Nonlinear Inversion from Partial EIT Data: Computational Experiments

TL;DR

This paper explores extending the D-bar method for electrical impedance tomography to cases with partial boundary data, demonstrating through computational experiments that it can effectively approximate full data solutions in two dimensions.

Contribution

It introduces a novel approach using localized basis functions to adapt the D-bar method for partial boundary measurements in EIT, supported by computational evidence.

Findings

D-bar method can be applied to partial boundary data in 2D.

Partial data CGO traces approximate full data solutions on available boundary.

Numerical simulations support the feasibility of the approach.

Abstract

Electrical impedance tomography (EIT) is a non-invasive imaging method in which an unknown physical body is probed with electric currents applied on the boundary, and the internal conductivity distribution is recovered from the measured boundary voltage data. The reconstruction task is a nonlinear and ill-posed inverse problem, whose solution calls for special regularized algorithms, such as D-bar methods which are based on complex geometrical optics solutions (CGOs). In many applications of EIT, such as monitoring the heart and lungs of unconscious intensive care patients or locating the focus of an epileptic seizure, data acquisition on the entire boundary of the body is impractical, restricting the boundary area available for EIT measurements. An extension of the D-bar method to the case when data is collected only on a subset of the boundary is studied by computational simulation. The approach is based on solving a boundary integral equation for the traces of the CGOs using localized basis functions (Haar wavelets). The numerical evidence suggests that the D-bar method can be applied to partial-boundary data in dimension two and that the traces of the partial data CGOs approximate the full data CGO solutions on the available portion of the boundary, for the necessary small $k$ frequencies.