Electrical Impedance Tomography (EIT) is a non-invasive imaging modality where an unknown physical body is probed with harmless and painless electrical currents via electrodes attached to the surface of the body, and the resulting voltages are measured. The goal is to recover the internal conductivity distribution of the body based on current-to-voltage boundary measurements. EIT has various applications in medical imaging, like respiratory imaging (see picture) or stroke classification and monitoring.
Reconstructions from EIT measurements are typically of low spatial resolution, but are superior in term of contrast to established imaging modalities, such as CT (X-ray tomography) and MRI (magnetic resonance imaging). That means EIT is superior in applications where high differences in conductivities occur, such as stroke (excessive blood or no blood) and respiratory imaging (blood and air).
The reconstruction task is a highly ill-posed nonlinear inverse problem, which is very sensitive to noise. This means that small changes in boundary measurements can correspond to large changes in the internal conductivity distribution, and furthermore in the case of EIT, noise in the data is amplified exponentially. Therefore, regularization is needed for the noise-robust recovery of conductivities from the boundary measurements. The image reconstruction task is too nonlinear to be covered by the presently available theory of iterative regularization. A possibility to overcome these obstacles is given by the D-bar methodology in two dimensions, which is a direct inversion method and a proven regularization strategy for the full nonlinear problem.