Anomalous thresholds and edge singularities in Electrical Impedance Tomography

TL;DR

This paper applies advanced mathematical techniques to analyze the rapid variation of current density near electrodes in Electrical Impedance Tomography, revealing the structure of singularities and eigenfunctions to improve understanding of the underlying physics.

Contribution

It introduces a novel application of singularity theory from particle physics to describe edge effects in EIT, providing a comprehensive analytic framework.

Findings

Describes the analytic structure of current density near electrode edges.

Provides a complete description of the Riemann sheet manifold of eigenfunctions.

Demonstrates the extension of methods to other weakly singular kernels.

Abstract

Studies of models of current flow behaviour in Electrical Impedance Tomography (EIT) have shown that the current density distribution varies extremely rapidly near the edge of the electrodes used in the technique. This behaviour imposes severe restrictions on the numerical techniques used in image reconstruction algorithms. In this paper we have considered a simple two dimensional case and we have shown how the theory of end point/pinch singularities which was developed for studying the anomalous thresholds encountered in elementary particle physics can be used to give a complete description of the analytic structure of the current density near to the edge of the electrodes. As a byproduct of this study it was possible to give a complete description of the Riemann sheet manifold of the eigenfunctions of the logarithmic kernel. These methods can be readily extended to other weakly singular kernels.