Nick Polydorides

A Matlab toolkit for three-dimensional electrical impedance tomography: a contribution to the Electrical Impedance and Diffuse Optical Reconstruction Software project

The objective of the Electrical Impedance and Diffuse Optical Reconstruction Software project is to develop freely available software that can be used to reconstruct electrical or optical material properties from boundary measurements. Nonlinear and ill posed problems such as electrical impedance and optical tomography are typically approached using a finite element model for the forward calculations and a regularized nonlinear solver for obtaining a unique and stable inverse solution. Most of the commercially available finite element programs are unsuitable for solving these problems because of their conventional inefficient way of calculating the Jacobian, and their lack of accurate electrode modelling. A complete package for the two-dimensional EIT problem was officially released by Vauhkonen et al at the second half of 2000. However most industrial and medical electrical imaging problems are fundamentally three-dimensional. To assist the development we have developed and released a free toolkit of Matlab routines which can be employed to solve the forward and inverse EIT problems in three dimensions based on the complete electrode model along with some basic visualization utilities, in the hope that it will stimulate further development. We also include a derivation of the formula for the Jacobian (or sensitivity) matrix based on the complete electrode model.

Krylov subspace iterative techniques: on the detection of brain activity with electrical impedance tomography

In this paper, we review some numerical techniques based on the linear Krylov subspace iteration that can be used for the efficient calculation of the forward and the inverse electrical impedance tomography problems. Exploring their computational advantages in solving large-scale systems of equations, we specifically address their implementation in reconstructing localized impedance changes occurring within the human brain. If the conductivity of the head tissues is assumed to be real, the preconditioned conjugate gradients (PCGs) algorithm can be used to calculate efficiently the approximate forward solution to a given error tolerance. The performance and the regularizing properties of the PCG iteration for solving ill-conditioned systems of equations (PCGNs) is then explored, and a suitable preconditioning matrix is suggested in order to enhance its convergence rate. For image reconstruction, the nonlinear inverse problem is considered. Based on the Gauss-Newton method for solving nonlinear problems we have developed two algorithms that implement the PCGN iteration to calculate the linear step solution. Using an anatomically detailed model of the human head and a specific scalp electrode arrangement, images of a simulated impedance change inside brains white matter have been reconstructed.

Improving the forward solver for the complete electrode model in EIT using algebraic multigrid

Image reconstruction in electrical impedance tomography is an ill-posed nonlinear inverse problem. Linearization techniques are widely used and require the repeated solution of a linear forward problem. To account correctly for the presence of electrodes and contact impedances, the so-called complete electrode model is applied. Implementing a standard finite element method for this particular forward problem yields a linear system that is symmetric and positive definite and solvable via the conjugate gradient method. However, preconditioners are essential for efficient convergence. Preconditioners based on incomplete factorization methods are commonly used but their performance depends on user-tuned parameters. To avoid this deficiency, we apply black-box algebraic multigrid, using standard commercial and freely available software. The suggested solution scheme dramatically reduces the time cost of solving the forward problem. Numerical results are presented using an anatomically detailed model of the human head.

Electrode configurations for improved spatial resolution in electrical impedance tomography

This study addresses the numerical treatment applied to the singular values of the sensitivity matrix in the presence of noisy measurements, subsequently suggesting electrode configurations that provide sensitivities with improved characteristics. We begin by examining the impact of the individual singular values on the spatial resolution of the image and then proceed to express the generalized Tikhonov regularization in terms of the generalized singular value decomposition in order to demonstrate how the reconstructed image is synthesized from the individual energy components. The electrode segmentation scheme is then introduced as a feasible configuration offering efficient and improved resolution impedance imaging. Finally, the regularized total least squares algorithm is implemented to provide the linear step solution within Newtons iterative scheme. Images of several reconstructed inhomogeneities are presented, using simulated measurements obtained from the segmented electrodes system.

Damage identification in carbon fiber reinforced polymer plates using electrical resistance tomography mapping

This study addresses the issue of structural damage identification and location in carbon fiber reinforced polymer plates using electrical measurements. Electrical resistance tomography is presented as a method for structural damage localization in composite parts. A set of electrodes is fixed on the edges of the part and combinations of DC current injections and voltage measurements are applied to the system. The change of voltage between different times in the part’s service life (e.g. start and degraded) are monitored. These sets of measurements are used as input to inversely calculate conductivity maps for the complete composite part and thus indirectly assess its structural health. Such processes are inherently ill-posed. Data post-processing approaches are proposed here to diminish this uncertainty and to conclude to an optimally converge solution of the inverse problem. To assist the process, a material-originating mathematical constraint is introduced. The method is applied on carbon fiber reinforced polymer plates for different damage modes. Experimental recordings show that the analysis of electrical fields allows detecting the presence of damage. Discontinuities as small as 0.1% of the inspected area can be sensed. The proposed data post-processing techniques were applied and conductivity maps were calculated. The results show that using these techniques locating damage is possible with less than 10% error. Material-based constraints greatly enhance the prediction of the data post-processing techniques. It is believed that by overcoming certain implementation issues, electrical resistance tomography could evolve in the direction of a non-destructive evaluation or a structural health monitoring technique for composite structures.

Progress towards non-intrusive optical measurement of gas turbine exhaust species distributions

We report on the development of three systems intended to provide fast, non-intrusive measurement of cross-sectional distributions of pollutant species within gas turbine exhaust flows, during ground-based testing. This research is motivated by the need for measurement systems to support the introduction of technologies for reducing the environmental impact of civil aviation. Tomographic techniques will allow estimation of the distributions of CO2, unburnt hydrocarbons (UHC), and soot, without obstruction of the exhaust, bypass or entrained flows, from measurements made in a plane immediately aft of the engine.

Effect of structured packing on EIT image reconstruction

Broadly used in chemical engineering, structured packing is beneficial for maximizing liquid spreading. In this paper, image reconstruction for electrical impedance tomography (EIT) sensors with structured packing is investigated. To evaluate the effect of structured packing on image quality, characterization of EIT sensor with structured packing is firstly carried out. Furthermore, fringe effect in image reconstruction using these sensors is presented. Based on the study, it is found that compared with standard Tikhonov regularization and Landweber iteration, using 2-order difference operators, such as Gaussian-Laplace operator, can effectively decrease the degradation of image quality. The study also discusses the feasibility of using homogeneous sensitivity matrix in image reconstruction. Numerical analysis is carried out to demonstrate the validation of the study.

The linearization error in electrical impedance tomography

Abstract—In borehole electromagnetic tomography and resistivity survey a linearized model approximation is often used, in the context of regularized regression, to image the conductivity distribution in a domain of interest. Due to the error introduced by the simplified model, quantitative image reconstruction becomes challenging without implementing a nonlinear algorithm. We derive a closed form expression of the linearization error in electrical impedance tomography based on the complete electrode model. The error term is expressed in an integral form involving the gradient of the perturbed electric potential in the interior of the domain and renders itself readily available for analytical or numerical computation. For real isotropic conductivity inhomogeneities with piecewise uniform characteristic functions the perturbed potential field can be shown to satisfy Poisson’s equation with Robin boundary conditions and interior point sources positioned at the interfaces of the inclusions. Simulation experiments using a finite element method have been performed to validate these results.

An efficient approach for limited-data chemical species tomography and its error bounds.

We present a computationally efficient reconstruction method for the limited-data chemical species tomography problem that incorporates projection of the unknown gas concentration function onto a low-dimensional subspace, and regularization using prior information obtained from a simple flow model. In this context, the contribution of this work is on the analysis of the projection-induced data errors and the calculation of bounds for the overall image error incorporating the impact of projection and regularization errors as well as measurement noise. As an extension to this methodology, we present a variant algorithm that preserves the positivity of the concentration image.

Subsecond observations of EIT voltage changes on the human scalp due to brain stimulus

A pilot study has investigated the feasibility of imaging human brain function using an electrical impedance tomography (EIT) system time-locked to an evoked response (ER) system. A sixteen-electrode planar EIT configuration was used with polar current injection. We report here measurements on two volunteer patients who were fully awake in all tests. Reference data (with no applied stimulus) yield EIT nearest-neighbour voltage differences over the range 5-28 mV. In comparison with forward calculations [C.M. Towers et al. (2000)], these data suggest a value of skull conductivity of the order of 0.05 S/m. Visual and auditory stimuli were applied as single discrete events to evoke neural responses (VER and AER respectively). In each case, EIT data acquisition commenced at a time between 70 and 740 ms later, taking 308 ms to complete. Average values of voltage pair data over many frames are presented here, with the emphasis on the VER data. When comparing data taken under stimulus conditions against the reference data, voltage differences of up to approximately 3 mV are observed in both AER and VER cases. We attribute these voltage changes to synaptic activity. Preliminary reconstructed images of conductivity are discussed.