Determining anisotropic real-analytic conductivities by boundary measurements
TL;DR: In this article, the inverse problem of determining the conductivity of a solid body is considered, where the current flux across the surface depends on the conductivities in the interior of the body.
Abstract: If an electrical potential is applied to the surface of a solid body, the current flux across the surface depends on the conductivity in the interior of the body. We want to consider the inverse problem of determining the conductivity by these boundary measurements
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TL;DR: A survey of the work in electrical impedance tomography can be found in this article, where the authors survey some of the most important works in the field. Butt.t.
Abstract: t. This paper surveys some of the work our group has done in electrical impedance tomography.
1,726 citations
TL;DR: In this article, the authors review theoretical and numerical studies of the inverse problem of electrical impedance tomography, which seeks the electrical conductivity and permittivity inside a body, given simultaneous measurements of electrical currents and potentials at the boundary.
Abstract: We review theoretical and numerical studies of the inverse problem of electrical impedance tomography which seeks the electrical conductivity and permittivity inside a body, given simultaneous measurements of electrical currents and potentials at the boundary.
687 citations
Cites background from "Determining anisotropic real-analyt..."
...Since uniqueness does not hold for anisotropic EIT, we rephrase the question as: does the DtN map γ determine γ , up to a diffeomorphism ? The answer is affirmative for real-valued γ = σ , in two dimensions, if σ ∈ C2,α( ), 0 < α < 1 and ∂ is C3,α (see [141]1) and in three dimensions, if σ is analytic [111] (see also [146, 148, 149])....
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TL;DR: In this article, anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity were constructed, which are singular close to a surface inside the body.
Abstract: We construct anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity. These conductivities are singular close to a surface inside the body.
482 citations
TL;DR: In this paper, the authors survey mathematical developments in the inverse method of electrical impedance tomography which consists in determining the electrical properties of a medium by making voltage and current measurements at the boundary of the medium.
Abstract: We survey mathematical developments in the inverse method of electrical impedance tomography which consists in determining the electrical properties of a medium by making voltage and current measurements at the boundary of the medium. In the mathematical literature, this is also known as Calderon's problem from Calderon's pioneer contribution (Calderon 1980 Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) p 65 (Soc. Brasil. Mat.)). We concentrate this review around the topic of complex geometrical optics solutions that have led to many advances in the field. In the last section, we review some counterexamples to Calderon's problems that have attracted a lot of interest because of connections with cloaking and invisibility.
473 citations
TL;DR: Anisotropic conductivities in dimension 3 are constructed that give rise to the same voltage and current measurements at the boundary of a body as a homogeneous isotropic conductivity.
Abstract: We construct anisotropic conductivities in dimension 3 that give rise to the same voltage and current measurements at the boundary of a body as a homogeneous isotropic conductivity. These conductivities are non-zero, but degenerate close to a surface inside the body.
393 citations
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TL;DR: In this paper, the single smooth coefficient of the elliptic operator LY = v yv can be determined from knowledge of its Dirichlet integrals for arbitrary boundary values on a fixed region 2 C R', n? 3.
Abstract: In this paper, we show that the single smooth coefficient of the elliptic operator LY = v yv can be determined from knowledge of its Dirichlet integrals for arbitrary boundary values on a fixed region 2 C R', n ? 3. From a physical point of view, we show that an isotropic conductivity can be determined by steady state measurements at the boundary.
1,608 citations
Book•
31 Dec 1978
TL;DR: CW-complexes generalities on homotopy classes of mappings homology homology of fibre spaces elementary theory the homology suspension Postnikov systems on mappings into group-like spaces homotropic operations stable homology stable homomorphism stable homological homology and homology in fibre spaces compact Lie groups additive relations as discussed by the authors.
Abstract: CW-complexes generalities on homotopy classes of mappings homotopy groups homotopy theory of CW-complexes homotopy with local coefficients homology of fibre spaces elementary theory the homology suspension Postnikov systems on mappings into group-like spaces homotopy operations stable homotopy and homology homology of fibre spaces compact Lie groups additive relations.
1,447 citations
01 Jan 1980
1,131 citations
TL;DR: In this article, it was shown that a three times continuously differentiable conductivity is identifiable by boundary measurements, and that a similar result holds for piecewise real-analytic conductivities.
Abstract: : In a recent paper the authors showed that an unknown real-analytic conductivity gamma may be determined from static boundary measurements. In this document they extend this analysis by demonstrating that a similar result holds for piecewise real-analytic conductivities. In addition, for the special case of a layered structure it is shown that a three times continuously differentiable conductivity is identifiable by boundary measurements. Originator-supplied keywords include: Inversion, Convergence, Algorithms, Estimates, Coefficients.
428 citations
TL;DR: In this article, the authors considered the impedance tomography problem for anisotropic conductivities and showed that if γ1 and γ2 produce the same boundary measurements, then γ 1, γ 2 = γ*γ2 for an appropriate Ψ, then they can construct a new conductivity Ψ*γ which produces the same voltage and current measurements on ∂ Ω.
Abstract: We consider the impedance tomography problem for anisotropic conductivities. Given a bounded region Ω in space, a diffeomorphism Ψ from Ω to itself which restricts to the identity on ∂ Ω, and a conductivity γ on Ω, it is easy to construct a new conductivity Ψ*γ which will produce the same voltage and current measurements on ∂ Ω.
We prove the converse in two dimensions (i.e., if γ1 and γ2 produce the same boundary measurements, then γ1, = Ψ*γ2 for an appropriate Ψ) for conductivities which are near a constant.
254 citations