Exact shape-reconstruction by one-step linearization in electrical impedance tomography

Bastian Harrach, Jin Keun Seo

Research output: Contribution to journalArticle

35 Citations (Scopus)

Abstract

For electrical impedance tomography (EIT), the linearized reconstruction method using the Fréchet derivative of the Neumann-to-Dirichlet map with respect to the conductivity has been widely used in the last three decades. However, few rigorous mathematical results are known regarding the errors caused by the linear approximation. In this work we prove that linearizing the inverse problem of EIT does not lead to shape errors for piecewise-analytic conductivities. If a solution of the linearized equations exists, then it has the same outer support as the true conductivity change, no matter how large the latter is. Under an additional definiteness condition we also show how to approximately solve the linearized equation so that the outer support converges toward the right one. Our convergence result is global and also applies for approximations by noisy finitedimensional data. Furthermore, we obtain bounds on how well the linear reconstructions and the true conductivity difference agree on the boundary of the outer support.

Original languageEnglish
Pages (from-to)1505-1518
Number of pages14
JournalSIAM Journal on Mathematical Analysis
Volume42
Issue number4
DOIs
Publication statusPublished - 2010 Sep 2

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Shape Reconstruction
Electrical Impedance Tomography
Acoustic impedance
Linearization
Conductivity
Tomography
Inverse problems
Derivatives
Dirichlet-to-Neumann Map
Linear Approximation
Convergence Results
Inverse Problem
Converge
Derivative
Approximation

All Science Journal Classification (ASJC) codes

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

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Exact shape-reconstruction by one-step linearization in electrical impedance tomography. / Harrach, Bastian; Seo, Jin Keun.

In: SIAM Journal on Mathematical Analysis, Vol. 42, No. 4, 02.09.2010, p. 1505-1518.

Research output: Contribution to journalArticle

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