Direct inversion from partial-boundary data in electrical impedance tomography

TL;DR

This paper develops a linearized D-bar method for electrical impedance tomography that reconstructs conductivity from partial boundary data, providing error estimates and numerical validation for realistic measurement scenarios.

Contribution

It introduces a novel approach to reconstruct conductivities from partial boundary data with proven linear error dependence and numerical validation.

Findings

Error depends linearly on missing boundary size

Reconstruction accuracy is validated numerically

Partial data can effectively approximate full boundary results

Abstract

In Electrical Impedance Tomography (EIT) one wants to image the conductivity distribution of a body from current and voltage measurements carried out on its boundary. In this paper we consider the underlying mathematical model, the inverse conductivity problem, in two dimensions and under the realistic assumption that only a part of the boundary is accessible to measurements. In this framework our data are modeled as a partial Neumann-to-Dirichlet map (ND map). We compare this data to the full-boundary ND map and prove that the error depends linearly on the size of the missing part of the boundary. The same linear dependence is further proved for the difference of the reconstructed conductivities -- from partial and full boundary data. The reconstruction is based on a truncated and linearized D-bar method. Auxiliary results include an extrapolation method to obtain the full-boundary data from the measured one, an approximation of the complex geometrical optics solutions computed directly from the ND map as well as an approximate scattering transform for reconstructing the conductivity. Numerical verification of the convergence results and reconstructions are presented for simulated test cases.