L(Ω)}, solution to  −∇ · (σ(x, ω)∇u(x, ω)) = 0 in Ω, σ(x, ω)∂νΩu(x, ω)(x) = f(x) on ∂Ω, ∫ ∂Ω u(x, ω)ds = 0, (1) where ω denotes the frequency, νΩ(x) is the outward normal vector to ∂Ω, σ(x, ω) is the conductivity distribution, and f ∈ H 1 2 (∂Ω) := {g ∈ H 1 2 (∂Ω) : ∫ ∂Ω g ds = 0} is the input current. In this paper we are interested in the case where the frequency dependent conductivity distribution takes the form σ(x, ω) = k0 + (k(ω)− k0)χD(x) (2) with χD(x) being the characteristic function of a smooth inclusion D in Ω (D ⊂ Ω), k(ω) : R+ → C \ R−, being a continuous complex-valued function, and k0 being a fixed positive constant (the conductivity of the background medium). The mfEIT inverse problem is to recover the shape and the position of the inclusion D from measurements of the boundary voltages u(x, ω) on ∂Ω for ω ∈ (ω, ω), 0 ≤ ω < ω. It has many important applications in biomedical imaging. Experimental research has found that the conductivity of many biological tissues varies strongly with respect to the frequency within certain frequency ranges [GPG]. In [AGGJS], using homogenization techniques, the authors analytically exhibit the fundamental mechanisms underlying the fact that effective biological tissue electrical properties and their frequency dependence reflect the tissue composition and physiology. There have been also several numerical studies on frequency-difference imaging. It was numerically shown that the approach can accommodate geometrical errors, including imperfectly known boundary [AAJS, JS, MSHA]. 1991 Mathematics Subject Classification. Primary: 35R30.