Deep D-Bar: Real-Time Electrical Impedance Tomography Imaging With Deep Neural Networks

The mathematical problem for electrical impedance tomography (EIT) is a highly nonlinear ill-posed inverse problem requiring carefully designed reconstruction procedures to ensure reliable image generation. D-bar methods are based on a rigorous mathematical analysis and provide robust direct reconstructions by using a low-pass filtering of the associated nonlinear Fourier data. Similarly to low-pass filtering of linear Fourier data, only using low frequencies in the image recovery process results in blurred images lacking sharp features, such as clear organ boundaries. Convolutional neural networks provide a powerful framework for post-processing such convolved direct reconstructions. In this paper, we demonstrate that these CNN techniques lead to sharp and reliable reconstructions even for the highly nonlinear inverse problem of EIT. The network is trained on data sets of simulated examples and then applied to experimental data without the need to perform an additional transfer training. Results for absolute EIT images are presented using experimental EIT data from the ACT4 and KIT4 EIT systems

Nonlinear Inversion from Partial EIT Data: Computational Experiments

Electrical impedance tomography (EIT) is a non-invasive imaging method in which an unknown physical body is probed with electric currents applied on the boundary, and the internal conductivity distribution is recovered from the measured boundary voltage data. The reconstruction task is a nonlinear and ill-posed inverse problem, whose solution calls for special regularized algorithms, such as D-bar methods which are based on complex geometrical optics solutions (CGOs). In many applications of EIT, such as monitoring the heart and lungs of unconscious intensive care patients or locating the focus of an epileptic seizure, data acquisition on the entire boundary of the body is impractical, restricting the boundary area available for EIT measurements. An extension of the D-bar method to the case when data is collected only on a subset of the boundary is studied by computational simulation. The approach is based on solving a boundary integral equation for the traces of the CGOs using localized basis functions (Haar wavelets). The numerical evidence suggests that the D-bar method can be applied to partial-boundary data in dimension two and that the traces of the partial data CGOs approximate the full data CGO solutions on the available portion of the boundary, for the necessary small $k$ frequencies.Comment: 24 pages, 12 figure

New K-Indices from South Atlantic Observatories : Port Stanley and Ascension Island

Port Stanley and Ascension Island magnetic observatories have been in continuous operation since the early 1990's. These remote South Atlantic locations provide much needed coverage in the global network of geomagnetic observatories and help to monitor the South Atlantic Anomaly. To enhance the production of longitude-sector planetary magnetic activity indices there is a requirement for local 3-hourly K-index values from Port Stanley (PST) observatory. We describe the process followed to establish an automated routine for the derivation of the indices and we assess the congruence of the indices to those available from other suitably located observatories. A similar procedure has been followed for Ascension Island observatory although this is not shown here

A Data-Driven Edge-Preserving D-bar Method for Electrical Impedance Tomography

In Electrical Impedance Tomography (EIT), the internal conductivity of a body is recovered via current and voltage measurements taken at its surface. The reconstruction task is a highly ill-posed nonlinear inverse problem, which is very sensitive to noise, and requires the use of regularized solution methods, of which D-bar is the only proven method. The resulting EIT images have low spatial resolution due to smoothing caused by low-pass filtered regularization. In many applications, such as medical imaging, it is known \emph{a priori} that the target contains sharp features such as organ boundaries, as well as approximate ranges for realistic conductivity values. In this paper, we use this information in a new edge-preserving EIT algorithm, based on the original D-bar method coupled with a deblurring flow stopped at a minimal data discrepancy. The method makes heavy use of a novel data fidelity term based on the so-called {\em CGO sinogram}. This nonlinear data step provides superior robustness over traditional EIT data formats such as current-to-voltage matrices or Dirichlet-to-Neumann operators, for commonly used current patterns.Comment: 24 pages, 11 figure

Direct EIT Reconstructions of Complex Admittivities on a Chest-Shaped Domain in 2-D

Electrical impedance tomography (EIT) is a medical imaging technique in which current is applied on electrodes on the surface of the body, the resulting voltage is measured, and an inverse problem is solved to recover the conductivity and/or permittivity in the interior. Images are then formed from the reconstructed conductivity and permittivity distributions. In the 2-D geometry, EIT is clinically useful for chest imaging. In this work, an implementation of a D-bar method for complex admittivities on a general 2-D domain is presented. In particular, reconstructions are computed on a chest-shaped domain for several realistic phantoms including a simulated pneumothorax, hyperinflation, and pleural effusion. The method demonstrates robustness in the presence of noise. Reconstructions from trigonometric and pairwise current injection patterns are included

Comparing D-Bar and Common Regularization-Based Methods for Electrical Impedance Tomography

Objective: To compare D-bar difference reconstruction with regularized linear reconstruction in electrical impedance tomography. Approach: A standard regularized linear approach using a Laplacian penalty and the GREIT method for comparison to the D-bar difference images. Simulated data was generated using a circular phantom with small objects, as well as a \u27Pac-Man\u27 shaped conductivity target. An L-curve method was used for parameter selection in both D-bar and the regularized methods. Main results: We found that the D-bar method had a more position independent point spread function, was less sensitive to errors in electrode position and behaved differently with respect to additive noise than the regularized methods. Significance: The results allow a novel pathway between traditional and D-bar algorithm comparison

A Direct D-Bar Method for Partial Boundary Data Electrical Impedance Tomography With a Priori Information

Electrical Impedance Tomography (EIT) is a non-invasive imaging modality that uses surface electrical measurements to determine the internal conductivity of a body. The mathematical formulation of the EIT problem is a nonlinear and severely ill-posed inverse problem for which direct D-bar methods have proved useful in providing noise-robust conductivity reconstructions. Recent advances in D-bar methods allow for conductivity reconstructions using EIT measurement data from only part of the domain (e.g., a patient lying on their back could be imaged using only data gathered on the accessible part of the body). However, D-bar reconstructions suffer from a loss of sharp edges due to a nonlinear low-pass filtering of the measured data, and this problem becomes especially marked in the case of partial boundary data. Including a priori data directly into the D-bar solution method greatly enhances the spatial resolution, allowing for detection of underlying pathologies or defects, even with no assumption of their presence in the prior. This work combines partial data D-bar with a priori data, allowing for noise-robust conductivity reconstructions with greatly improved spatial resolution. The method is demonstrated to be effective on noisy simulated EIT measurement data simulating both medical and industrial imaging scenarios

MEME08: A global magnetic field model with satellite data weighting

A new data weighting scheme is introduced for satellite geomagnetic survey data. This scheme allows vector samples of the field to be used at all magnetic latitudes and results in an improved lithospheric model, particularly in the auroral regions. Data weights for 20-second spaced satellite samples are derived from two noise estimators for the sample. Firstly the standard deviation along the 20 seconds of satellite track, centred on each sample, is computed as a measure of local magnetic activity. Secondly a larger-scale noise estimator is defined in terms of a ‘local area vector activity’ (LAVA) index for the sample. This is derived from activity estimated from the geographically nearest magnetic observatories to the sample point. Weighting of satellite data by the inverse-sum-of-squares of these noise estimators leads to a robust model of the field (called ‘Model of Earth’s Magnetic Environment 2008, or ‘MEME08’ - to rhyme with ‘beam’) to about spherical harmonic degree 60. In particular we find that vector data may be used at all latitudes and that there is no need to use particularly complex model parameterizations, regularisation, or prior data correction to remove estimates of un-modelled source fields