Sweep data of electrical impedance tomography

Abstract

This work considers electrical impedance tomography in the special case that the boundary measurements of current and voltage are carried out with two (infinitely) small electrodes. One of the electrodes lies at a fixed position while the other is moved along the object boundary in a sweeping motion, with the corresponding measurement being the (relative) potential difference required for maintaining a unit current between the two electrodes. Assuming that the two-dimensional object of interest has constant background conductivity but is contaminated by compactly supported inhomogeneities, it is shown that such sweep data represent the boundary value of a holomorphic function defined in the exterior of the embedded inclusions. This observation makes it possible to use the sweep data as the input for the convex source support method in order to localize conductivity inhomogeneities. The functionality of the resulting algorithm is demonstrated by numerical experiments both with idealized point electrode data and with simulated complete electrode model measurements.

Point Measurements for a Neumann-to-Dirichlet Map and the Calderón Problem in the Plane

“…where δ z denotes the delta distribution at z on ∂D. This is a generalization of the concept of (standard) sweep data from [11], or the other way around, sweep data are the restriction of bisweep data onto ∂D × {y 0 } for some fixed y 0 ∈ ∂D. What is more, the bisweep data can be approximated by two-electrode measurements in the framework of the realistic complete electrode model [13].…”

“…Notice that σ * is a feasible conductivity, i.e., it is strictly positive definite and equals one in some interior neighborhood of ∂B, due to the basic properties of conformal mappings (cf. [11,Section 3]). By definition,ς σ * =ς σ whereς σ * is the angular sweep data for σ * on ∂B defined by (15) andς σ is given by (16).…”

“…Our main analytic tool is the so-called bisweep data, which are the (relative) potential differences between two points on ∂D when delta distribution current is driven between the very same points (cf. [11]). We demonstrate that the bisweep data uniquely determine the whole Neumann-to-Dirichlet map for any symmetric anisotropic σ ∈ L ∞ (D, R n×n ), σ ≥ cI > 0, conductivity as long as it satisfies the isotropic homogeneity assumption near the object boundary.…”

“…The main reason for choosing to work with the Neumann-to-Dirichlet map instead of the Dirichlet-to-Neumann map is that in practical electrical impedance tomography, which is the imaging modality corresponding to the Calderón problem, the natural boundary condition on nonaccessible parts of the object boundary is the homogeneous Neumann, not the homogeneous Dirichlet condition [8,9,28]. In particular, the bisweep data can be approximated by real-life measurements performed by two small movable electrodes [11,13,16].…”

Construction of Indistinguishable Conductivity Perturbations for the Point Electrode Model in Electrical Impedance Tomography

“…i · ∇u 0 j } 1≤i≤j≤N is a family of linearly independent functions on any nonempty Lipschitz domain Ω D. Proof. In this simple geometry, it is known that (see, e.g., [12])…”