Reconstruction of piecewise constant layered conductivities in electrical impedance tomography

Research output: Working paperResearch

Abstract

This work presents a new constructive uniqueness proof for Calderón's inverse problem of electrical impedance tomography, subject to local Cauchy data, for a large class of piecewise constant conductivities that we call "piecewise constant layered conductivities" (PCLC). The resulting reconstruction method only relies on the physically intuitive monotonicity principles of the local Neumann-to-Dirichlet map, and therefore the method lends itself well to efficient numerical implementation and generalization to electrode models. Several direct reconstruction methods exist for the related problem of inclusion detection, however they share the property that "holes in inclusions" or "inclusions-within-inclusions" cannot be determined. One such method is the monotonicity method of Harrach, Seo, and Ullrich, and in fact the method presented here is a modified variant of the monotonicity method which overcomes this problem. More precisely, the presented method abuses that a PCLC type conductivity can be decomposed into nested layers of positive and/or negative perturbations that, layer-by-layer, can be determined via the monotonicity method. The conductivity values on each layer are found via basic one-dimensional optimization problems constrained by monotonicity relations.
Original languageEnglish
PublisherarXiv.org
Number of pages10
Publication statusPublished - 2019

Fingerprint

electrical impedance
tomography
conductivity
inclusions
uniqueness
perturbation
optimization
electrodes

Keywords

  • electrical impedance tomography
  • partial data reconstruction
  • piecewise constant coefficient
  • monotonicity principle

Cite this

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title = "Reconstruction of piecewise constant layered conductivities in electrical impedance tomography",
abstract = "This work presents a new constructive uniqueness proof for Calder{\'o}n's inverse problem of electrical impedance tomography, subject to local Cauchy data, for a large class of piecewise constant conductivities that we call {"}piecewise constant layered conductivities{"} (PCLC). The resulting reconstruction method only relies on the physically intuitive monotonicity principles of the local Neumann-to-Dirichlet map, and therefore the method lends itself well to efficient numerical implementation and generalization to electrode models. Several direct reconstruction methods exist for the related problem of inclusion detection, however they share the property that {"}holes in inclusions{"} or {"}inclusions-within-inclusions{"} cannot be determined. One such method is the monotonicity method of Harrach, Seo, and Ullrich, and in fact the method presented here is a modified variant of the monotonicity method which overcomes this problem. More precisely, the presented method abuses that a PCLC type conductivity can be decomposed into nested layers of positive and/or negative perturbations that, layer-by-layer, can be determined via the monotonicity method. The conductivity values on each layer are found via basic one-dimensional optimization problems constrained by monotonicity relations.",
keywords = "electrical impedance tomography, partial data reconstruction, piecewise constant coefficient, monotonicity principle",
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AU - Garde, Henrik

PY - 2019

Y1 - 2019

N2 - This work presents a new constructive uniqueness proof for Calderón's inverse problem of electrical impedance tomography, subject to local Cauchy data, for a large class of piecewise constant conductivities that we call "piecewise constant layered conductivities" (PCLC). The resulting reconstruction method only relies on the physically intuitive monotonicity principles of the local Neumann-to-Dirichlet map, and therefore the method lends itself well to efficient numerical implementation and generalization to electrode models. Several direct reconstruction methods exist for the related problem of inclusion detection, however they share the property that "holes in inclusions" or "inclusions-within-inclusions" cannot be determined. One such method is the monotonicity method of Harrach, Seo, and Ullrich, and in fact the method presented here is a modified variant of the monotonicity method which overcomes this problem. More precisely, the presented method abuses that a PCLC type conductivity can be decomposed into nested layers of positive and/or negative perturbations that, layer-by-layer, can be determined via the monotonicity method. The conductivity values on each layer are found via basic one-dimensional optimization problems constrained by monotonicity relations.

AB - This work presents a new constructive uniqueness proof for Calderón's inverse problem of electrical impedance tomography, subject to local Cauchy data, for a large class of piecewise constant conductivities that we call "piecewise constant layered conductivities" (PCLC). The resulting reconstruction method only relies on the physically intuitive monotonicity principles of the local Neumann-to-Dirichlet map, and therefore the method lends itself well to efficient numerical implementation and generalization to electrode models. Several direct reconstruction methods exist for the related problem of inclusion detection, however they share the property that "holes in inclusions" or "inclusions-within-inclusions" cannot be determined. One such method is the monotonicity method of Harrach, Seo, and Ullrich, and in fact the method presented here is a modified variant of the monotonicity method which overcomes this problem. More precisely, the presented method abuses that a PCLC type conductivity can be decomposed into nested layers of positive and/or negative perturbations that, layer-by-layer, can be determined via the monotonicity method. The conductivity values on each layer are found via basic one-dimensional optimization problems constrained by monotonicity relations.

KW - electrical impedance tomography

KW - partial data reconstruction

KW - piecewise constant coefficient

KW - monotonicity principle

M3 - Working paper

BT - Reconstruction of piecewise constant layered conductivities in electrical impedance tomography

PB - arXiv.org

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