Monotonicity-Based Shape Reconstruction in Electrical Impedance Tomography

Abstract

Current-voltage measurements in electrical impedance tomography can be partially ordered with respect to definiteness of the associated self-adjoint Neumann-to-Dirichlet operators (NtD). With this ordering, a point-wise larger conductivity leads to smaller current-voltage measurements, and smaller conductivities lead to larger measurements.We present a converse of this simple monotonicity relation and use it to solve the shape reconstruction (aka inclusion detection) problem in EIT. The outer shape of a region where the conductivity differs from a known background conductivity can be found by simply comparing the measurements to that of smaller or larger test regions. † Birth name: Bastian Gebauer,

Recent Progress on the Factorization Method for Electrical Impedance Tomography

“…Roughly speaking, under this general assumption, the Factorization Method then characterizes the support of σ − σ 0 up to holes in the support that have no connections to Σ. For a precise formulation, we use the concept of the inner and outer support from [28] that has been inspired by the use of the infinity support of Kusiak and Sylvester [29]; see also [25, 30]. …”

“…So far, there are no rigorous convergence results for numerical implementations of this range criterion (see, however, Lechleiter [32] for a first step in this direction). As a promising approach to overcome both problems, we would like to point out the recent work on monotony-based methods [28]. …”

Electrical Impedance Tomography

“…In the presence of noise, of course, one needs a sufficiently negative eigenvalue to be sure of the result of the test, and the method does assume that the conductivity is of the given form (96). Recently a partial converse of the monotonicity result has been found [54] and this promises more accurate fast methods of reconstructing the shape of an inclusion.…”

The monotonicity method for the inverse crack scattering problem

“…The feature of this method is to understand the inclusion relation of an unknown onject and artificial one by comparing the data operator with some operator corresponding to an artificial object. For recent works of the monotonicity method, we refer to [2,3,4,5,6,10].…”