Bounds on the volume of an inclusion in a body from a complex conductivity measurement

TL;DR

This paper develops bounds on the volume of an inclusion within a body using complex conductivity measurements from electrical impedance tomography, providing exact volume estimates for specific geometries.

Contribution

It introduces a method to estimate inclusion volume bounds from boundary measurements with tight bounds that are exact in certain cases.

Findings

Bounds are derived from boundary electrical measurements.

Bounds are tight and can give exact volume in specific geometries.

Method applies to complex conductivities in 2D and 3D.

Abstract

We derive bounds on the volume of an inclusion in a body in two or three dimensions when the conductivities of the inclusion and the surrounding body are complex and assumed to be known. The bounds are derived in terms of average values of the electric field, current, and certain products of the electric field and current. All of these average values are computed from a single electrical impedance tomography measurement of the voltage and current on the boundary of the body. Additionally, the bounds are tight in the sense that at least one of the bounds gives the exact volume of the inclusion for certain geometries and boundary conditions.