Learning from electric X-ray images: the new EIT

Abstract: A fundamental connection between Electrical Impedance Tomography (EIT) and classical X-ray tomography was found in [Greenleaf et al 2018]. There it was shown that a one-dimensional Fourier transform applied to the spectral parameter of Complex Geometric Optics (CGO) solutions to a Beltrami equation is a useful technique. Microlocal analysis of the involved complex principal type operators reveals singularities propagating in curious ways. They enable a novel filtered back-projection type nonlinear reconstruction algorithm for EIT. This approach is called Virtual Hybrid Edge Detection (VHED).

One of the medically most promising applications of EIT is stroke imaging. There are two main types of stroke: (1) brain hemorrhage and (2) ischemic stroke caused by a blood clot. The symptoms for those two conditions are the same, but the treatments are completely the opposite. There are two main uses for EIT here: (a) classifying the type of stroke already in the ambulance with a cost-effective portable device, and (b) monitoring the state of recovering stroke patients in the intensive care unit.

The main difficulty in using EIT for head imaging is the resistive skull. Because of that, the relevant signal from the brain is weak and almost buried in noise. Given the extreme ill-posedness of the inverse conductivity problem, it is quite a challenge to design a robust EIT algorithm for either (a) or (b).

VHED offers a way to divide the information in EIT measurements into geometrically understood pieces. One could wish that those pieces are less sensitive to noise than a full reconstructed image of the conductivity. This presentation shows how machine learning can be used for classifying stroke (problem (a)) above based on VHED profiles. Examined are fully connected neural networks (FCNN), convolutional neural networks (CNN) and recurrent neural networks (RNN). Perhaps surprisingly, CNNs offer the worst performance, while RNNs are slightly better than FCNNs.

Mathematics

Audience: researchers in the topic

International Zoom Inverse Problems Seminar, UC Irvine