Inverse Problems and Imaging (IPI)

A direct D-bar method for partial boundary data electrical impedance tomography with a priori information

Pages: 427 - 454, Volume 11, Issue 3, June 2017      doi:10.3934/ipi.2017020

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Melody Alsaker - Gonzaga University, Mathematics Department, 502 E. Boone Ave. MSC 2615, Spokane, WA 99258-0072, United States (email)
Sarah Jane Hamilton - Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI 53233, United States (email)
Andreas Hauptmann - Department of Computer Science, University College London, WC1E 6BT London, United Kingdom (email)

Abstract: Electrical Impedance Tomography (EIT) is a non-invasive imaging modality that uses surface electrical measurements to determine the internal conductivity of a body. The mathematical formulation of the EIT problem is a nonlinear and severely ill-posed inverse problem for which direct D-bar methods have proved useful in providing noise-robust conductivity reconstructions. Recent advances in D-bar methods allow for conductivity reconstructions using EIT measurement data from only part of the domain (e.g., a patient lying on their back could be imaged using only data gathered on the accessible part of the body). However, D-bar reconstructions suffer from a loss of sharp edges due to a nonlinear low-pass filtering of the measured data, and this problem becomes especially marked in the case of partial boundary data. Including a priori data directly into the D-bar solution method greatly enhances the spatial resolution, allowing for detection of underlying pathologies or defects, even with no assumption of their presence in the prior. This work combines partial data D-bar with a priori data, allowing for noise-robust conductivity reconstructions with greatly improved spatial resolution. The method is demonstrated to be effective on noisy simulated EIT measurement data simulating both medical and industrial imaging scenarios.

Keywords:  Electrical impedance tomography, partial boundary data, Neumann-to-Dirichlet map, D-bar method, a priori information.
Mathematics Subject Classification:  Primary: 65N21; Secondary: 94A08.

Received: September 2016;      Revised: October 2016;      Available Online: April 2017.