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408
Multiscale scientific computation: Review 2001
 Multiscale and Multiresolution Methods
, 2001
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The Calderón problem with partial data in two dimensions
 J. Amer. Math. Soc
"... Abstract. We prove for a two dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an ..."
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Cited by 55 (18 self)
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Abstract. We prove for a two dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can determine uniquely the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results. 1.
Electrical impedance tomography and Calderón problem
 INVERSE PROBLEMS
, 2009
"... 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 Calderón’ ..."
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Cited by 52 (1 self)
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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 Calderón’s problem from Calderón’s pioneer contribution [23]. We concentrate this article 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 Calderón’s problems that have attracted a lot of interest because of connections with cloaking and invisibility.
Uniqueness in the Inverse Conductivity Problem for Conductivites with 3/2 Derivatives in L^p, p > 2n
"... this paper, we contribute nothing to the analysis of G # . The estimates used are from the paper of Sylvester and Uhlmannn [16]. It is possible that some improvement can be made here. We expect that one should be able to prove uniqueness for conductivities which have 3/2 derivatives in L with p & ..."
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Cited by 46 (0 self)
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this paper, we contribute nothing to the analysis of G # . The estimates used are from the paper of Sylvester and Uhlmannn [16]. It is possible that some improvement can be made here. We expect that one should be able to prove uniqueness for conductivities which have 3/2 derivatives in L with p > 2n/3. However, the straightforward generalization of the argument presented below would require that f # #G # f map functions which are compactly supported to functions which are locally in L with p and r satisfying 1/p 1/r = 1/n. Many such estimates fail, see [2] for further discussion
An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem. Inverse Problems
, 2000
"... Abstract. The 2D inverse conductivity problem requires one to determine the unknown electrical conductivity distribution inside a bounded domain ⊂ R2 from knowledge of the DirichlettoNeumann map. The problem has geophysical, industrial, and medical imaging (electrical impedance tomography) appli ..."
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Cited by 44 (12 self)
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Abstract. The 2D inverse conductivity problem requires one to determine the unknown electrical conductivity distribution inside a bounded domain ⊂ R2 from knowledge of the DirichlettoNeumann map. The problem has geophysical, industrial, and medical imaging (electrical impedance tomography) applications. In 1996 A Nachman proved that the DirichlettoNeumann map uniquely determines C2 conductivities. The proof, which is constructive, outlines a direct method for reconstructing the conductivity. In this paper we present an implementation of the algorithm in Nachman’s proof. The paper includes numerical results obtained by applying the general algorithms described to two radially symmetric cases of small and large contrast. (Some figures in this article are in colour only in the electronic version; see www.iop.org) 1.
2007 On uniqueness in the inverse conductivity problem with local data Inverse Probl
 Imaging
"... The inverse condictivity problem with many boundary measurements consists of recovery of conductivity coefficient a (principal part) of an elliptic equation in a domain Ω ⊂ Rn, n = 2, 3 from the Neumann data given for all Dirichlet data (DirichlettoNeumann map). Calderon [5] proposed the ..."
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Cited by 44 (0 self)
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The inverse condictivity problem with many boundary measurements consists of recovery of conductivity coefficient a (principal part) of an elliptic equation in a domain Ω ⊂ Rn, n = 2, 3 from the Neumann data given for all Dirichlet data (DirichlettoNeumann map). Calderon [5] proposed the
Electric Field Imaging
, 1999
"... The physical user interface is an increasingly significant factor limiting the effectiveness of our interactions with and through technology. This thesis introduces Electric Field Imaging, a new physical channel and inference framework for machine perception of human action. Though electric field se ..."
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Cited by 43 (6 self)
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The physical user interface is an increasingly significant factor limiting the effectiveness of our interactions with and through technology. This thesis introduces Electric Field Imaging, a new physical channel and inference framework for machine perception of human action. Though electric field sensing is an important sensory modality for several species of fish, it has not been seriously explored as a channel for machine perception. Technological applications of field sensing, from the Theremin to the capacitive elevator button, have been limited to simple proximity detection tasks. This thesis presents a solution to the inverse problem of inferring geometrical information about the configuration and motion of the human body from electric field measurements. It also presents simple, inexpensive hardware and signal processing techniques for making the field measurements, and several new applications of electric field sensing. The signal
A global stability estimate for the Gel’fandCalderón inverse problem in two dimensions
, 2010
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