A parametric level set-based approach to difference imaging in electrical impedance tomography

This paper presents a novel difference imaging approach based on the recently developed parametric level set (PLS) method for estimating the change in a target conductivity from electrical impedance tomography measurements. As in conventional difference imaging, the reconstruction of conductivity change is based on data sets measured from the surface of a body before and after the change. The key feature of the proposed approach is that the conductivity change to be reconstructed is assumed to be piecewise constant, while the geometry of the anomaly is represented by a shape-based PLS function employing Gaussian radial basis functions (GRBFs). The representation of the PLS function by using GRBF provides flexibility in describing a large class of shapes with fewer unknowns. This feature is advantageous, as it may significantly reduce the overall number of unknowns, improve the condition number of the inverse problem, and enhance the computational efficiency of the technique. To evaluate the proposed PLS-based difference imaging approach, results obtained via simulation, phantom study, and in vivo pig data are studied. We find that the proposed approach tolerates more modeling errors and leads to a significant improvement in image quality compared with the conventional linear approach

Parametric Level Set Methods for Inverse Problems

In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of parameterizing the level set function results a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, re-initialization and use of signed distance function. Moreover, we show that from a computational point of view, low order representation of the problem paves the path for easier use of Newton and quasi-Newton methods. Specifically for the purposes of this paper, we parameterize the level set function in terms of adaptive compactly supported radial basis functions, which used in the proposed manner provides flexibility in presenting a larger class of shapes with fewer terms. Also they provide a "narrow-banding" advantage which can further reduce the number of active unknowns at each step of the evolution. The performance of the proposed approach is examined in three examples of inverse problems, i.e., electrical resistance tomography, X-ray computed tomography and diffuse optical tomography

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

Nonstationary shape estimation in electrical impedance tomography using a parametric level set-based extended Kalman filter approach

This paper presents a parametric level set based reconstruction method for non-stationary applications using electrical impedance tomography (EIT). Owing to relatively low signal to noise ratios in EIT measurement systems and the diffusive nature of EIT, reconstructed images often suffer from low spatial resolution. In addressing these challenges, we propose a computationally efficient shape-estimation approach where the conductivity distribution to be reconstructed is assumed to be piecewise constant, and the region boundaries are assumed to be non-stationary in the sense that the characteristics of region boundaries change during measurement time. The EIT inverse problem is formulated as a state estimation problem in which the system is modeled with a state equation and an observation equation. Given the temporal evolution model of the boundaries and observation model, the objective is to estimate a sequence of states for the nonstationary region boundaries. The implementation of the approach is based on the finite element method and a parametric representation of the region boundaries using level set functions. The performance of the proposed approach is evaluated with simulated examples of thorax imaging, using noisy synthetic data and experimental data from a laboratory setting. In addition, robustness studies of the approach w.r.t the modeling errors caused by inaccurately known boundary shape, non-homogeneous background and varying conductivity values of the targets are carried out and it is found that the proposed approach tolerates such kind of modeling errors, leading to good reconstructions in non-stationary situations

Empirical validation of statistical parametric mapping for group imaging of fast neural activity using electrical impedance tomography

Electrical impedance tomography (EIT) allows for the reconstruction of internal conductivity from surface measurements. A change in conductivity occurs as ion channels open during neural activity, making EIT a potential tool for functional brain imaging. EIT images can have  >10 000 voxels, which means statistical analysis of such images presents a substantial multiple testing problem. One way to optimally correct for these issues and still maintain the flexibility of complicated experimental designs is to use random field theory. This parametric method estimates the distribution of peaks one would expect by chance in a smooth random field of a given size. Random field theory has been used in several other neuroimaging techniques but never validated for EIT images of fast neural activity, such validation can be achieved using non-parametric techniques. Both parametric and non-parametric techniques were used to analyze a set of 22 images collected from 8 rats. Significant group activations were detected using both techniques (corrected p  <  0.05). Both parametric and non-parametric analyses yielded similar results, although the latter was less conservative. These results demonstrate the first statistical analysis of such an image set and indicate that such an analysis is an approach for EIT images of neural activity

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

Improving the forward model for electrical impedance tomography of brain function through rapid generation of subject specific finite element models

Electrical Impedance Tomography (EIT) is a non-invasive imaging method which allows internal electrical impedance of any conductive object to be imaged by means of current injection and surface voltage measurements through an array of externally applied electrodes. The successful generation of the image requires the simulation of the current injection patterns on either an analytical or a numerical model of the domain under examination, known as the forward model, and using the resulting voltage data in the inverse solution from which images of conductivity changes can be constructed. Recent research strongly indicates that geometric and anatomical conformance of the forward model to the subject under investigation significantly affects the quality of the images. This thesis focuses mainly on EIT of brain function and describes a novel approach for the rapid generation of patient or subject specific finite element models for use as the forward model. After introduction of the topic, methods of generating accurate finite element (FE) models using commercially available Computer-Aided Design (CAD) tools are described and show that such methods, though effective and successful, are inappropriate for time critical clinical use. The feasibility of warping or morphing a finite element mesh as a means of reducing the lead time for model generation is then presented and demonstrated. This leads on to the description of methods of acquiring and utilising known system geometry, namely the positions of electrodes and registration landmarks, to construct an accurate surface of the subject, the results of which are successfully validated. The outcome of this procedure is then used to specify boundary conditions to a mesh warping algorithm based on elastic deformation using well-established continuum mechanics procedures. The algorithm is applied to a range of source models to empirically establish optimum values for the parameters defining the problem which can successfully generate meshes of acceptable quality in terms of discretization errors and which more accurately define the geometry of the target subject. Further validation of the algorithm is performed by comparison of boundary voltages and image reconstructions from simulated and laboratory data to demonstrate that benefits in terms of image artefact reduction and localisation of conductivity changes can be gained. The processes described in the thesis are evaluated and discussed and topics of further work and application are described

Iterative Updating of Model Error for Bayesian Inversion

In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when optimization algorithms are used to find a single estimate, or to speed up Markov chain Monte Carlo (MCMC) calculations in the Bayesian framework. The use of an approximate model introduces a discrepancy, or modeling error, that may have a detrimental effect on the solution of the ill-posed inverse problem, or it may severely distort the estimate of the posterior distribution. In the Bayesian paradigm, the modeling error can be considered as a random variable, and by using an estimate of the probability distribution of the unknown, one may estimate the probability distribution of the modeling error and incorporate it into the inversion. We introduce an algorithm which iterates this idea to update the distribution of the model error, leading to a sequence of posterior distributions that are demonstrated empirically to capture the underlying truth with increasing accuracy. Since the algorithm is not based on rejections, it requires only limited full model evaluations. We show analytically that, in the linear Gaussian case, the algorithm converges geometrically fast with respect to the number of iterations. For more general models, we introduce particle approximations of the iteratively generated sequence of distributions; we also prove that each element of the sequence converges in the large particle limit. We show numerically that, as in the linear case, rapid convergence occurs with respect to the number of iterations. Additionally, we show through computed examples that point estimates obtained from this iterative algorithm are superior to those obtained by neglecting the model error.Comment: 39 pages, 9 figure