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Electrical Impedance Tomography
 SIAM REVIEW
, 1999
"... This paper surveys some of the work our group has done in electrical impedance tomography. ..."
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Cited by 83 (1 self)
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This paper surveys some of the work our group has done in electrical impedance tomography.
The Calderón problem with partial data
 Ann. of Math. (to
"... In this paper we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension n≥3, the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of [1] but ..."
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Cited by 48 (16 self)
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In this paper we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension n≥3, the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of [1] but use a richer set of solutions to the Dirichlet problem.
High contrast impedance tomography
 INVERSE PROBLEMS
, 1996
"... We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The ..."
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Cited by 44 (6 self)
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We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The smoothly varying part of the conductivity is recovered by a linearization process as is usual. We present the results of several numerical experiments that illustrate
Multiscale scientific computation: Review 2001
 Multiscale and Multiresolution Methods
, 2001
"... ..."
Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction.
, 1998
"... We derive an asymptotic formula for the electrostatic voltage potential in the presence of a finite number of diametrically small inhomogeneities with conductivity different from the background conductivity. We use this formula to establish continuous dependence estimates and to design an effective ..."
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Cited by 23 (5 self)
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We derive an asymptotic formula for the electrostatic voltage potential in the presence of a finite number of diametrically small inhomogeneities with conductivity different from the background conductivity. We use this formula to establish continuous dependence estimates and to design an effective computational identification procedure. 1. Introduction 2. The electrostatic problem 3. An energy estimate 4. Some additional preliminary estimates 5. An asymptotic formula for the voltage potential 6. Properties of the polarization tensor 7. The continuous dependence of the inhomogeneities 8. Computational results. 9. References 1 Introduction The nondestructive inspection technique known as electrical impedance imaging has recently received considerable attention in the mathematical as well as in the engineering literature [2, 4, 10, 14, 17]. Using this technique one seeks to determine information about the internal conductivity (or impedance) profile of an object based on boundary i...
Enhanced Electrical Impedance Tomography via the MumfordShah Functional
 ESAIM: Control, Optimization and Calculus of Variations
, 2001
"... We consider the problem of electrical impedance tomography where conductivity distribution in a domain is to be reconstructed from boundary measurements of voltage and currents. It is wellknown that this problem is highly illposed. In this work, we propose the use of the MumfordShah functional, dev ..."
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Cited by 14 (0 self)
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We consider the problem of electrical impedance tomography where conductivity distribution in a domain is to be reconstructed from boundary measurements of voltage and currents. It is wellknown that this problem is highly illposed. In this work, we propose the use of the MumfordShah functional, developed for segmentation and denoising of images, as a regularization. After establishing existence properties of the resulting variational problem, we proceed by demonstrating the approach in several numerical examples. Our results indicate that this is an eective approach for overcoming the illposedness. Moreover, it has the capability of enhancing the reconstruction while at the same time segmenting the conductivity image. 1 Introduction and formulation of the problem The purpose of this work is to demonstrate that the MumfordShah functional from image processing can be used eectively to regularize the classical problem of electrical impedance tomography. In electrical impedance tomogr...
Inverse Problems For A Perturbed Dissipative HalfSpace
 INVERSE PROBLEMS
, 1995
"... This paper addresses the scattering of acoustic and electromagnetic waves from a perturbed dissipative halfspace. For simplicity, the perturbation is assumed to have compact support. Section 1 discusses the application that motivated this work and explains how the scalar model used here is related ..."
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Cited by 11 (3 self)
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This paper addresses the scattering of acoustic and electromagnetic waves from a perturbed dissipative halfspace. For simplicity, the perturbation is assumed to have compact support. Section 1 discusses the application that motivated this work and explains how the scalar model used here is related to Maxwell's equations. Section 2 introduces three formulations for direct and inverse problems for the halfspace geometry. Two of these formulations relate to scattering problems, and the third to a boundary value problem. Section 3 shows how the scattering problems can be related to the boundary value problem. This shows that the three inverse problems are equivalent in a certain sense. In section 4, the boundary value problem is used to outline a simple way to formulate a multidimensional layer stripping procedure. This procedure is unstable and does not constitute a practical algorithm for solving the inverse problem. The paper concludes with three appendices, the first two of which car...
The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction
 Comm. Pure Appl. Math
"... The goal of this paper is to establish global uniqueness and obtain reconstruction, in dimensions n ≥ 3, for the Calderón problem in the class of potentials conormal to a smooth submanifold H in R n. In the case of hypersurfaces, the potentials considered here may have any singularity weaker than th ..."
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Cited by 11 (10 self)
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The goal of this paper is to establish global uniqueness and obtain reconstruction, in dimensions n ≥ 3, for the Calderón problem in the class of potentials conormal to a smooth submanifold H in R n. In the case of hypersurfaces, the potentials considered here may have any singularity weaker than that of the delta function
Recovering the conductivity at the boundary from the Dirichlet to Neumann map: a pointwise result
"... A formula is given for recovering the boundary values of the coefficient of an elliptic operator, divr, from the Dirichlet to Neumann map. The main point is that one may recover without any a priori smoothness assumptions. The formula allows one to recover the value of pointwise. Let... ..."
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Cited by 10 (4 self)
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A formula is given for recovering the boundary values of the coefficient of an elliptic operator, divr, from the Dirichlet to Neumann map. The main point is that one may recover without any a priori smoothness assumptions. The formula allows one to recover the value of pointwise. Let...
Matching Pursuit for Imaging High Contrast Conductivity
, 1999
"... We show that imaging an isotropic, high contrast conducting medium is asymptotically equivalent to the identification of a unique resistor network, given measurements of currents and voltages at the boundary. We show that a matching pursuit approach can be used effectively towards the numerical solu ..."
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Cited by 9 (4 self)
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We show that imaging an isotropic, high contrast conducting medium is asymptotically equivalent to the identification of a unique resistor network, given measurements of currents and voltages at the boundary. We show that a matching pursuit approach can be used effectively towards the numerical solution of the high contrast imaging problem, if the library of functions is constructed carefully and in accordance with the asymptotic theory. We also show how other libraries of functions that at first glance seem reasonable, in fact, do not work well. When the contrast in the conductivity is not so high, we show that wavelets can be used, especially nonorthogonal wavelet libraries. However, the library of functions that is based on the high contrast asymptotic theory is more robust, even for intermediate contrasts, and especially so in the presence of noise. Key words. Impedance tomography, high contrast, asymptotic resistor network, imaging. Contents 1 Introduction 1 2 The Neumann to Dir...