Full nonlinear 2D EIT D-bar reconstructions of Images for Electrical Impedance Tomography using the Method D-bar 2D

Abstract

This work proposes to develop a version of the D-Bar method without linearization for Electrical Impedance Tomography. The D-Bar method is an algorithm capable of generating absolute images for Electrical Impedance Tomography (EIT) without iteration, the only method of its kind at present. In the D-Bar method there is a change of variables that takes the generalized Laplace equation, which represents the electrical conductivity problem, to the Schrödinger equation, which has special solutions known as Faddeev solutions. The EIT inverse problem is ill-posed and leads to ill-conditioned algebraic equations. The determination of the conductivity by the D-Bar method based on the existence and uniqueness proof of Adrian Nachman [Ann. of Math. 143 (1996)] relies on the use of an intermediate object called the scattering transform. The ill-posedness becomes evident in the determination of the scattering transform, more specifically in the determination of the Faddeev solutions. There is a simplification in the current implementation of the determination of Faddeev solutions psi(x,k) in the D-Bar method in which the asymptotic behavior of the Faddeev solution is used in place of psi(x,k) in calculation of the scattering transform. This is a kind of linearization in one step of the overall method.This work proposes to calculate the scattering transform from exponentially growing solutions and therefore obtain images with higher resolution because the linearization is avoided. The method will generate a system of ill-conditioned algebraic equations. The solution of this system requires the use of regularizations. Several regularization techniques are compared. The feasibility to incorporate a priori knowledge of the conductivity in this stage of the D-Bar method is researched. The resulting images are compared with those obtained by approximating e^{ikx} in the scattering transform's equation.