A remark on the enclosure method for a body with an unknown homogeneous background conductivity

TL;DR

This paper extends the enclosure method in electrical impedance tomography to cases where the background conductivity is unknown, demonstrating that cavity locations can still be inferred under certain conditions in two dimensions.

Contribution

It shows that the enclosure method can be applied without known background conductivity if the boundary voltage Fourier series is band-limited and meets specific non-vanishing criteria.

Findings

Locating cavities without known background conductivity in 2D.

Enclosure method applicability under band-limited boundary data.

Conditions for extracting cavity information in unknown homogeneous backgrounds.

Abstract

Previous applications of the enclosure method with a finite set of observation data to a mathematical model of electrical impedance tomography are based on the assumption that the conductivity of the background body is homogeneous and known. This paper considers the case when the conductivity is homogeneous and unknown. It is shown that, in two dimensions if the domain occupied by the background body is enclosed by an ellipse, then it is still possible to extract some information about the location of unknown cavities or inclusions embedded in the body without knowing the background conductivity provided the Fourier series expansion of the voltage on the boundary does not contain high frequency parts (band limited) and satisfies a non vanishing condition of a quantity involving the Fourier coefficients.