Distinguishability revisited: depth dependent bounds on reconstruction quality in electrical impedance tomography

TL;DR

This paper rigorously quantifies how the ability to distinguish inclusions in electrical impedance tomography depends on their depth, providing bounds that support the observed resolution differences near the boundary.

Contribution

It derives explicit depth-dependent bounds on the distinguishability of inclusions, supported by numerical verification using a novel characterization of the forward map.

Findings

Bounds depend explicitly on inclusion depth

Reconstruction resolution is better near the boundary

Numerical verification confirms theoretical bounds

Abstract

The reconstruction problem in electrical impedance tomography is highly ill-posed, and it is often observed numerically that reconstructions have poor resolution far away from the measurement boundary but better resolution near the measurement boundary. The observation can be quantified by the concept of distinguishability of inclusions. This paper provides mathematically rigorous results supporting the intuition. Indeed, for a model problem lower and upper bounds on the distinguishability of an inclusion are derived in terms of the boundary data. These bounds depend explicitly on the distance of the inclusion to the boundary, i.e. the depth of the inclusion. The results are obtained for disk inclusions in a homogeneous background in the unit disk. The theoretical bounds are verified numerically using a novel, exact characterization of the forward map as a tridiagonal matrix.