Problem Of Impedance Tomography Research Articles

An iterative thresholding algorithm for the inverse problem of electrical resistance tomography

Image reconstruction in Electrical Resistance Tomography (ERT) is an ill-posed nonlinear inverse problem. Considering the sparsity property of ERT model, in this paper, we replace the conventional l2 regularization penalty term by weighted lp(1≤p<2) penalty term. To overcome the non-quadratic property, a surrogate term is added to the objective function. An interesting condition is that the classical methods (e.g. SVD, Landweber iteration) can be used to solve the lp(1≤p<2) least squares problems. Both typical and complicated distributions (e.g. annular and cross-shape) have been examined using a 16-electrode configuration based on the finite element method (FEM) software COMSOL. The simulated results demonstrate the feasibility of the proposed algorithm, and compared to the l2 regularization method, the proposed algorithms can produce images of higher quality, which are evaluated both qualitatively and quantitatively.

Hybrid Topological Derivative-Gradient Based Methods for Nondestructive Testing

This paper is devoted to the reconstruction of objects buried in a medium and their material properties by hybrid topological derivative-gradient based methods. After illustrating the techniques in time-harmonic acoustic problems with different boundary conditions and in electrical impedance tomography problems with continuous Neumann conditions, we extend the hybrid method for a realistic model in tomography where the boundary conditions are given at a discrete set of electrodes.

Study of noise effects in electrical impedance tomography with resistor networks

We present a study of the numerical solution of the two dimensional electrical impedance tomography problem, with noisy measurements of the Dirichlet to Neumann map. The inversion uses parametrizations of the conductivity on optimal grids. The grids are optimal in the sense that finite volume discretizations on them give spectrally accurate approximations of the Dirichlet to Neumann map. The approximations are Dirichlet to Neumann maps of special resistor networks, that are uniquely recoverable from the measurements. Inversion on optimal grids has been proposed and analyzed recently, but the study of noise effects on the inversion has not been carried out. In this paper we present a numerical study of both the linearized and the nonlinear inverse problem. We take three different parametrizations of the unknown conductivity, with the same number of degrees of freedom. We obtain that the parametrization induced by the inversion on optimal grids is the most efficient of the three, because it gives the smallest standard deviation of the maximum a posteriori estimates of the conductivity, uniformly in the domain. For the nonlinear problem we compute the mean and variance of the maximum a posteriori estimates of the conductivity, on optimal grids. For small noise, we obtain that the estimates are unbiased and their variance is very close to the optimal one, given by the Cramer-Rao bound. For larger noise we use regularization and quantify the trade-off between reducing the variance and introducing bias in the solution. Both the full and partial measurement setups are considered.

Inverse Problems for Partial Differential Equations

This workshop brought together mathematicians engaged in different aspects of inverse problems for partial differential equations. Classical topics such as the inverse problems of impedance tomography and scattering theory as well as new developments such as interior transmission eigenvalues were discussed.

Iterative method for solving a three-dimensional electrical impedance tomography problem in the case of piecewise constant conductivity and one measurement on the boundary

The problem of electrical impedance tomography in a bounded three-dimensional domain with a piecewise constant electrical conductivity is considered. The boundary of the inhomogeneity is assumed to be unknown. The inverse problem is to determine the surface that is the boundary of the inhomogeneity from given measurements of the potential and its normal derivative on the outer boundary of the domain. An iterative method for solving the inverse problem is proposed, and numerical results are presented.

Hybrid tomography for conductivity imaging

Hybrid imaging techniques utilize couplings of physical modalities—they are called hybrid, because typically, the excitation and measurement quantities belong to different modalities. Recently there has been an enormous research interest in this area because these methods promise very high resolution. In this paper, we give a review on hybrid tomography methods for electrical conductivity imaging. The reviewed imaging methods utilize couplings between electric, magnetic and ultrasound modalities. By this it is possible to perform high-resolution electrical impedance imaging and to overcome the low-resolution problem of electric impedance tomography.

Iterative sinc-convolution method for solving planar D-bar equation with application to EIT.

The numerical solution of D-bar integral equations is the key in inverse scattering solution of many complex problems in science and engineering including conductivity imaging. Recently, a couple of methodologies were considered for the numerical solution of D-bar integral equation, namely product integrals and multigrid. The first one involves high computational complexity and other one has low convergence rate disadvantages. In this paper, a new and efficient sinc-convolution algorithm is introduced to solve the two-dimensional D-bar integral equation to overcome both of these disadvantages and to resolve the singularity problem not tackled before effectively. The method of sinc-convolution is based on using collocation to replace multidimensional convolution-form integrals- including the two-dimensional D-bar integral equations - by a system of algebraic equations. Separation of variables in the proposed method allows elimination of the formulation of the huge full matrices and therefore reduces the computational complexity drastically. In addition, the sinc-convolution method converges exponentially with a convergence rate of O(e-cN). Simulation results on solving a test electrical impedance tomography problem confirm the efficiency of the proposed sinc-convolution-based algorithm.

Adaptive and Stochastic Algorithms for Electrical Impedance Tomography and DC Resistivity Problems with Piecewise Constant Solutions and Many Measurements

This article develops fast numerical methods for the practical solution of the famous electrical impedance tomography and DC resistivity problems in the presence of discontinuities and potentially many experiments or data. Based on a Gauss–Newton (GN) approach coupled with preconditioned conjugate gradient (PCG) iterations, we propose two algorithms. One determines adaptively the number of inner PCG iterations required to stably and effectively carry out each GN iteration. The other algorithm, useful especially in the presence of many experiments, employs a randomly chosen subset of experiments at each GN iteration that is controlled using a cross validation approach. Numerical examples demonstrate the efficacy of our algorithms.

Second-order topological expansion for electrical impedance tomography

Second-order topological expansions in electrical impedance tomography problems with piecewise constant conductivities are considered. First-order expansions usually consist of local terms typically involving the state and the adjoint solutions and their gradients estimated at the point where the topological perturbation is performed. In the case of second-order topological expansions, non-local terms which have a higher computational cost appear. Interactions between several simultaneous perturbations are also considered. The study is aimed at determining the relevance of these non-local and interaction terms from a numerical point of view. A level set based shape algorithm is proposed and initialized by using topological sensitivity analysis.

A reconstruction algorithm for electrical impedance tomography based on sparsity regularization

SUMMARYThis paper develops a novel sparse reconstruction algorithm for the electrical impedance tomography problem of determining a conductivity parameter from boundary measurements. The sparsity of the ‘inhomogeneity’ with respect to a certain basis is a priori assumed. The proposed approach is motivated by a Tikhonov functional incorporating a sparsity‐promoting ℓ1‐penalty term, and it allows us to obtain quantitative results when the assumption is valid. A novel iterative algorithm of soft shrinkage type was proposed. Numerical results for several two‐dimensional problems with both single and multiple convex and nonconvex inclusions were presented to illustrate the features of the proposed algorithm and were compared with one conventional approach based on smoothness regularization. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Explicit characterization of the support of non-linear inclusions

We study inverse problems for non-linear penetrable media in the context of scattering theory and impedance tomography. Using a general description of the range of the non-linear far-field operator we show an explicit characterization of the support of a weakly non-linear inhomogeneous scattering object. Application of the same technique to the impedance tomography problem for a monotonic non-linear inclusion yields a characterization of the inclusion's support from the non-linear Neumann-to-Dirichlet operator.

Galerkin boundary element method for the forward problem of ERT

Electrical resistance tomography (ERT) is a novel non-intrusive method that determines the cross-sectional electric conductivity distribution. Currently, the ERT forward problem solvers are mainly based on a finite element method, which require a large number of mesh elements. A Galerkin boundary element method (GBEM) is presented for the ERT forward problem in both 2D and 3D. The complete electrode model is adopted. The contact resistivity of electrodes is measured by rectangular cells of different lengths, as well as the conductivity of the background medium. GBEM formulations solving the multi-domain problems are derived. Numerical results show that the method has a better performance on precision compared by both the finite element method and the collocation boundary element method.

Direct localization of poles of a meromorphic function from measurements on an incomplete boundary

This paper proposes an algebraic method to reconstruct the positions of multiple poles in a meromorphic function field from measurements on an arbitrary simple arc in it. A novel issue is the exactness of the algorithm depending on whether the arc is open or closed, and whether it encloses or does not enclose the poles. We first obtain a differential equation that can equivalently determine the meromorphic function field. From it, we derive linear equations that relate the elementary symmetric polynomials of the pole positions to weighted integrals of the field along the simple arc and end-point terms of the arc when it is an open one. Eliminating the end-point terms based on an appropriate choice of weighting functions and a combination of the linear equations, we obtain a simple system of linear equations for solving the elementary symmetric polynomials. We also show that our algorithm can be applied to a 2D electric impedance tomography problem. The effects of the proximity of the poles, the number of measurements and noise on the localization accuracy are numerically examined.

Subspace-Based Optimization Method in Electric Impedance Tomography

This paper extends the application of the subspace-based optimization method, which was originally proposed to solve inverse scattering problems, to two dimensional electric impedance tomography (EIT) problems with inclusions embedded in a known homogeneous background. Although the EIT problem is both physically and mathematically different from the inverse scattering problem, the EIT problem can be cast into an optimization problem whose objective function is in the same format as that of the inverse scattering problem. Numerical simulations validate the fast convergence and robustness of the proposed inverse method.

Electrical Impedance Tomography Problem With Inaccurately Known Boundary and Contact Impedances

In electrical impedance tomography (EIT) electric currents are injected into a body with unknown electromagnetic properties through a set of contact electrodes at the boundary of the body. The resulting voltages are measured on the same electrodes and the objective is to reconstruct the unknown conductivity function inside the body based on these data. All the traditional approaches to the reconstruction problem assume that the boundary of the body and the electrode-skin contact impedances are known a priori. However, in clinical experiments one usually lacks the exact knowledge of the boundary and contact impedances, and therefore, approximate model domain and contact impedances have to be used in the image reconstruction. However, it has been noticed that even small errors in the shape of the computation domain or contact impedances can cause large systematic artefacts in the reconstructed images, leading to loss of diagnostically relevant information. In a recent paper (Kolehmainen , 2006), we showed how in the 2-D case the errors induced by the inaccurately known boundary can be eliminated as part of the image reconstruction and introduced a novel method for finding a deformed image of the original isotropic conductivity using the theory of TeichmUller mappings. In this paper, the theory and reconstruction method are extended to include the estimation of unknown contact impedances. The method is implemented numerically and tested with experimental EIT data. The results show that the systematic errors caused by inaccurately known boundary and contact impedances can efficiently be eliminated by the reconstruction method.

INTEGRAL REPRESENTATIONS IN ELECTRICAL IMPEDANCE TOMOGRAPHY USING BOUNDARY INTEGRAL OPERATORS

Electrical impedance tomography (EIT) problem with anisotropic anomalous region is formulated in a few different ways using boundary integral operators. The Frechet derivative of Neumann-to-Dirichlet map is computed also by using boundary integral operators and the boundary of the anomalous region is approximated by trigonometric expansion with Lagrangian basis. The numerical reconstruction is done in case that the conductivity of the anomalous region is isotropic.

The Factorization Method for Electrical Impedance Tomography in the Half-Space

We consider the inverse problem of electrical impedance tomography in a conducting half-space, given electrostatic measurements on its boundary, i.e., a hyperplane. We first provide a rigorous weak analysis of the corresponding forward problem and then develop a numerical algorithm to solve an associated inverse problem. This inverse problem consists of the reconstruction of certain inclusions within the half-space which have a different conductivity than the background. To solve the inverse problem we employ the so-called factorization method of Kirsch, which so far has only been considered for the impedance tomography problem in bounded domains. Our analysis of the forward problem makes use of a Liouville-type argument which says that a harmonic function in the entire two-dimensional plane must be a constant if some weighted $L^2$-norm of this function is bounded.

A projection-based level-set approach to enhance conductivity anomaly reconstruction in electrical resistance tomography

In this paper we consider level-set-based methods for anomaly reconstruction for applications involving low-sensitivity data concentrating specifically on the problem of electrical resistance tomography (ERT). Typical descent-based inversion methods suffer from poor reconstruction precision in low-sensitivity regions and typically display quite slow convergence. In this paper, we develop a method for constructing the level-set speed function that is capable of overcoming these problems. In the case of the gradient-descent-based level-set evolution, the speed function is defined in terms of the dot products of residual and sensitivity functions. For Gauss–Newton-type methods, however, the speed function is the solution of the linearized inverse problem at each iteration. Here we propose a projection-based approach where the displacement of the level zero, i.e. estimated anomaly contour, at each point depends on the correlation coefficients between the sensitivity of the data to conductivity perturbations and the residual error. In other words, the proposed velocity field is invariant to the absolute amplitudes of both residual and sensitivity, but rather is a reflection of the angle between these two quantities. We demonstrate that our method is a descent-based reconstruction and we relate the mathematical formulation of the projection-based speed function to that of the gradient-descent method and the Gauss–Newton-type approach. The comparison suggests that the proposed technique can be seen as a diagonal approximation of the Gauss–Newton formulation. Using a quasi-linear source-type formulation of the forward problem, we describe an efficient implementation of the projection-based approach for finite-domain imaging problems. The proposed algorithm is simulated with numerical 2D and 3D data and its performance and efficiency are compared to those of the gradient-descent method.

Regularization by dynamic programming

We investigate continuous regularization methods for linear inverse problems of static and dynamic type. These methods are based on dynamic programming approaches for linear quadratic optimal control problems. We prove regularization properties and also obtain rates of convergence for our methods. A numerical example concerning a dynamical electrical impedance tomography (EIT) problem is used to illustrate the theoretical results.

Principal component analysis and artificial neural network approach to electrical impedance tomography problems approximated by multi-region boundary element method

The idea of electrical impedance tomography (EIT) is to evaluate conductivity or permittivity distribution inside the examined object by measuring the voltages between electrodes placed on its surface. In this paper, EIT as a default 3D diagnostic method of the breast cancer is suggested. The breast was modelled as a hemisphere consists of two spatially homogenous areas with different conductivity. In order to determine the distribution of potential in the breast model, a multi-region boundary element method (BEM) was implemented. In this paper, a multi-region BEM with quadratic interpolation function for the flat, triangular surface elements was introduced. The inverse problem solution provided the identification of the size and the position of the anomalies in the breast tissue. For this purpose the efficient method based on principal component analysis (PCA) and the artificial neural network (ANN) was used. PCA applied to EIT data allows reducing dimensionality of measured data for 3D space and removing the unused part of information, usually corresponding to noise and interrelated variables. ANN method allows to obtain the results of inverse problem solution in real-time.