Electrical impedance tomography and Mittag-Leffler's function

Abstract

Given a two-dimensional bounded domain and a conductivity distribution, how to extract information about unknown inclusions in conductivity from finitely many noisy Cauchy data? (In practice, the data are given by electrical impedance tomography (EIT) measurements.) First, a direct approach is presented, based on an extraction formula of the set of all points in the domain that can be connected with infinity by a straight line without intersecting the closure of the inclusions. The formula uses an indicator function that depends on a large parameter τ and can be calculated from infinitely many non-noisy Cauchy data. Second, a modified indicator function is defined. It can be calculated from finitely many noisy Cauchy data. Its properties are studied; the results suggest how many data are needed and how to choose τ. Third, a regularized algorithm that is based on the theoretical study is given for extracting inclusions approximately. The choice of all parameters is described explicitly. Numerical examples using simulated continuum data show that a reliable estimate for the number and location of inclusions is achieved. However, the shape of the inclusions is not recovered.