Problem Of Impedance Tomography Research Articles

A piecewise-constant binary model for electrical impedance tomography

In this paper, we consider the electrical impedance tomography problem in a computational approach. This inverse problem is the recovery of the electrical conductivity $\sigma$ in a domain from boundary measurements, given in the form of the Neumann-to-Dirichlet map. We formulate the inverse problem as a variational one, with a fitting term and a regularization term. We restrict the minimization with respect to the unknown $\sigma$ to piecewise-constant functions defined on rectangular domains in two dimensions. We borrow image segmentation techniques to solve the minimization problem. Several experimental results of conductivity reconstruction from synthetic data are shown, with and without noise, that validate the proposed method.

Newton regularizations for impedance tomography: a numerical study**Financial support by the Institut für Wissenschaftliches Rechnen und Mathematische Modellbildung, Universität Karlsruhe (TH), and the Mathematical Department of the University of Delaware, USA, is gratefully acknowledged.

The inexact Newton iteration REGINN for regularizing nonlinear ill-posed problems consists of two components: the (outer) Newton iteration, stopped by a discrepancy principle, and the inner iteration, which computes the Newton update by solving approximately a linearized system. The second author proved the convergence of REGINN furnished with the conjugate gradients method as inner iteration (Rieder 2005 SIAM J. Numer. Anal. 43 604–22). Amongst others the following feature distinguishes REGINN from other Newton-like regularization schemes: the regularization level for the locally linearized systems can be adapted dynamically incorporating information on the local degree of ill-posedness gained during the iteration. Of course, the potential of this feature can be fully explored only by meaningful numerical experiments in a realistic setting. Therefore, we apply REGINN to the 2D-electrical impedance tomography problem with the complete electrode model. This inverse problem is known to be severely ill-posed. The achieved reconstructions are compared qualitatively and quantitatively with reconstructions from a one-step method which is closely related to the NOSER algorithm (Cheney et al 1990 Int. J. Imag. Syst. Technol. 2 66–75), an often used solver in impedance tomography. Our detailed numerical comparison reveals REGINN to be a competitive solver outperforming the one-step method when noise corrupts the data and/or a moderately large number of electrodes is used.

Effect of stray capacitances on bio-impedances in quasi-static electric field

Evaluating bio-impedances, inevitable stray capacitances have considerable impact on the measurement data. If the stray capacitances are not considered when simulating the electric impedance tomography problem, incorrect conductivities are reconstructed. To study the effect of the stray capacitances, a finite-element model of a test person in surroundings exposed to a quasi-static electric field is used to simulate the measuring setup. Results clearly demonstrate that the displacement currents at high frequencies in air have to be modeled appropriately.

Identification of simple poles via boundary measurements and an application of EIT**This work is partly supported by grant R02-2003-000-10012-0 from the Korea Science and Engineering Foundation.

We consider the problem of identifying simple poles of a meromorphic function by means of the value of the function measured on a circle enclosing those poles. We propose an algorithm for this problem in a mathematically rigorous way with a stability estimate. Results of numerical testing are provided to show validity of the algorithm. We then apply the method to an electrical impedance tomography problem to detect diametrically small inclusions of disc shape via boundary measurements.

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Focus on Software

The impedance tomography problem

Using an operator theoretic framework and pseudo-spectral methods, we provide a simple and explicit formula for the conductivity coefficient in terms of the Dirichlet to Neumann map and the eigenvalues of the Laplacian operator.

Anisotropic Polarization Tensors and Detection of an Anisotropic Inclusion

We introduce the notion of (generalized) anisotropic polarization tensors and prove that they are symmetric and positive-definite. We also estimate the eigenvalues of the anisotropic polarization tensor in terms of the volume of the given domain. We apply these properties to an electrical impedance tomography problem to detect a single anisotropic inhomogeneity of small size. The goal is to detect an unknown inclusion when anisotropic conductivities of the background and the inclusion are known. Three results of computational experiments for the detection of an inclusion are given. One experiment is to assure the validity of the algorithm, while the other two are to see how anisotropy plays a role in the detection procedure.

Equipotential line method for magnetic resonance electrical impedancetomography

We consider magnetic resonance electrical impedance tomography, which aimsto reconstruct the conductivity distribution using the internal currentdensity furnished by magnetic resonance imaging. We show the uniquenessof the conductivity reconstruction with one measurement imposing theDirichlet boundary condition. We also propose a fast non-iterative numericalalgorithm for the conductivity reconstruction using the internal currentvector information. The algorithm is mainly based on efficient numericalconstruction of equipotential lines. The resulting numerical method isstable in the sense that the error of the computed conductivity is linearlyproportional to the input noise level and the introduction of internal currentdata makes the impedance tomography problem well-posed. We presentvarious numerical examples to show the feasibility of using our method.

Application of a Hybrid Genetic/Powell Algorithm and a Boundary Element Method to Electrical Impedance Tomography

An optimization method based on a genetic algorithm (GA) and a boundary element method is applied to solve an electrical impedance tomography problem. The scheme is applied to reconstruct highly irregular shapes and to image and count objects inside a host medium of different impedance. A Pareto multiobjective optimization method is applied to improve the performance of the GA. Comparisons between the GA and a calculus-based method for selected test problems show that the calculus-based method outperforms the GA in simple cases but that for more complex cases the GA reaches the correct solution whereas the calculus-based method does not. A hybrid scheme that we developed combining a calculus-based method and the GA is shown to be the most efficient and robust even when applied to the complex cases we tested. The sensitivity of the current scheme is evaluated in the presence of noise.

Statistical inversion and Monte Carlo sampling methodsin electrical impedance tomography

This paper discusses the electrical impedance tomography (EIT)problem: electric currents are injected into a body with unknownelectromagnetic properties through a set of contact electrodes.The corresponding voltages that are needed to maintain thesecurrents are measured. The objective is to estimate the unknownresistivity, or more generally the impedivity distribution ofthe body based on this information. The most commonly usedmethod to tackle this problem in practice is to use gradient-based locallinearizations. We give a proof for the differentiability of theelectrode boundary data with respect to the resistivity distribution andthe contact impedances. Due to the ill-posedness of the problem,regularization has to be employed. In this paper, we consider the EITproblem in the framework of Bayesian statistics, where the inverseproblem is recast into a form of statistical inference. The problem is toestimate the posterior distribution of the unknown parametersconditioned on measurement data. From the posterior density,various estimates for the resistivity distribution can becalculated as well as a posteriori uncertainties. The search ofthe maximum a posteriori estimate is typically anoptimization problem, while the conditional expectation iscomputed by integrating the variable with respect to theposterior probability distribution. In practice, especially whenthe dimension of the parameter space is large, this integrationmust be done by Monte Carlo methods such as the Markov chainMonte Carlo (MCMC) integration. These methods can also be usedfor calculation of a posteriori uncertainties for theestimators. In this paper, we concentrate on MCMC integrationmethods. In particular, we demonstrate by numerical examples thestatistical approach when the prior densities arenon-differentiable, such as the prior penalizing the totalvariation or the L1 norm of the resistivity.

Electrical Impedance Tomography

t. This paper surveys some of the work our group has done in electrical impedance tomography.

Approximation of the inverse electrical impedance tomography problem by an inverse transmission problem

The inverse problem in electrical impedance tomography (EIT) is severely ill-posed. Therefore, it is desirable to include any available a priori information in a numerical algorithm for its solution. If the conductivity inside the object is known to be piecewise constant this information can be directly incorporated in the algorithm by using boundary integral methods for the computation of the forward map. In this paper we will describe an implementation of this idea and compare the results with standard methods using both synthetic and measured data from the clinical applications of EIT.

Electrical impedance tomography with basis constraints

In this paper, we consider the impedance tomography problem of estimating the conductivity distribution within the body from static current/voltage measurements on the body's surface. We present a new method of implementing prior information of the conductivities in the optimization algorithm. The method is based on the approximation of the prior covariance matrix by simulated samples of feasible conductivities. The reduction of the dimensionality of the optimization problem is performed by principal component analysis (PCA).

High-contrast impedance tomography

We introduce an output least-squares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modelled on network approximation results from an asymptotic analysis and its recovery is based on this model. The smoothly varying part of the conductivity is recovered by a linearization process as is usual. We present the results of several numerical experiments that illustrate the performance of the method.

An image-enhancement technique for electrical impedance tomography

We propose an idea for reconstructing 'blocky' conductivity profiles in electrical impedance tomography. By 'blocky' profiles, we mean functions that are piecewise constant, and hence have sharply defined edges. The method is based on selecting a conductivity distribution that has the least total variation from all conductivities that are consistent with the measured data. We provide some motivation for this approach and formulate a computationally feasible problem for the linearized version of the impedance tomography problem. A simple gradient descent-type minimization algorithm, closely related to recent work on noise and blur removal in image processing via non-linear diffusion is described. The potential of the method is demonstrated in several numerical experiments.

An anisotropic inverse boundary value problem

AbstractWe consider the impedance tomography problem for anisotropic conductivities. Given a bounded region Ω in space, a diffeomorphism Ψ from Ω to itself which restricts to the identity on ∂ Ω, and a conductivity γ on Ω, it is easy to construct a new conductivity Ψ*γ which will produce the same voltage and current measurements on ∂ Ω.We prove the converse in two dimensions (i.e., if γ1 and γ2 produce the same boundary measurements, then γ1, = Ψ*γ2 for an appropriate Ψ) for conductivities which are near a constant.