Electrical Impedance Tomography Problem Research Articles

Quantification of measurement error effects on conductivity reconstruction in electrical impedance tomography

Electrical impedance tomography (EIT) is a boundary measurement inverse technique targeting reconstruction of the conductivity distribution of the interior of a physical body based on boundary measurement data. Typically, the measured data are uncertain because of various error sources; thus, there are many uncertainties in the reconstructed image. This study attempts to quantify the effects of these measurement errors on EIT reconstruction. A comprehensive framework that combines uncertainty quantification techniques and EIT reconstruction techniques is proposed. In this framework, a polynomial chaos expansion method is used to construct a surrogate model of the conductivity field with respect to the measurement errors. Two shape detection indices are introduced to show the EIT reconstruction quality. Finally, under certain detection index constraints, statistical and sensitivity analyses are performed using the properties of the surrogate model. Several EIT problems are examined in this study, involving one or two anomalies in a circular domain or two asymmetric anomalies in a body-like domain. The results show that the proposed framework can quantify the effects of measurement errors on EIT reconstruction at reasonable cost. Further, for the test cases, the measurement errors at the electrodes close to the anomalies are shown to have the greatest influence on the image reconstruction.

Reconstruction of piecewise constant layered conductivities in electrical impedance tomography

This work presents a new constructive uniqueness proof for Calderón’s inverse problem of electrical impedance tomography, subject to local Cauchy data, for a large class of piecewise constant conductivities that we call piecewise constant layered conductivities (PCLC). The resulting reconstruction method only relies on the physically intuitive monotonicity principles of the local Neumann-to-Dirichlet map, and therefore the method lends itself well to efficient numerical implementation and generalization to electrode models. Several direct reconstruction methods exist for the related problem of inclusion detection, however they share the property that “holes in inclusions” or “inclusions-within-inclusions” cannot be determined. One such method is the monotonicity method of Harrach, Seo, and Ullrich, and in fact the method presented here is a modified variant of the monotonicity method which overcomes this problem. More precisely, the presented method abuses that a PCLC type conductivity can be decomposed into nested layers of positive and/or negative perturbations that, layer-by-layer, can be determined via the monotonicity method. The conductivity values on each layer are found via basic one-dimensional optimization problems constrained by monotonicity relations.

A Bayesian filtering approach to layer stripping for electrical impedance tomography

Layer stripping is a method for solving inverse boundary value problems for elliptic PDEs, originally proposed in the literature for solving the Calderón problem of electrical impedance tomography (EIT), where the data consist of the Neumann-to-Dirichlet operator on the boundary. Defining a tangent–normal coordinate system near the boundary, the data are extended to a family of boundary operators on tangential surfaces inside the body, and it is shown that the operators satisfy a non-linear Riccati type differential equation with respect to the normal coordinate. The layer stripping process consists of a sequence of two alternating steps: the conductivity near the current boundary is estimated from the spatial high-frequency limit of the boundary data, and the boundary operator is propagated through a thin layer further into the domain via the Riccati equation. This way, the unknown conductivity in the interior of the domain is estimated layer by layer starting from the boundary and moving inward. The ill-posedness of the EIT problem manifests itself in such high sensitivity of the backwards Riccati equation to errors in the boundary data to cause the solutions to blow up in finite time, thus requiring regularization. In this article, we formulate the layer stripping process in the framework of Bayesian inverse problems, and we revisit the implementation in the light of Bayesian filtering. More specifically, we recast the related inverse boundary value problem as a state estimation problem, and propose an algorithm for its numerical solution based on ensemble Kalman filtering (EnKF). The new Bayesian layer stripping approach that we propose is quite robust, derivative-free and intrinsically suited for the quantification of uncertainties in the estimate. Furthermore, we show that the algorithm can be extended to realistic data collected by using a finite number of contact electrodes.

A Fast Inclusion Boundary Reconstruction Framework for Electrical Impedance Tomography With Parametric Snake Model

Electrical impedance tomography (EIT) is a noninvasive technique estimating the conductivity inside an observation domain from its peripheral measurements and owns many industrial and biomedical applications. Inclusion reconstruction is one of the key problems in EIT. However, the high-resolution inclusion reconstruction is challenging due to the ill-posedness and nonlinearity of EIT. The shape reconstruction methods focusing on recovering the shape of the inclusion is one kind of advanced methods to enhance the EIT resolution. To achieve high-performance shape reconstruction, a parametric geometric constraint boundary reconstruction (PGCBR) framework is proposed. The shape reconstruction is implemented by minimizing the energy function of the inclusion boundary represented by the parametric snake model. To accelerate the convergence, the Newton method is adopted to solve the energy minimization problem. The first- and second-order partial derivatives of the energy terms are calculated with the EIT shape sensitivity formula. A series of numerical tests evaluated the influences of the key factors, e.g., the basis function and the number of the knot and interpolating points, of PGCBR on its reconstruction performance and proved its robustness against the measurement noise. The following experimental tests, including both single- and multiphase conductivity cases, further proved the advantages of PGCBR over the nonparametric method.

Monotonicity-Based Reconstruction of Extreme Inclusions in Electrical Impedance Tomography

The monotonicity-based approach has become one of the fundamental methods for reconstructing inclusions in the inverse problem of electrical impedance tomography. Thus far the method has not been proven to be able to handle extreme inclusions that correspond to some parts of the studied domain becoming either perfectly conducting or perfectly insulating. The main obstacle has arguably been establishing suitable monotonicity principles for the corresponding Neumann-to-Dirichlet boundary maps. In this work, we tackle this shortcoming by first giving a convergence result in the operator norm for the Neumann-to-Dirichlet map when the conductivity coefficient decays to zero and/or grows to infinity in some given parts of the domain. This allows passing the necessary monotonicity principles to the limiting case. Subsequently, we show how the monotonicity method generalizes to the definite case of reconstructing either perfectly conducting or perfectly insulating inclusions, as well as to the indefinite case where the perturbed conductivity can take any values between, and including, zero and infinity.

Optimal Depth-Dependent Distinguishability Bounds for Electrical Impedance Tomography in Arbitrary Dimension

The inverse problem of electrical impedance tomography is severely ill-posed. In particular, the resolution of images produced by impedance tomography deteriorates as the distance from the measurement boundary increases. Such depth dependence can be quantified by the concept of distinguishability of inclusions. This paper considers the distinguishability of perfectly conducting ball inclusions inside a unit ball domain, extending and improving known two-dimensional results to an arbitrary dimension $d \geq 2$ with the help of Kelvin transformations. The obtained depth-dependent distinguishability bounds are also proven to be optimal.

Integral formulation of the complete electrode model of electrical impedance tomography

We model electrical impedance tomography (EIT) based on the minimum energy principle. It results in a constrained minimization problem in terms of current density. The new formulation is proved to have a unique solution within appropriate function spaces. By characterizing its solution with the Lagrange multiplier method, we relate the new formulation to the so-called shunt model and the complete electrode model (CEM) of EIT. Based on the new formulation, we also propose a new numerical method to solve the forward problem of EIT. The new solver is formulated in terms of current. It was shown to give similar results to that of the traditional finite element method, with simulations on a 2D EIT model.

Methods of Forming the Control Voltage in the Current Generation Problem for Electrical Impedance Tomography

The paper considers questions of control signal for injection current generation in electric impedance tomography devices. The most common injection current generation scheme is voltage controlled current sources. In these circuits, the shape, frequency and amplitude of the current are set by the control voltage. The most common control voltage generation schemes are considered; their advantages and disadvantages are indicated. A circuit using direct digital synthesis and an amplitude-tuning digital-to-analog converter was selected, its block diagram was developed, and an operation algorithm was described. The developed scheme will improve the efficiency of the research by the method of electrical impedance tomography due to the complete automation of the process of controlling its parameters.

A Two-Stage Deep Learning Method for Robust Shape Reconstruction With Electrical Impedance Tomography

As a noninvasive and radiation-free imaging modality, electrical impedance tomography (EIT) has attracted much attention in the last two decades and owns many industry and biomedical applications. However, due to the nonlinearity and ill-posedness of its inverse problem, the EIT images always suffer from low spatial resolution and are sensitive to the modeling errors. To achieve high resolution and modeling error robust EIT image, a two-stage deep learning (TSDL) method is proposed. The proposed method consists of a prereconstruction block and a convolutional neural network (CNN). The prereconstruction block learns the regularization pattern from the training data set and provides a rough reconstruction of the target. The CNN postprocesses the prereconstruction result in a multilevel feature analysis strategy and eliminates the modeling errors with prior information of the observation domain shape. The prereconstruction and CNN blocks are trained together by using a minimum square approach. To evaluate the performance of the TSDL method, the lung EIT problem was studied. The training data set is calculated from more than 100 000 EIT simulation models generated from computed tomography (CT) scans across 792 patients. Lung injury, measurement noise, and model errors are randomly simulated during the model generation process. The trained TSDL model is evaluated with simulation testes, as well as the experimental tests from a laboratory setting. According to the results, the TSDL method could achieve high accuracy shape reconstructions and is robust against measurement noise and modeling errors.

Solving electrical impedance tomography with deep learning

This paper introduces a new approach for solving electrical impedance tomography (EIT) problems using deep neural networks. The mathematical problem of EIT is to invert the electrical conductivity from the Dirichlet-to-Neumann (DtN) map. Both the forward map from the electrical conductivity to the DtN map and the inverse map are high-dimensional and nonlinear. Motivated by the linear perturbative analysis of the forward map and based on a numerically low-rank property, we propose compact neural network architectures for the forward and inverse maps for both 2D and 3D problems. Numerical results demonstrate the efficiency of the proposed neural networks.

A modified L1/2 regularization algorithm for electrical impedance tomography

A more approximate solution can be obtained when the regularization technique is used to solve the ill-posed and nonlinear problem of electrical impedance tomography (EIT). However, due to the distribution of the ill-posed sensitivity matrix and sharp changes in preserving boundary information, commonly used regularization algorithms have their own limitations, such as filtering out the effective data and reconstructing images with unclear edges. In this paper, a modified algorithm based on L1/2 regularization (MA1/2) was proposed with the L1/2-norm as the form of the penalty term due to its better sparse characteristic. In addition, characteristics of the gradient in eight neighborhoods were utilized to retrieve reasonable data which might be filtered out. Two different dimension operators for the reconstruction quality were adopted accordingly. Both simulation and static experimental results demonstrated that the MA1/2 method significantly decreased the effect of measurement noise and improved the quality of reconstructed images.

Induced-Current Learning Method for Nonlinear Reconstructions in Electrical Impedance Tomography

Electrical impedance tomography (EIT) is an attractive technique that aims to reconstruct the unknown electrical property in a domain from the surface electrical measurements. In this work, the induced-current learning method (ICLM) is proposed to solve nonlinear electrical impedance tomography (EIT) problems. Specifically, the cascaded end-to-end convolutional neural network (CEE-CNN) architecture is designed to implement the ICLM. The CEE-CNN greatly decreases the nonlinearities in EIT problems by designing a combined objective function and introducing multiple labels. A noticeable characteristic of the proposed CNN scheme is that the input parameters are chosen as both induced contrast current (ICC) and the updated electrical field from a spectral analysis and the output is chosen as ICC, which is fundamentally different from prevailing CNN schemes. Further, several skip connections are introduced to focus on learning only the unknown part of ICC. ICLM is verified with both numerical and experimental tests on typical EIT problems, and it is found that ICLM is able to solve typical EIT problems in less than 1 second with high image qualities. More importantly, it is also highly robust to measurement noises and modeling errors, such as inaccurate boundary data.

Depth Analysis of Planar Array for 3D Electrical Impedance Tomography

Electrical impedance tomography (EIT) imaging modality has great potential on industrial applications with the advantages of being high temporal resolution. It is especially useful in cases, such as geophysical detection, landmine detection, and detections on non-transparent region, where the measurement data are only available from single surface, for data acquisition. Instead of the circular EIT model that uses the traditional circular electrode model, in this paper, planar array EIT is implemented, aiming to visualize a pipeline transporting a two-phase flow. The planar array can explore spatial information within its detectable region by producing 3D images, which have a higher spatial resolution in the axis direction than a traditional EIT with a dual-plane electrode sensor. However, in solving the inverse problem of a 3D subsurface EIT using a planar array, the images may be degraded, especially in cases where the location of the target is relatively deep. The total variation (TV) algorithm as block prior assumption-based regularization method has the potential to improve the image quality, and some works have shown that TV reconstructs sharper images, which provides an advantage when representing spatial information. In this paper, the performance of subsurface EIT using the TV algorithm for 3D visualization is presented based on simulations and experiments, and the results of quantitative measurement of depth are discussed.

Improved Imaging Resolution of Electrical Impedance Tomography Using Artificial Neural Networks for Image Reconstruction.

Electrical impedance tomography (EIT) is a noninvasive and non-radiative medical imaging technique based on detecting the inhomogeneous electrical properties of the tissue. The inverse problem of EIT is a highly nonlinear ill-posed problem, which is the main reason that affects image quality. Our goal is to solve the EIT inverse problem using the nonlinear mapping properties of artificial neural networks (ANNs) and convolutional neural networks (CNNs). In this paper, the adaptive moment estimation (ADAM) optimization method and mean-square-error (MSE) function are used to train an ANN to solve the inverse problem and a CNN to process the ANN image. The networks are trained on datasets of simulated data, and tested on datasets of simulated data and experimental data. Results for time-difference EIT (td-EIT) images are presented using simulated EIT data from EIDORS and experimental EIT data from our EIT systems. The results are used to compare the proposed method with the one-step Gauss-Newton linear method and RBFNN method. The proposed method offers improved resolution (RES), low position error (PE) and excellent artefact removal compared to the existing methods. The experimental results show that our method can improve the RES by 50 to 70 percent and reduce the PE by 60 to 70 percent. The improvements in RES and processing speed are essential for clinical EIT measurement of dynamic physiological processes.

Reconstruction of Small Inclusions in Electrical Impedance Tomography Problems

Reconstruction of Small Inclusions in Electrical Impedance Tomography Problems

Comparison of statistical inversion with iteratively regularized Gauss Newton method for image reconstruction in electrical impedance tomography

In this paper, we investigate image reconstruction from the Electrical Impedance Tomography (EIT) problem using a statistical inversion method based on Bayes’ theorem and an Iteratively Regularized Gauss Newton (IRGN) method. We compare the traditional IRGN method with a new Pilot Adaptive Metropolis algorithm that (i) enforces smoothing constraints and (ii) incorporates a sparse prior. The statistical algorithm reduces the reconstruction error in terms of ℓ2 and ℓ1 norm in comparison to the IRGN method for the synthetic EIT reconstructions presented here. However, there is a trade-off between the reduced computational cost of the deterministic method and the higher resolution of the statistical algorithm. We bridge the gap between these two approaches by using the IRGN method to provide a more informed initial guess to the statistical algorithm. Our coupling procedure improves convergence speed and image resolvability of the proposed statistical algorithm.

Focusing Sensor Design for Open Electrical Impedance Tomography Based on Shape Conformal Transformation.

Electrical Impedance Tomography (EIT) is a non-invasive detection method to image the conductivity changes inside an observation region by using the electrical measurements at the boundary of this region. In some applications of EIT, the observation domain is infinite and is only accessible from one side, which leads to the so-called open EIT (OEIT) problem. Compared with conventional EIT problems, the observation region in OEIT can only be measured from limited projection directions, which makes high resolution imaging much more challenging. To improve the imaging quality of OEIT, a focusing sensor design strategy is proposed based on shape conformal theory. The conformal bijection is used to map a standard EIT sensor defined at a unit circle to a focusing OEIT sensor defined at an upper half plane. A series of numerical and experimental testes are conducted. Compared with the traditional sensor structure, the proposed focusing sensor has higher spatial resolution at the near-electrode region and is good at distinguishing multi-inclusions which are close to each other.

Approximation of continuous EIT data from electrode measurements with Bayesian methods

The electrical impedance tomography (EIT) in its classical formulation seeks to estimate the electric conductivity distribution inside the body from the knowledge of the Dirichlet-to-Neumann (DtN) map of the conductivity equation at the boundary. Numerical methods for the solution of the EIT problem have been developed based on this formulation, most notably the d-bar method and the layer stripping algorithm. In practice, however, the EIT data (electrode data), collected by using a fixed number of contact electrodes, is tantamount to knowledge of the resistance matrix, a mapping between given current configuration and the corresponding vector of measured electrode voltages. Forward models corresponding to the DtN data and the electrode data differ in terms of the boundary values and no direct connection between them has been established. In this article, we analyze the relation between the two boundary data types, and propose to approximate the DtN data from the measured resistance matrix for solving the EIT inverse problem within the Bayesian framework, leveraging a sample based prior and a principal component model reduction.

Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes

For the linearized reconstruction problem in electrical impedance tomography with the complete electrode model, Lechleiter and Rieder (2008 Inverse Problems 24 065009) have shown that a piecewise polynomial conductivity on a fixed partition is uniquely determined if enough electrodes are being used. We extend their result to the full non-linear case and show that measurements on a sufficiently high number of electrodes uniquely determine a conductivity in any finite-dimensional subset of piecewise-analytic functions. We also prove Lipschitz stability, and derive analogue results for the continuum model, where finitely many measurements determine a finite-dimensional Galerkin projection of the Neumann-to-Dirichlet operator on a boundary part.

Shape optimizationviaa levelset and a Gauss-Newton method

In the context of shape optimization via level-set methods, we propose a general framework for a Gauss-Newton method to optimize quadratic functionals. Our approach provides a natural extension of the shape derivative as a vector field defined in the whole working domain. We implement and discuss this method in two cases: first a least-square error minimization reminiscent of the Electrical Impedance Tomography problem, and second the compliance problem with volume constraints.