Regularized D-bar method for the inverse conductivity problem
Kim Knudsen Matti Lassas Jennifer L. Mueller Samuli Siltanen
Inverse Problems & Imaging 2009, 3(4): 599-624 doi: 10.3934/ipi.2009.3.599
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
Inverse Problems & Imaging 2011, 5(3): 531-549 doi: 10.3934/ipi.2011.5.531
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
Inverse Problems & Imaging 2014, 8(4): 1013-1031 doi: 10.3934/ipi.2014.8.1013
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
The born approximation and Calderón's method for reconstruction of conductivities in 3-D
Kim Knudsen Jennifer L. Mueller
Conference Publications 2011, 2011(Special): 844-853 doi: 10.3934/proc.2011.2011.844
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
Inverse Problems & Imaging 2008, 2(1): 43-61 doi: 10.3934/ipi.2008.2.43
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
Use of an optimized spatial prior in D-bar reconstructions of EIT tank data
Melody Alsaker Jennifer L. Mueller
Inverse Problems & Imaging 2018, 12(4): 883-901 doi: 10.3934/ipi.2018037

The aim of this paper is to demonstrate the feasibility of using spatial a priori information in the 2-D D-bar method to improve the spatial resolution of EIT reconstructions of experimentally collected data. The prior consists of imperfectly known information about the spatial locations of inclusions and the assumption that the conductivity is a mollified piecewise constant function. The conductivity values for the prior are constructed using a novel method in which a nonlinear constrained optimization routine is used to select the values for the piecewise constant function that give the best fit to the scattering transform computed from the measured data in a disk. The prior is then included in the high-frequency components of the scattering transform and in the computation of the solution of the D-bar equation, with weights to control the influence of the prior. In addition, a new technique is described for selecting regularization parameters to truncate the measured scattering data, in which complex scattering frequencies for which the values of the scattering transform differ greatly from those in the scattering prior are omitted. The effectiveness of the method is demonstrated on EIT data collected on saline-filled tanks with agar heart and lungs with various added inhomogeneities.

keywords: Electrical impedance tomography D-bar methods reconstruction algorithm a priori data nonlinear optimization

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