Resolution-Controlled Conductivity Discretization in Electrical Impedance Tomography

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

EIT Reconstruction Algorithms: Pitfalls, Challenges and Recent Developments

We review developments, issues and challenges in Electrical Impedance Tomography (EIT), for the 4th Workshop on Biomedical Applications of EIT, Manchester 2003. We focus on the necessity for three dimensional data collection and reconstruction, efficient solution of the forward problem and present and future reconstruction algorithms. We also suggest common pitfalls or ``inverse crimes'' to avoid.Comment: A review paper for the 4th Workshop on Biomedical Applications of EIT, Manchester, UK, 200

An investigation of planar array system artefacts generated within an electrical impedance mammography system developed for breast cancer detection

An Electrical Impedance Mammography (EIM) planar array imaging system is being developed at the University of Sussex for the detection of breast cancers. Investigations have shown that during data collection, systematic errors and patient artefacts are frequently introduced during signal acquisition from different electrodes pairs. This is caused, in particular, by the large variations in the electrode-skin contact interface conditions occurring between separate electrode positions both with the same and different patients. As a result, the EIM image quality is seriously affected by these errors. Hence, this research aims to experimentally identify, analyse and propose effective methods to reduce the systematic errors at the electrode-skin interface. Experimental studies and subsequent analysis is presented to determine what ratio of electrode blockage seriously affects the acquired raw data which may in turn compromise the reconstruction. This leads to techniques for the fast and accurate detection of any such occurrences. These methodologies can be applied to any planar array based EIM system

The regularized monotonicity method: detecting irregular indefinite inclusions

In inclusion detection in electrical impedance tomography, the support of perturbations (inclusion) from a known background conductivity is typically reconstructed from idealized continuum data modelled by a Neumann-to-Dirichlet map. Only few reconstruction methods apply when detecting indefinite inclusions, where the conductivity distribution has both more and less conductive parts relative to the background conductivity; one such method is the monotonicity method of Harrach, Seo, and Ullrich. We formulate the method for irregular indefinite inclusions, meaning that we make no regularity assumptions on the conductivity perturbations nor on the inclusion boundaries. We show, provided that the perturbations are bounded away from zero, that the outer support of the positive and negative parts of the inclusions can be reconstructed independently. Moreover, we formulate a regularization scheme that applies to a class of approximative measurement models, including the Complete Electrode Model, hence making the method robust against modelling error and noise. In particular, we demonstrate that for a convergent family of approximative models there exists a sequence of regularization parameters such that the outer shape of the inclusions is asymptotically exactly characterized. Finally, a peeling-type reconstruction algorithm is presented and, for the first time in literature, numerical examples of monotonicity reconstructions for indefinite inclusions are presented.Comment: 28 pages, 7 figure

Comparison of linear and non-linear monotononicity-based shape reconstruction using exact matrix characterizations

Detecting inhomogeneities in the electrical conductivity is a special case of the inverse problem in electrical impedance tomography, that leads to fast direct reconstruction methods. One such method can, under reasonable assumptions, exactly characterize the inhomogeneities based on monotonicity properties of either the Neumann-to-Dirichlet map (non-linear) or its Fr\'echet derivative (linear). We give a comparison of the non-linear and linear approach in the presence of measurement noise, and show numerically that the two methods give essentially the same reconstruction in the unit disk domain. For a fair comparison, exact matrix characterizations are used when probing the monotonicity relations to avoid errors from numerical solution to PDEs and numerical integration. Using a special factorization of the Neumann-to-Dirichlet map also makes the non-linear method as fast as the linear method in the unit disk geometry.Comment: 18 pages, 5 figures, 1 tabl

Further investigation of a contactless patient-electrode interface of an Electrical Impedance Mammography system

The Sussex Mk4 Electrical Impedance Mammography (EIM) system is a novel instrument, designed for the detection of early breast cancer, based upon Electrical Impedance Tomography (EIT). Many innovations in the field have been incorporated in the design improving both signal distribution and response. This paper investigates the behaviour of the contactless patient-electrode interface. The interface was studied in detail using phantom and healthy volunteer, in-vivo, data. Our findings show the necessity for the careful design of electrode enclosure so that the response of the system is not affected by the unpredictable positioning of the breast; it closely mimics those conditions seen when using the phantom. The paper includes a number of possible designs and their individual characteristics. In addition an explanation on the unanticipated effects and solutions for such are described. © 2010 IOP Publishing Ltd

Distinguishability revisited: depth dependent bounds on reconstruction quality in electrical impedance tomography

The reconstruction problem in electrical impedance tomography is highly ill-posed, and it is often observed numerically that reconstructions have poor resolution far away from the measurement boundary but better resolution near the measurement boundary. The observation can be quantified by the concept of distinguishability of inclusions. This paper provides mathematically rigorous results supporting the intuition. Indeed, for a model problem lower and upper bounds on the distinguishability of an inclusion are derived in terms of the boundary data. These bounds depend explicitly on the distance of the inclusion to the boundary, i.e. the depth of the inclusion. The results are obtained for disk inclusions in a homogeneous background in the unit disk. The theoretical bounds are verified numerically using a novel, exact characterization of the forward map as a tridiagonal matrix.Comment: 25 pages, 6 figure

Sound speed uncertainty in acousto-electric tomography

The goal in acousto-electric tomography is to reconstruct an image of the unknown electric conductivity inside an object from boundary measurements of electrostatic currents and voltages collected while the object is penetrated by propagating ultrasound waves. This problem is a coupled-physics inverse problem. Accurate knowledge of the propagating ultrasound wave is usually assumed and required, but in practice tracking the propagating wave is hard due to inexact knowledge of the interior acoustic properties of the object. In this work, we model uncertainty in the sound speed of the acoustic wave, and formulate a suitable reconstruction method for the interior power density and conductivity. We also establish theoretical error bounds, and show that the suggested approach can be understood as a regularization strategy for the inverse problem. Finally, we numerically simulate the sound speed variations from a numerical breast tissue model, and computationally explore the effect of using an inaccurate sound speed on the error in reconstructions. Our results show that with reasonable uncertainty in the sound speed reliable reconstruction is still possible.Peer reviewe