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The backprojection method for electrical impedance image reconstruction has been adapted for the opposing current drive configuration implemented in the second generation of Aberdeen impedance imaging systems. The logarithmic conformal transformation is used to solve the Forward problem for a two-dimensional homogeneous medium of circular cross-section. Pixel weights of backprojection are calculated from the normalised distances of the pixel centres from the boundary side of backprojection. An experimental solution to the Forward problem is a homogeneous medium of irregular cross-section and three-dimensional boundary is proposed and implemented. A thorax phantom was built for this purpose using radiotherapy moulding techniques. The potential distribution in this phantom was measured using a tetrapolar inpedance measuring device and the equipotential lines falling on the electrodes were plotted. A reconstruction matrix capable of reconstructing dynamic impedance images of the thorax was formulated. Images representing resistivity change distributions between maximum inspiration and maximum expiration have been reconstructed. These thorax cross-section images show the most faithful representation of the expected resistivity changes due to respiration.

Thesis (MEng)--University of South Australia, 1995

Thesis (MEng)--University of South Australia, 1995

In Electrical Impedance Tomography the conductivity of an object is estimated from boundary measurements. An array of electrodes is attached to the surface of the object and current stimuli are applied via these electrodes. The resulting volt ages are measured. The process of estimating the conductivity as a function of space inside the object from voltage measurements at the surface is called reconstruction. Mathematically the ElT reconstruction is a non linear inverse problem, the stable solution of which requires regularisation nwthods. Most common regularisation methods impose that the reconstructed image should be smooth. Such methods confer stability to the reconstruction process, but limit the capability of describing sharp variations in the sought parameter. In this thesis two new methods of regularisation are proposed. The first method, Gallssian anisotropic regularisation, enhances the reconstruction of sharp conductivity changes occurring at the interface between a contrasting object and the background. As such changes are step changes, reconstruction with traditional smoothing regularisation techniques is unsatisfactory. The Gaussian anisotropic filtering works by incorporating prior structural information. The approximate knowledge of the shapes of contrasts allows us to relax the smoothness in the direction normal to the expected boundary. The construction of Gaussian regularisation filters that express such directional properties on the basis of the structural information is discussed, and the results of numerical experiments are analysed. The method gives good results when the actual conductivity distribution is in accordance with the prior information. When the conductivity distribution violates the prior information the method is still capable of properly locating the regions of contrast. The second part of the thesis is concerned with regularisation via the total variation functional. This functional allows the reconstruction of discontinuous parameters. The properties of the functional are briefly introduced, and an application in inverse problems in image denoising is shown. As the functional is non-differentiable, numerical difficulties are encountered in its use. The aim is therefore to propose an efficient numerical implementation for application in ElT. Several well known optimisation methods arc analysed, as possible candidates, by theoretical considerations and by numerical experiments. Such methods are shown to be inefficient. The application of recent optimisation methods called primal- dual interior point methods is analysed be theoretical considerations and by numerical experiments, and an efficient and stable algorithm is developed. Numerical experiments demonstrate the capability of the algorithm in reconstructing sharp conductivity profiles.

Introduction: Electrical Impedance Tomography (EIT) is an emerging clinical imaging technique. Functional EIT by Evoked Response (fEITER) was developed at the University of Manchester as a high-speed, functional brain imaging device for use at the bedside. This 32-electrode EIT system applies an injection frequency of 10kHz and captures data using a 10ms temporal resolution. This thesis reports on the first volunteer and patient trials undertaken using fEITER for the following conditions: (a) flashing visual sequence - 14 awake volunteers; (b) a voluntary Valsalva manoeuvre (VM) - 15 awake volunteers and (c) during the induction of anaesthesia - 16 elective surgical patients. Aims: The research presented in this thesis was undertaken to differentiate between noise and physiological changes in raw fEITER data signals. Methods: SNR was determined for fEITER. Raw fEITER signals were pre-processed to reduce noise and dominant trends before multiple comparisons between reference and stimulus data were undertaken. Histograms and ROC curves were produced to illustrate the difference between reference and stimulus fEITER data. AUC values for single-subject and pooled ROC curves were calculated to determine whether fEITER data can be reliably differentiated between reference and stimulus conditions. Approximate Entropy (ApEn) was applied to evaluate the regularity of high frequency components within fEITER data for each trial condition. Results: Average SNR values for fEITER acquired using mesh and physical phantoms ranged from 62.94dB to 63.58dB, and 28.29dB to 31.45dB respectively. The following AUC values were acquired: Visual stimulus-frontal electrode pairs and electrode pairs overlying the visual cortex 0.520 and 0.505 respectively; VM: 0.658; and induction of anaesthesia: 0.547. The VM induced the greatest difference between pooled reference and stimulus data. Visual stimulation and induction of anaesthesia data showed poor distinction between pooled reference and stimulus data, although some single subject data did show a significant response. No significant differences were acquired for the comparison of ApEn-reference and ApEn-stimulus data for all trial conditions using a Wilcoxon's signed ranks test (visual stimulus-frontal electrode pairs: upper p = 0.998, visual stimulus-electrode overlying the visual cortex: upper p = 0.980; the VM: upper p = 0.976, and induction of anaesthesia: p = 0.912). Discussion: Although single-subject and pooled fEITER data recorded during the VM produced the greatest differences between reference and stimulus measurements, stimuli such as visual flashes and induction of anaesthesia may not be large enough to induce quantifiable changes between reference and stimulus data recorded from single electrode pairs. Collectively, these results provide little evidence to show that pre-processing of raw fEITER data amplifies features in fEITER waveforms which may be representative of physiological changes induced by an applied stimulus.

Electrical impedance tomography (EIT) is an imaging modality with many possible practical applications. It is mainly used for geophysical applications, for which it is called electrical resistivity tomography. There have also been many proposed medical applications such as respiratory monitoring and breast tumour screening. Although there have been many uniqueness and stability results published over the last few decades, most of the results are in the context of the theoretical continuous problem. In practice however, we almost always have to solve a discretised problem for which very few theoretical results exist. In this thesis we aim to bridge the gap between the continuous and discrete problems. The first problem we solve is the three-dimensional triangulation problem of uniquely embedding a tetrahedral mesh in R3. We parameterise the problem in terms of dihedral angles and we provide a constructive procedure for identifying the independent angles and the independent set of constraints that the dependent angles must satisfy. We then use the implicit function theorem to prove that the embedding is locally unique. We also present a numerical example to illustrate that the result works in practice. Without the understanding of the geometric constraints involved in embedding a three-dimensional triangulation, we cannot solve more complex problems involving embeddings of finite element meshes. We next investigate the discrete EIT problem for anisotropic conductivity. It is well known that the entries of the finite element system matrix for piecewise linear potential and piecewise constant conductivity are equivalent to conductance values of resistors defined on the edges of the finite element mesh. We attempt to tackle the problem of embedding a finite element mesh in R3, such that it is consistent with some known edge conductance values. It is a well known result that for the anisotropic conductivity problem, the boundary data is invariant under diffeomorphisms that fix the boundary. Before investigating this effect on the discrete case, we define the linear map from conductivities to edge conductances and investigate the injectivity of this map for a simplistic example. This provides an illustrative example of how a poor choice of finite element mesh can result in a non-unique solution to the discrete inverse problem of EIT. We then extend the investigation to finding interior vertex positions and conductivity distributions that are consistent with the known edge conductances. The results show that if the total number of interior vertex coordinates and anisotropic conductivity variables is larger than the number of edges in the mesh, then there exist discrete diffeomorphisms that perturb the vertices and conductivities such that no change in the edge conductances is observed. We also show that the non-uniqueness caused by the non-injectivity of the linear map has a larger effect than the non-uniqueness caused by diffeomorphism invariance.

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Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (p. [113]-116).