Abstract. The aim in electric impedance tomography is to recover the conductivity inside a physical body from boundary measurements of current and voltage. In many situations of practical importance, the investigated object has known background conductivity but is contaminated by inhomogeneities. In this work, we try to extract all possible information about the support of such inclusions inside a twodimensional object from only one pair of measurements of impedance tomography. Our noniterative and computationally cheap method is based on the concept of the convex source support, which stems from earlier works of Kusiak, Sylvester, and the authors. The functionality of our algorithm is demonstrated by various numerical experiments.
This paper reinvestigates a recently introduced notion of backscattering for the inverse obstacle problem in impedance tomography. Under mild restrictions on the topological properties of the obstacles, it is shown that the corresponding backscatter data are the boundary values of a function that is holomorphic in the exterior of the obstacle(s), which allows to reformulate the obstacle problem as an inverse source problem for the Laplace equation. For general obstacles, the convex backscattering support is then defined to be the smallest convex set that carries an admissible source, i.e., a source that yields the given (backscatter) data as the trace of the associated potential. The convex backscattering support can be computed numerically; numerical reconstructions are included to illustrate the viability of the method.
We investigate the inverse source problem of electrostatics in a bounded and convex domain with compactly supported source. We try to extract all information about the unknown source support from the given Cauchy data of the associated potential, adopting by this previous work of Kusiak and Sylvester to the case of electrostatics. We introduce, and for the unit disk we also compute numerically, what we call the discoidal source support, i.e., the smallest set made up by the intersection of disks within the domain, which carries a source compatible with the given data. (2000): 35R30, 65N21. AMS subject classification
Abstract. We fix an incorrect statement from our paper [M. Hanke, N. Hyvönen, and S. Reusswig, SIAM J. Math. Anal., 41 (2009), pp. 1948-1966 claiming that two different perfectly conducting inclusions necessarily have different backscatter in impedance tomography. We also present a counterexample to show that this kind of nonuniqueness does indeed occur.Key words. electric impedance tomography, backscatter, uniqueness theorem AMS subject classifications. 35R30, 65N21 DOI. 10.1137/110821780In the paper [1] we claimed (in the concluding remarks) that the same arguments that we have used in the rest of the paper for insulating cavities can also be applied to establish that two (simply connected) perfectly conducting inclusions with the same backscatter data of impedance tomography are necessarily the same.Unfortunately, while our uniqueness result for insulating cavities is absolutely correct, the corresponding statement on perfect conductors fails to be true, and we will provide a counterexample below. The correct statement is as follows. (Throughout, we say that a perfect conductor is supported in the closure of a domain Ω if the homogeneous Neumann condition on Γ = ∂Ω in the forward problem associated with the backscatter data, i.e., [1, (2.16)], is replaced by a homogeneous Dirichlet condition, and the normalizing condition on T = ∂D is deleted.) Theorem 1. Assume that Ω is a simply connected domain with C 2 -boundary, and that a perfect conductor is supported in Ω ⊂ D, where D is the unit disk. Let Φ be a conformal map that takes D \ Ω onto a concentric annulus {x ∈ D : R < |x| < 1}, and define Proof. Note that β R is a continuous and monotonic function of R ∈ [0, 1), with β 0 = 0 and lim R→1 β R = −∞; in fact, β R is the (constant) backscatter data corresponding to a discoidal perfect conductor of radius R centered at the origin (cf. [1, Example 2.1]). Now, for a fixed Ω as in the statement of the theorem, R, and thus *
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