Elliptic Cylinder Geometry for Distinguishability Analysis in Impedance Tomography

Abstract

Electrical impedance tomography (EIT) is a technique that computes the cross-sectional impedance distribution within the body by using current and voltage measurements made on the body surface. It has been reported that the image reconstruction is distorted considerably when the boundary shape is considered to be more elliptical than circular as a more realistic shape for the measurement boundary. This paper describes an alternative framework for determining the distinguishability region with a finite measurement precision for different conductivity distributions in a body modeled by elliptic cylinder geometry. The distinguishable regions are compared in terms of modeling error for predefined inhomogeneities with elliptical and circular approaches for a noncircular measurement boundary at the body surface. Since most objects investigated by EIT are noncircular in shape, the analytical solution for the forward problem for the elliptical cross section approach is shown to be useful in order to reach a better assessment of the distinguishability region defined in a noncircular boundary. This paper is concentrated on centered elliptic inhomogeneity in the elliptical boundary and an analytic solution for this type of forward problem. The distinguishability performance of elliptical cross section with cosine injected current patterns is examined for different parameters of elliptical geometry.

Application of conformal transformation to elliptic geometry for electric impedance tomography

“…These coefficients are multiplied with the eigenvalues of resistivity functions to find the Fourier coefficients of the voltage distribution. As a result, voltage distribution for both cases can be calculated by the formulae given below which are derived from the Laplace equation [11]:…”

“…The general problems of EIT such as solving the forward problem to obtain the voltage distribution at the surface of interest, determining the best current pattern to be injected through electrodes and searching the smallest detectable object size have been addressed and reported in many studies in the literature [8][9][10][11][12][13][14]. Studying geometric models for analytic solutions of forward problems in EIT has recently drawn particular attention [10][11][12][13][14][15]. Specifically, one of the studies by Cheney and Isaacson [16] aimed at finding the voltage distribution at the boundary of a cross-section presented in a circular geometry when a current pattern is applied through a set of electrodes equally spaced at the boundary.…”

“…Another significant study [15] was on the voltage distribution calculated for eccentric inhomogeneities in the body for the circular case. Based on the concern about more efficient geometric representation, there are also studies [11][12][13] in which the elliptic geometry was analyzed under the analytical solution in two-dimensional and three-dimensional geometries since the elliptic geometry is relatively a more realistic model considering the cross-sections associated with the human thorax. Moreover, it is reported [10,11] that the assumptions made in the circular model cause some important problems such as inaccuracy in detecting inhomogeneities inside the body and low performance in the reconstruction phase.…”

“…The accuracy in prediction of the smallest size of inhomogeneities has been improved further by developing the forward solution for the elliptic geometry [11]. An analytical solution has been offered for distinguishability analysis similar to one suggested by Cheney and Isaacson [16].…”

“…An analytical solution has been offered for distinguishability analysis similar to one suggested by Cheney and Isaacson [16]. Although the theory behind the derivations showed immediate impact on the assessment of distinguishability criteria, there is still much remaining to extend the analytic solutions proposed and their results in comparison with numerical solutions [11].…”