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

Andrew E. Thaler (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.; and Schlumberger-Doll Research Center, Cambridge, Massachusetts, U.S.A.)

Graeme W. Milton (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)

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.

Keywords

volume fraction bounds, electrical impedance tomography, size estimation

2010 Mathematics Subject Classification

31A25, 31B20, 35J57, 35Q60

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Published 12 March 2015