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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
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...
Low Frequency Electromagnetic Fields in High Contrast Media
"... . Using variational principles we construct discrete network approximations for the Dirichlet to Neumann or Neumann to Dirichlet maps of high contrast, low frequency electromagnetic media. 1 Introduction Imaging of the electrical conductivity and permittivity of a heterogeneous body by means of low ..."
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Cited by 2 (1 self)
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. Using variational principles we construct discrete network approximations for the Dirichlet to Neumann or Neumann to Dirichlet maps of high contrast, low frequency electromagnetic media. 1 Introduction Imaging of the electrical conductivity and permittivity of a heterogeneous body by means of lowfrequency electrical or electromagnetic field measurements is an inverse problem, often called "impedance tomography", "electromagnetic induction tomography", "magnetotellurics" and so on. Applications arise in many areas, for example in medicine with diagnostic imaging, in nondestructive testing, in oil recovery, in subsurface flow monitoring, in underground contaminant detection, etc. In this paper we will focus attention on imaging heterogeneous media with large variations in the magnitude of their electrical properties. This is relevant in many geophysical applications where the conductivity can vary over several orders of magnitude. For example, a dry rock matrix is insulating compared...
A Multiscattering Series for Impedance Tomography in Layered Media
, 1998
"... We introduce an inversion algorithm for tomographic images of layered media. The algorithm is based on a multiscattering series expansion of the Green's function that, unlike the Born series, converges unconditionally. Our inversion algorithm obtains images of the medium that improve iteratively, as ..."
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Cited by 1 (0 self)
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We introduce an inversion algorithm for tomographic images of layered media. The algorithm is based on a multiscattering series expansion of the Green's function that, unlike the Born series, converges unconditionally. Our inversion algorithm obtains images of the medium that improve iteratively, as we use more and more terms in the multiscattering series. We present the derivation of the multiscattering series, formulate the inversion algorithm and demonstrate its performance through numerical experiments. Key words: multiscattering, imaging, electrical conductivity. Contents 1 Introduction 1 2 Derivation of the multiscattering series 3 2.1 Pathintegral representation of the Green's function . . . . . . . . . . . . . . . . . . 3 2.2 Isotropic, homogeneous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Medium with one plane interface of discontinuity of the conductivity . . . . . . . . . 5 2.4 Multiscattering series expansion of the Green's function in a ...
Modeling Across Scales: Discrete Geometric Structures in Homogenization and Inverse Homogenization
"... Imaging and simulation methods are typically constrained to resolutions much coarser than the scale of physical microstructures present in body tissues or geological features. Mathematical and numerical homogenization address this practical issue by identifying and computing appropriate spatial ave ..."
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Cited by 1 (1 self)
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Imaging and simulation methods are typically constrained to resolutions much coarser than the scale of physical microstructures present in body tissues or geological features. Mathematical and numerical homogenization address this practical issue by identifying and computing appropriate spatial averages that result in accuracy and consistency between the macroscales we observe and the underlying microscale models we assume. Among the various applications benefiting from homogenization, Electric Impedance Tomography (EIT) images the electrical conductivity of a body by measuring electrical potentials consequential to electric currents applied to the exterior of the body. EIT is routinely used in breast cancer detection and cardiopulmonary imaging, where current flow in finescale tissues underlies the resulting coarsescale images. In this paper, we introduce a geometric approach for the homogenization (simulation) and inverse homogenization (imaging) of divergenceform
On the Continuum Limit of a . . .
, 2004
"... We consider finite difference approximations of solutions of inverse SturmLiouville problems in bounded intervals. Using threepoint finite difference schemes, we discretize the equations on socalled optimal grids constructed as follows: For a staggered grid with 2k points, we ask that the finite ..."
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We consider finite difference approximations of solutions of inverse SturmLiouville problems in bounded intervals. Using threepoint finite difference schemes, we discretize the equations on socalled optimal grids constructed as follows: For a staggered grid with 2k points, we ask that the finite difference operator (a k × k Jacobi matrix) and the SturmLiouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval. This requirement determines uniquely the entries in the Jacobi matrix, which are grid cell averages of the coefficients in the continuum problem. If these coefficients are known, we can find the grid, which we call optimal because it gives, by design, a finite difference operator with a prescribed spectral measure. We focus attention on the inverse problem, where neither the coefficients nor the grid are known. A key question in inversion is how to parametrize the coefficients, i.e., how to choose the grid. It is clear that, to be successful, this grid must be close to the optimal
A nonlinear multigrid . . . conductivity and permittivity at low frequency
, 2001
"... We propose a nonlinear multigrid approach for imaging the electrical conductivity and permittivity of a body �, given partial, usually noisy knowledge of the NeumanntoDirichlet map at the boundary. The algorithm is a nested iteration, where the image is constructed on a sequence of grids in �, sta ..."
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We propose a nonlinear multigrid approach for imaging the electrical conductivity and permittivity of a body �, given partial, usually noisy knowledge of the NeumanntoDirichlet map at the boundary. The algorithm is a nested iteration, where the image is constructed on a sequence of grids in �, starting from the coarsest grid and advancing towards the finest one. We show various numerical examples that demonstrate the effectiveness and robustness of the algorithm and prove local convergence. (Some figures in this article are in colour only in the electronic version; see www.iop.org)