Probing for electrical inclusions with complex spherical waves

Abstract

Let a physical body in R 2 or R 3 be given. Assume that the electric conductivity distribution inside consists of conductive inclusions in a known smooth background. Further, assume that a subset ⊂ ∂ is available for boundary measurements. It is proved using hyperbolic geometry that certain information about the location of the inclusions can be exactly recovered from static electric measurements on . More precisely: given a ball B with center outside the convex hull of and satisfying (B ∩ ∂ ) ⊂ , boundary measurements on with explicitly given Dirichlet data are enough to determine whether B intersects the inclusion. An approximate detection algorithm is introduced based on the theory. Numerical experiments in dimension two with simulated noisy data suggest that the algorithm finds the inclusion-free domain near and is robust against measurement noise.

Inverse problems: seeing the unseen

“…See [164] for Dirac systems, [40] for Maxwell and [91] for elasticity. Using methods of hyperbolic geometry similar to [101] it is shown in [82] that one can reconstruct inclusions from the local DN map using CGO solutions that decay exponentially inside a ball and grow exponentially outside. These are called complex spherical waves.…”

“…These are called complex spherical waves. A numerical implementation of this method has been done in [82]. The construction of complex spherical waves can also be done using the CGO solutions constructed in [107].…”

Complex geometrical optics solutions for anisotropic equations and applications

“…This inverse problem has been extensively studied both theoretically and numerically. We refer to [8] Then we have γ(x) ≥ δI almost everywhere in Ω. Let v be the solution of…”

“…When γ 0 and γ 1 are isotropic, derivations of (3.4) and (3.5) can be found in [8] (or in [9]). Here we adapt their arguments to treat anisotropic conductivities.…”

“…For example, one can reconstruct the convex hull of the object with Calderón type solutions [11], [12]. With complex spherical waves, one may be able to reconstruct some nonconvex parts of the object [8], [19]. In the two-dimensional case, we are able to get much more information of the object by using Mittag-Leffler functions [14], [15] or the special solutions constructed in [21].…”