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Feasibility of hybrid inverse problems in limited view

Hjørdis Schlüter

Abstract

Hybrid inverse problems such as Acousto-Electric Tomography, Current Density Imaging or Magnetic Resonance Electric Impedance Tomography are concerned with reconstructing the electrical conductivity from interior measurements. For a two-dimensional object the measurements correspond to two different functions imposed as the Neumann boundary condition to an elliptic PDE. In limited view these functions are only non-zero on the part of the boundary that one can control. In this paper we address how to choose such boundary functions in limited view such that the reconstruction procedure is feasible. This is related to the corresponding Jacobian being non-zero. We supplement the theoretical results by numerical reconstructions following the approach of Acousto-Electric Tomography.

Feasibility of hybrid inverse problems in limited view

Abstract

Hybrid inverse problems such as Acousto-Electric Tomography, Current Density Imaging or Magnetic Resonance Electric Impedance Tomography are concerned with reconstructing the electrical conductivity from interior measurements. For a two-dimensional object the measurements correspond to two different functions imposed as the Neumann boundary condition to an elliptic PDE. In limited view these functions are only non-zero on the part of the boundary that one can control. In this paper we address how to choose such boundary functions in limited view such that the reconstruction procedure is feasible. This is related to the corresponding Jacobian being non-zero. We supplement the theoretical results by numerical reconstructions following the approach of Acousto-Electric Tomography.

Paper Structure

This paper contains 9 sections, 3 theorems, 36 equations, 61 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Reconstruction procedure
  4. Reconstruction of $\mathbf{S}_i=\sqrt{\sigma} \nabla u_i$
  5. Reconstruction of $\sigma$
  6. Numerical examples
  7. Reconstruction from noisy measurements
  8. Discussion
  9. Acknowledgments

Key Result

Proposition 2.1

Let $\Omega \subset \mathbb{R}^2$ be a simply connected bounded Lipschitz domain and let $\sigma \in L^{\infty}(\Omega)$ satisfy $\lambda \leq \sigma$ for some $\lambda \in (0,1)$. Let $g \in H^{-\frac{1}{2}}(\partial \Omega)$ with $\int_{\partial \Omega} g \, \mathrm{d}S=0$ and $u \in H^1(\Omega)$ Denote by $2M$ the number of closed arcs $\Gamma_1,...,\Gamma_{2M}$ into which $\partial \Omega$ ca

Figures (61)

  • Figure 9: True conductivity $\sigma_1$ used for reconstruction from noisy measurements (note the difference in the range of the colorbar relative to Figure \ref{['fig:phantoms']}).
  • Figure : $t$
  • Figure : $\sigma_{\text{1}}$
  • Figure : $\log(H_{11})$
  • Figure : $\Gamma_8$
  • ...and 56 more figures

Theorems & Definitions (8)

  • Proposition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 3.1
  • Remark 3.2