Inverse Problems and Imaging (IPI)

A data-driven edge-preserving D-bar method for electrical impedance tomography

Pages: 1053 - 1072, Volume 8, Issue 4, November 2014      doi:10.3934/ipi.2014.8.1053

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Sarah Jane Hamilton - Department of Mathematics, Statistics, and Computer Science, Marquette University, Milwaukee, WI 53233, United States (email)
Andreas Hauptmann - Department of Mathematics and Statistics, University of Helsinki, Helsinki, 00014, Finland (email)
Samuli Siltanen - University of Helsinki, Department of Mathematics and Statistics, FI-00014 Helsinki, Finland (email)

Abstract: In Electrical Impedance Tomography (EIT), the internal conductivity of a body is recovered via current and voltage measurements taken at its surface. The reconstruction task is a highly ill-posed nonlinear inverse problem, which is very sensitive to noise, and requires the use of regularized solution methods, of which D-bar is the only proven method. The resulting EIT images have low spatial resolution due to smoothing caused by low-pass filtered regularization. In many applications, such as medical imaging, it is known a priori that the target contains sharp features such as organ boundaries, as well as approximate ranges for realistic conductivity values. In this paper, we use this information in a new edge-preserving EIT algorithm, based on the original D-bar method coupled with a deblurring flow stopped at a minimal data discrepancy. The method makes heavy use of a novel data fidelity term based on the so-called CGO sinogram. This nonlinear data step provides superior robustness over traditional EIT data formats such as current-to-voltage matrices or Dirichlet-to-Neumann operators, for commonly used current patterns.

Keywords:  Inverse conductivity problem, electrical impedance tomography, image segmentation, complex geometrical optics solutions, D-bar method.
Mathematics Subject Classification:  Primary: 65N21; Secondary: 94A08.

Received: December 2013;      Revised: October 2014;      Available Online: November 2014.