t. This paper surveys some of the work our group has done in electrical impedance tomography.

Electrical Impedance Tomography

We review the current state-of-the-art of diffuse optical imaging, which is an emerging technique for functional imaging of biological tissue. It involves generating images using measurements of visible or near-infrared light scattered across large (greater than several centimetres) thicknesses of tissue. We discuss recent advances in experimental methods and instrumentation, and examine new theoretical techniques applied to modelling and image reconstruction. We review recent work on in vivo applications including imaging the breast and brain, and examine future challenges.

https://iopscience.iop.org/article/10.1088/0031-9155/50/4/R01/pdf

Recent advances in diffuse optical imaging.

We review theoretical and numerical studies of the inverse problem of electrical impedance tomography which seeks the electrical conductivity and permittivity inside a body, given simultaneous measurements of electrical currents and potentials at the boundary.

/pdf/electrical-impedance-tomography-3po2ct3t0v.pdf

Electrical impedance tomography

We show that the Dirichlet to Neumann map for the equation ∇·σ∇u =0 in a two-dimensional domain uniquely determines the bounded measurable �

/pdf/calderon-s-inverse-conductivity-problem-in-the-plane-1snio4yhcy.pdf

Calderon's inverse conductivity problem in the plane

This book presents the main results and methods on inverse spectral problems for Sturm-Liouville differential operators and their applications. Inverse problems of spectral analysis consist in recovering operators from their spectral characteristics. Such problems often appear in mathematics, mechanics, physics, electronics, geophysics, meteorology and other branches of natural sciences. Inverse problems also play an important role in solving non-linear evolution equations in mathematical physics. Interest in this subject has been increasing permanently because of the appearance of new important applications, resulting in intensive study of inverse problem theory all over the world.

http://www.gbv.de/dms/goettingen/326677410.pdf

Inverse Sturm-Liouville problems and their applications

The 2D inverse conductivity problem requires one to determine the unknown electrical conductivity distribution inside a bounded domain ??2 from knowledge of the Dirichlet-to-Neumann map. The problem has geophysical, industrial, and medical imaging (electrical impedance tomography) applications. In 1996 A Nachman proved that the Dirichlet-to-Neumann map uniquely determines C2 conductivities. The proof, which is constructive, outlines a direct method for reconstructing the conductivity. In this paper we present an implementation of the algorithm in Nachman's proof. The paper includes numerical results obtained by applying the general algorithms described to two radially symmetric cases of small and large contrast.

https://iopscience.iop.org/article/10.1088/0266-5611/17/5/501/pdf

An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem

The problem this paper addresses is how to use the two-dimensional D-bar method for electrical impedance tomography with experimental data collected on finitely many electrodes covering a portion of the boundary of a body. This requires an approximation of the Dirichlet-to-Neumann, or voltage-to-current density map, defined on the entire boundary of the region, from a finite number of matrix elements of the current-to-voltage map. Reconstructions from experimental data collected on a saline filled tank containing agar heart and lung phantoms are presented, and the results are compared to reconstructions by the NOSER algorithm on the same data.

Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography

A strategy for regularizing the inversion procedure for the two-dimensional D-bar reconstruction algorithm
based on the global uniqueness proof of Nachman [Ann. Math. 143 (1996)] for the ill-posed inverse conductivity problem is presented. The strategy utilizes truncation of the boundary integral equation and the scattering transform. It is shown that this leads to a bound on the error in the scattering transform and a stable reconstruction of the conductivity; an explicit rate of convergence in appropriate Banach spaces is derived as well. Numerical results are also included, demonstrating the convergence of the reconstructed conductivity to the true conductivity as the noise level tends to zero. The results provide a link between two traditions of inverse problems research: theory of regularization and inversion methods based on complex geometrical optics. Also, the procedure is a novel regularized imaging method for electrical impedance tomography.

Regularized d-bar method for the inverse conductivity problem

A three-dimensional reconstruction algorithm in electrical impedance imaging is presented for determining the conductivity distribution beneath the surface of a medium, given surface voltage data measured on a rectangular array of electrodes. Such an electrode configuration may be desirable for using electrical impedance tomography to detect tumors in the human breast. The algorithm is based on linearizing the conductivity about a constant value. Here, the authors describe a simple implementation of the algorithm on a four electrode-by-four-electrode array and the reconstructions obtained from numerical and experimental tank data. The results demonstrate significantly better spatial resolution in the plane of the electrodes than with respect to depth.

A reconstruction algorithm for electrical impedance tomography data collected on rectangular electrode arrays

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.

http://iopscience.iop.org/article/10.1088/0967-3334/27/5/S04/pdf

Jennifer L. Mueller

Papers

An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem

Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography

Regularized d-bar method for the inverse conductivity problem

A reconstruction algorithm for electrical impedance tomography data collected on rectangular electrode arrays

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