The born approximation and Calderón's method for reconstruction of conductivities in 3-D
Kim Knudsen Jennifer L. Mueller
Two algorithms for the direct reconstruction of conductivities in a bounded domain in $\mathbb{R}^3$ from surface measurements of the solutions to the conductivity equation are presented. The algorithms are based on complex geometrical optics solutions and a nonlinear scattering transform. We test the algorithms on three numerically simulated examples, including an example with a complex coefficient. The spatial resolution and amplitude of the examples are well-reconstructed.
keywords: Reconstruction algorithm Electrical Impedance Tomography Inverse conductivity problem Calderón problem
2D EIT reconstructions using Calderon's method
Jutta Bikowski Jennifer L. Mueller
The pioneering work ''On an inverse boundary value problem'' by A. Calderón has inspired a multitude of research, both theoretical and numerical, on the inverse conductivity problem (ICP). The problem has an important application in a medical imaging technique known as electrical impedance tomography (EIT) in which currents are applied on electrodes on the surface of a body, the resulting voltages are measured, and the ICP is solved to determine the conductivity distribution in the interior of the body, which is then displayed to form an image. In this article, the reconstruction method proposed by Calderón is implemented in 2D for both simulated and experimental data including perfusion data collected on a human chest.
keywords: electrical impedance tomography Calderón's method. inverse conductivity problem
Regularized D-bar method for the inverse conductivity problem
Kim Knudsen Matti Lassas Jennifer L. Mueller Samuli Siltanen
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.
keywords: ill-posed problem electrical impedance tomography inverse problem regularization. inverse conductivity problem
Direct electrical impedance tomography for nonsmooth conductivities
Kari Astala Jennifer L. Mueller Lassi Päivärinta Allan Perämäki Samuli Siltanen
A new reconstruction algorithm is presented for eit in dimension two, based on the constructive uniqueness proof given by Astala and Päivärinta in [Ann. of Math. 163 (2006)]. The method is non-iterative, provides a noise-robust solution of the full nonlinear eit problem, and applies to more general conductivities than previous approaches. In particular, the new algorithm applies to piecewise smooth conductivities. Reconstructions from noisy and non-noisy simulated data from conductivity distributions representing a cross-sections of a chest and a layered medium such as stratified flow in a pipeline are presented. The results suggest that the new method can recover useful and reasonably accurate eit images from data corrupted by realistic amounts of measurement noise. In particular, the dynamic range in medium-contrast conductivities is reconstructed remarkably well.
keywords: Electrical impedance tomography. Nonlinear Fourier transform Complex geometrical optics solution Beltrami equation Conductivity equation Inverse conductivity problem Inverse problem Quasiconformal map Numerical solver
A real-time D-bar algorithm for 2-D electrical impedance tomography data
Melody Dodd Jennifer L. Mueller
The aim of this paper is to show the feasibility of the D-bar method for real-time 2-D EIT reconstructions. A fast implementation of the D-bar method for reconstructing conductivity changes on a 2-D chest-shaped domain is described. Cross-sectional difference images from the chest of a healthy human subject are presented, demonstrating what can be achieved in real time. The images constitute the first D-bar images from EIT data on a human subject collected on a pairwise current injection system.
keywords: inverse problems. real-time imaging D-bar algorithm electrical impedance tomography

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