Jennifer L. Mueller

A direct reconstruction algorithm for electrical impedance tomography

A direct (noniterative) reconstruction algorithm for electrical impedance tomography in the two-dimensional (2-D), cross-sectional geometry is reviewed. New results of a reconstruction of a numerically simulated phantom chest are presented. The algorithm is based on the mathematical uniqueness proof by A.I. Nachman [1996] for the 2-D inverse conductivity problem. In this geometry, several of the clinical applications include monitoring heart and lung function, diagnosis of pulmonary embolus, diagnosis of pulmonary edema, monitoring for internal bleeding, and the early detection of breast cancer.

Imaging cardiac activity by the D-bar method for electrical impedance tomography

A practical D-bar algorithm for reconstructing conductivity changes from EIT data taken on electrodes in a 2D geometry is described. The algorithm is based on the global uniqueness proof of Nachman (1996 Ann. Math. 143 71-96) for the 2D inverse conductivity problem. Results are shown for reconstructions from data collected on electrodes placed around the circumference of a human chest to reconstruct a 2D cross-section of the torso. The images show changes in conductivity during a cardiac cycle.

D-bar method for electrical impedance tomography with discontinuous conductivities

The effects of truncating the (approximate) scattering transform in the D‐bar reconstruction method for two‐dimensional electrical impedance tomography are studied. The method is based on the uniqueness proof of Nachman [Ann. of Math. (2), 143 (1996), pp. 71–96] that applies to twice differentiable conductivities. However, the reconstruction algorithm has been successfully applied to experimental data, which can be characterized as piecewise smooth conductivities. The truncation is shown to stabilize the method against measurement noise and to have a smoothing effect on the reconstructed conductivity. Thus the truncation can be interpreted as regularization of the D‐bar method. Numerical reconstructions are presented demonstrating that features of discontinuous high contrast conductivities can be recovered using the D‐bar method. Further, a new connection between Calderon’s linearization method and the D‐bar method is established, and the two methods are compared numerically and analytically.

Reconstruction of conductivity changes due to ventilation and perfusion from EIT data collected on a rectangular electrode array.

In this paper we demonstrate that conductivity changes caused by ventilation and perfusion in a human subject can be reconstructed from electrical impedance tomography data collected on a rectangular array of electrodes placed on a subjects chest. Currents are applied on the electrodes and the resulting voltages on the electrodes are measured. A 3D reconstruction algorithm is used to reconstruct the conductivity distribution in the region beneath the array. Time traces of the reconstructed conductivity distribution demonstrate the detected changes in conductivity due to ventilation and perfusion.

A numerical method for backward parabolic problems with non-selfadjoint elliptic operators

A method of solution of backward parabolic problems with non-selfadjoint elliptic operators is presented. The method employs a quasisolution approach and is based on the separation of the problem into a sequence of well-posed forward problems on the entire mesh and an ill-posed system of algebraic equations on a coarser submesh. For the corresponding forward problem the continuous dependence of the solution on the initial profile is proved. From this result a stability estimate on the final time T is obtained. The estimate shows a decrease in stability of the forward (hence, the backward) problem, as the final time T is increased. Using the stability result the existence of a quasisolution of the backward problem is proved. For the solution of the intermediate non-selfadjoint forward problems a modified alternating-direction finite difference scheme is presented. The ill-conditioned system of algebraic equations is solved by using truncated singular value decomposition. The effectiveness of the method is demonstrated on a numerical test problem with exact and noisy data.

2D D-bar reconstructions of human chest and tank data using an improved approximation to the scattering transform

The D-bar method for electrical impedance tomography requires the computation of an intermediate function known as the scattering transform from the measured data. An approximation to the scattering transform utilizing the standard Greens function for the Laplacian was introduced for the 2D D-bar method in Mueller and Siltanen (2003 SIAM J. Sci. Comp. 24 1232-66) and tested on simple numerically simulated conductivity distributions. In this work, the approximation is implemented for experimental data for the first time. It is tested on both tank and human chest data, and the results demonstrate decreased blurring toward the boundary in the images than in images computed with the t(exp) approximation to the scattering transform.

Mapping properties of the nonlinear Fourier transform in dimension two

A class of compactly supported Schrödinger potentials in dimension two is given for which the inverse scattering method related to the Novikov–Veselov evolution equation is well-defined. There is no smallness assumption on the initial potential. Regularity results are proven for the direct and inverse scattering transforms, also called nonlinear Fourier transforms.

Effect of Domain Shape Modeling and Measurement Errors on the 2-D D-Bar Method for EIT

The D-bar algorithm based on Nachmans 2-D global uniqueness proof for the inverse conductivity problem (Nachman, 1996) is implemented on a chest-shaped domain. The scattering transform is computed on this chest-shaped domain using trigonometric and adjacent current patterns and the complete electrode model for the forward problem is computed with the finite element method in order to obtain simulated voltage measurements. The robustness and effectiveness of the method is demonstrated on a simulated chest with errors in input currents, output voltages, electrode placement, and domain modeling.

Direct numerical reconstruction of conductivities in three dimensions using scattering transforms

A direct three dimensional EIT reconstruction algorithm based on complex geometrical optics solutions and a nonlinear scattering transform is presented and implemented for spherically symmetric conductivity distributions. The scattering transform is computed both with a Born approximation and from the forward problem for purposes of comparison. Reconstructions are computed for several test problems. A connection to Calderons linear reconstruction algorithm is established, and reconstructions using both methods are compared.

Reconstructions of piecewise constant conductivities by the D-bar method for electrical impedance tomography

The importance of the solution of the boundary integral equation for the exponentially growing solutions to the Schrodinger equation arising from the 2-D inverse conductivity problems is demonstrated by a study of reconstructions of simple piecewise constant conductivities on a disk from two methods of approximating the scattering transform in the D-bar method and from Calderons linearization method.