Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map

Francini, Elisa

doi:10.1088/0266-5611/16/1/309

Cited by 47 publications

(92 citation statements)

References 10 publications

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“…This result also shows that complex conductivities can be determined uniquely from the DN map. Francini has shown in [57] that this was the case for conductivities with small imaginary part. It also implies unique determination of a potential from the fixed energy scattering amplitude in two dimensions.…”

Section: Bukhgeim's Resultsmentioning

confidence: 89%

Inverse problems: seeing the unseen

2014

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This survey article deals mainly with two inverse problems and the relation between them. The first inverse problem we consider is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This is called electrical impedance tomography and also Calderón's problem since the famous analyst proposed it in the mathematical literature (Calderón in On an inverse boundary value problem. Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro, 1980), Soc Brasil Mat. Rio de Janeiro, pp. [65][66][67][68][69][70][71][72][73] 1980). The second is on travel time tomography. The question is whether one can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as a geometry problem, the boundary rigidity problem. Can we determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points? These two inverse problems concern visibility, that is whether we can determine the internal properties of a medium by making measurements at the boundary. The last topic of this paper considers the opposite issue: invisibility: Can one make objects invisible to different types of waves, including light?Communicated by Ari Laptev.

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Section: Bukhgeim's Resultsmentioning

confidence: 89%

Inverse problems: seeing the unseen

2014

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“…This result also shows that complex conductivities can be determined uniquely from the DN map. Francini has shown in [58] that this was the case for conductivities with small imaginary part. It also implies unique determination of a potential from the fixed energy scattering amplitude in two dimensions.…”

Section: Bukhgeim's Resultsmentioning

confidence: 89%

Inverse Problems: Visibility and Invisibility

Uhlmann¹

2013

Journées Équations Aux Dérivées Partielles

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“…A second type of CGO solution is required for the mathematical reconstruction algorithm. These solutions were introduced in [17], and involve formulating the problem as an elliptic system. Define as a transformation of and a matrix operator by (12) Defining a vector in terms of the solution to (1), one sees that Francini shows in [17] that for sufficiently small, , and , there exists a unique 2 2 matrix for that is a solution to (13) with , and where the asymptotic condition is made precise in [17].…”

Section: Description Of the Algorithmmentioning

confidence: 99%

“…where (17) (18) and the matrix of CGO solutions are related to via (19) The values of the CGO solutions are found by solving the following D-bar equation, in the variable, derived in [17] (20) This completes the set of equations necessary to directly determine from . We clarify these steps in Fig.…”

Section: Description Of the Algorithmmentioning

confidence: 99%

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Direct EIT Reconstructions of Complex Admittivities on a Chest-Shaped Domain in 2-D

Hamilton

Mueller

2013

IEEE Trans. Med. Imaging

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Electrical impedance tomography (EIT) is a medical imaging technique in which current is applied on electrodes on the surface of the body, the resulting voltage is measured, and an inverse problem is solved to recover the conductivity and/or permittivity in the interior. Images are then formed from the reconstructed conductivity and permittivity distributions. In the 2-D geometry, EIT is clinically useful for chest imaging. In this work, an implementation of a D-bar method for complex admittivities on a general 2-D domain is presented. In particular, reconstructions are computed on a chest-shaped domain for several realistic phantoms including a simulated pneumothorax, hyperinflation, and pleural effusion. The method demonstrates robustness in the presence of noise. Reconstructions from trigonometric and pairwise current injection patterns are included.

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Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map

Abstract: We consider the conductivity problem in two dimensions. We show that a complexvalued coefficient γ , whose imaginary part is small, can be recovered from the knowledge of the Dirichlet-to-Neumann map.

Cited by 47 publications

References 10 publications

Inverse problems: seeing the unseen

Inverse problems: seeing the unseen

Inverse Problems: Visibility and Invisibility

Direct EIT Reconstructions of Complex Admittivities on a Chest-Shaped Domain in 2-D

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