Electrical Impedance Tomography Problem Research Articles

An optimal transport approach for 3D electrical impedance tomography

Abstract This work solves the three-dimensional inverse boundary value problem with the quadratic Wasserstein distance (W2), which originates from the optimal transportation (OT) theory. The computation of the W2 distance on the manifold surface is boiled down to solving the generalized Monge-Ampère equation, whose solution is directly related to the gradient of the W2 distance. An efficient first-order method based on iteratively solving Poisson's equation is introduced to solve the fully nonlinear elliptic equation. Combining with the adjoint-state technique, the optimization framework based on the W2 distance is developed to solve the three-dimensional electrical impedance tomography (EIT) problem. The proposed method is especially suitable for severely ill-posed and highly nonlinear inverse problems. Numerical experiments demonstrate that our method improves the stability and outperforms the traditional regularization methods.

Oracle-Net for Nonlinear Compressed Sensing in Electrical Impedance Tomography Reconstruction Problems

Sparse recovery principles play an important role in solving many nonlinear ill-posed inverse problems. We investigate a variational framework with learned support estimation for compressed sensing sparse reconstructions, where the available measurements are nonlinear and possibly corrupted by noise. A graph neural network, named Oracle-Net, is proposed to predict the support from the nonlinear measurements and is integrated into a regularized recovery model to enforce sparsity. The derived nonsmooth optimization problem is then efficiently solved through a constrained proximal gradient method. Error bounds on the approximate solution of the proposed Oracle-based optimization are provided in the context of the ill-posed Electrical Impedance Tomography problem (EIT). Numerical solutions of the EIT nonlinear inverse reconstruction problem confirm the potential of the proposed method which improves the reconstruction quality from undersampled measurements, under sparsity assumptions.

A mixed finite element method for the forward problem of electrical impedance tomography with the shunt model

We propose a mixed finite element method for the forward problem of electrical impedance tomography with the shunt model. The method is based on a constrained minimization formulation in terms of the current density, where parts of the Lagrange multiplier are interpreted as the voltages on the electrodes. We justify the well-posedness of this continuous formulation, estimate the regularity of its solution, and use Raviart–Thomas elements for its discretization. The resulting mixed method is then proved to be stable, and converge especially with respect to the voltages on the electrodes. Simulation on a 2D model also gives consistent results.

Tackling capacitive coupling in broad-band spectral electrical impedance tomography (sEIT) measurements by selecting electrode configurations

SUMMARY Electromagnetic (EM) coupling effects including both inductive and capacitive coupling have long been an essential problem in broad-band spectral electrical impedance tomography (sEIT) measurements at the field scale. Efforts have been made to remove EM coupling numerically or to suppress the effects by modified data acquisition strategies. For near-surface applications with relatively small survey layouts, inductive coupling can be well removed in the mHz to kHz frequency range. With the use of shielded coaxial cables and so-called active electrodes where the amplifiers are mounted at the electrodes, capacitive coupling in sEIT measurements can also be reduced. However, it remains challenging to cope with capacitive coupling between the cable shield and the ground, especially in resistive field conditions. The aim of this study is to deal with this type of capacitive coupling effect by identifying and filtering out sEIT measurements that are strongly affected by capacitive coupling. Based on a correction method for capacitive coupling proposed in a previous study, an approach to estimate measurement errors due to capacitive coupling is presented first. In the second step, a workflow was proposed to calculate the capacitive coupling strength (CCS) for each electrode configuration, which is defined as the ratio of the imaginary part of the impedance induced by capacitive coupling and the imaginary part of the impedance due to the subsurface electrical conductivity. In the final step, measurements with low CCS were selected for inversion and the results were compared with inversion results obtained using the previously developed correction approach. It was found that the filtering method based on CCS is more capable in tackling capacitive coupling compared to using model-based corrections. Spectrally consistent sEIT results up to kHz were obtained using the newly developed filtering method, which were not achieved in previous work using model-based correction.

Deep unrolling networks with recurrent momentum acceleration for nonlinear inverse problems

Combining the strengths of model-based iterative algorithms and data-driven deep learning solutions, deep unrolling networks (DuNets) have become a popular tool to solve inverse imaging problems. Although DuNets have been successfully applied to many linear inverse problems, their performance tends to be impaired by nonlinear problems. Inspired by momentum acceleration techniques that are often used in optimization algorithms, we propose a recurrent momentum acceleration (RMA) framework that uses a long short-term memory recurrent neural network (LSTM-RNN) to simulate the momentum acceleration process. The RMA module leverages the ability of the LSTM-RNN to learn and retain knowledge from the previous gradients. We apply RMA to two popular DuNets—the learned proximal gradient descent (LPGD) and the learned primal-dual (LPD) methods, resulting in LPGD-RMA and LPD-RMA, respectively. We provide experimental results on two nonlinear inverse problems: a nonlinear deconvolution problem, and an electrical impedance tomography problem with limited boundary measurements. In the first experiment we have observed that the improvement due to RMA largely increases with respect to the nonlinearity of the problem. The results of the second example further demonstrate that the RMA schemes can significantly improve the performance of DuNets in strongly ill-posed problems.

Efficient Jacobian Computations for Complex ECT/EIT Imaging

The reconstruction of the spatial complex conductivity σ+jωε0εr from complex valued impedance measurements forms the inverse problem of complex electrical impedance tomography or complex electrical capacitance tomography. Regularized Gauß-Newton schemes have been proposed for their solution. However, the necessary computation of the Jacobian is known to be computationally expensive, as standard techniques such as adjoint field methods require additional simulations. In this work, we show a more efficient way to computationally access the Jacobian matrix. In particular, the presented techniques do not require additional simulations, making the use of the Jacobian, free of additional computational costs.

A Graph Neural Network Approach with Improved Levenberg–Marquardt for Electrical Impedance Tomography

Electrical impedance tomography (EIT) is a non-invasive imaging method that allows for the acquisition of resistivity distribution information within an object without the use of radiation. EIT is widely used in various fields, such as medical imaging, industrial imaging, geological exploration, etc. Presently, most electrical impedance imaging methods are restricted to uniform domains, such as pixelated pictures. These algorithms rely on model learning-based image reconstruction techniques, which often necessitate interpolation and embedding if the fundamental imaging model is solved on a non-uniform grid. EIT technology still confronts several obstacles today, such as insufficient prior information, severe pathological conditions, numerous imaging artifacts, etc. In this paper, we propose a new electrical impedance tomography algorithm based on the graph convolutional neural network model. Our algorithm transforms the finite-element model (FEM) grid data from the ill-posed problem of EIT into a network graph within the graph convolutional neural network model. Subsequently, the parameters in the non-linear inverse problem of the EIT process are updated by using the improved Levenberg—Marquardt (ILM) method. This method generates an image that reflects the electrical impedance. The experimental results demonstrate the robust generalizability of our proposed algorithm, showcasing its effectiveness across different domain shapes, grids, and non-distributed data.

Damage detection of CFRP laminates based on Adamax-INN model for EIT

Carbon fiber reinforced polymer (CFRP) is electrically conductive, making it possible to detect structural damage of CFRP laminates with electrical impedance tomography (EIT). However, the inverse problem of EIT is severely non-linear, ill-posed, and underdetermined, thus limiting the resolution and accuracy of images reconstructed with EIT. This paper solves the inverse problem of EIT based on invertible neural networks (INN) and optimizes INN with Adamax gradient descent algorithm. Simulation and experimental results demonstrated that, compared with traditional EIT image reconstruction algorithms and radial basis function (RBF) neural network algorithm, the Adamax-INN model largely reduced the artifacts in reconstructed images, improved the damage recognition accuracy and edge clarity, and displayed good noise immunity.

Optimal transportation for electrical impedance tomography

This work establishes a framework for solving inverse boundary problems with the geodesic-based quadratic Wasserstein distance ( W 2 W_{2} ). A general form of the Fréchet gradient is systematically derived from the optimal transportation (OT) theory. In addition, a fast algorithm based on the new formulation of OT on S 1 \mathbb {S}^{1} is developed to solve the corresponding optimal transport problem. The computational complexity of the algorithm is reduced to O ( N ) O(N) from O ( N 3 ) O(N^{3}) of the traditional method. Combining with the adjoint-state method, this framework provides a new computational approach for solving the challenging electrical impedance tomography problem. Numerical examples are presented to illustrate the effectiveness of our method.

On the electric impedance tomography problem for nonorientable surfaces with internal holes

Let ( M , g ) (M,g) be a compact (in general, nonorientable) surface with boundary ∂ M \partial M and let Γ 0 \Gamma _0 , …, Γ m − 1 \Gamma _{m-1} be connected components of ∂ M \partial M . Let u = u f ( x ) u=u^{f}(x) be a solution to the problem Δ g u = 0 \Delta _{g}u=0 in M M , u | Γ 0 = f u\big |_{\Gamma _0}=f , u | Γ j = 0 u\big |_{\Gamma _j}=0 , j = 1 j=1 , …, m ′ m’ , ∂ ν u | Γ j = 0 \partial _{\nu }u\big |_{\Gamma _j}=0 , j = m ′ + 1 j=m’+1 , …, m − 1 m-1 , where ν \nu is the outward normal. With this problem, one associates the DN map Λ : f ↦ ∂ ν u f | Γ 0 \Lambda \colon f\mapsto \partial _{\nu }u^{f}\big |_{\Gamma _0} . The purpose is to determine M M from Λ \Lambda . To this end, an algebraic version of the boundary control method is applied. The key instrument is the algebra A \mathfrak {A} of functions holomorphic on the appropriate orientable double cover of M M . It is proved that A \mathfrak {A} is determined by Λ \Lambda up to isometric isomorphism. The spectrum of the algebra A \mathfrak {A} provides a relevant copy M ′ M’ of M M . This copy is conformally equivalent to M M while its DN map coincides with Λ \Lambda .

A cascaded convolutional neural networks for stroke detection imaging

In recent years, electrical impedance tomography has widely been used in stroke detection. To improve the prediction accuracy and anti-noise ability of the system, the inverse problem of electrical impedance tomography needs to be solved, for which cascade convolutional neural networks are used. The proposed network is divided into two parts so that the advantages can be compounded when parts of a network are cascaded together. To get high-resolution imaging, an optimized network based on encoding and decoding is designed in the first part. The second part is composed of a residual module, which is used to extract the characteristics of voltage information and ensure that no information is lost. The anti-noise performance of the network is better than other networks. In physical experiments, it is also proved that the algorithm can roughly restore the location of the object in the field.

Finite element modeling of the electrical impedance tomography technique driven by machine learning

To create a human-like skin for a robotic application, current touch sensor technologies have a few drawbacks. Electrical Impedance Tomography (EIT) is a candidate for this application due to its applicability over complex geometries; nevertheless, it has accuracy concerns. This study employs artificial neural networks (ANNs) to investigate the accuracy and capability of EIT-based touch sensors. A finite element (FE) model is utilized to solve the forward EIT problem while simultaneously determining the system’s comprehensive mechanical response. The FE model is comprised of a polyurethane (PU) foam domain, a conductive spray layer and a set of sixteen electrodes. To replicate the process of touching the sensor body, a punch of varying diameters and touch forces is utilized. The mechanical response of the sensor body is modeled using the hyperfoam material model calibrated through experimental uniaxial and shear test data, while the electric conductivity of the sprayed skin surface is obtained experimentally as function of applied strain. The viscoelastic behavior of the PU foam material is also obtained experimentally. These experimental data were implemented in the FE model through user subroutines to model the mechanical and electrical properties of the sensor in the EIT forward problem. The traditional EIT inverse problem image reconstruction was replaced utilizing ANNs as an alternative to extract mechanics based parameters. The ANNs were created to predict the spatial coordinates of the touch point, and they were proven to be extremely accurate. Using the EIT voltage readings as input, the ANNs were utilized to forecast the system’s mechanical behavior such as contact pressure, contact area, indentation depth, and touching force.

Comparison of Different Radial Basis Function Networks for the Electrical Impedance Tomography (EIT) Inverse Problem

This paper aims to determine whether regularization improves image reconstruction in electrical impedance tomography (EIT) using a radial basis network. The primary purpose is to investigate the effect of regularization to estimate the network parameters of the radial basis function network to solve the inverse problem in EIT. Our approach to studying the efficacy of the radial basis network with regularization is to compare the performance among several different regularizations, mainly Tikhonov, Lasso, and Elastic Net regularization. We vary the network parameters, including the fixed and variable widths for the Gaussian used for the network. We also perform a robustness study for comparison of the different regularizations used. Our results include (1) determining the optimal number of radial basis functions in the network to avoid overfitting; (2) comparison of fixed versus variable Gaussian width with or without regularization; (3) comparison of image reconstruction with or without regularization, in particular, no regularization, Tikhonov, Lasso, and Elastic Net; (4) comparison of both mean square and mean absolute error and the corresponding variance; and (5) comparison of robustness, in particular, the performance of the different methods concerning noise level. We conclude that by looking at the R2 score, one can determine the optimal number of radial basis functions. The fixed-width radial basis function network with regularization results in improved performance. The fixed-width Gaussian with Tikhonov regularization performs very well. The regularization helps reconstruct the images outside of the training data set. The regularization may cause the quality of the reconstruction to deteriorate; however, the stability is much improved. In terms of robustness, the RBF with Lasso and Elastic Net seem very robust compared to Tikhonov.

Level Set and Optimal Control for the Inverse Inclusion Reconstruction in Electrical Impedance Tomography Modeling

In this paper, we address the identification of an unknown inclusion in an elliptic equation, which arises in the context of electrical impedance tomography and multiphase problems. We present a novel approach by formulating the overdetermined system as an optimal design problem based on the unknown inclusion and two state solutions, which is derived from by introducing a least square functional. We establish the existence results and first optimality conditions, and employ the finite element method to discretize the involved variational equations. Furthermore, we update the unknown inclusion using the level-set method. To demonstrate the effectiveness and validity of our proposed algorithm, we investigate simple and complex geometries numerically. Our results showcase the efficiency and accuracy of the method, making it a promising tool for solving similar inverse problems in various engineering applications.

A modified Levenberg–Marquardt scheme for solving a class of parameter identification problems

Parameter identification problems in PDEs are special class of nonlinear inverse problems which has many applications in science and technology. One such application is the Electrical Impedance Tomography (EIT) problem. Although many methods are available in literature to tackle nonlinear problems, the computation of Fréchet derivative is often a bottle neck for deriving the solution. Moreover, many assumptions are required to establish the convergence of such methods. In this paper, we propose a modified form of Levenberg–Marquardt scheme which does not require the knowledge of exact Fréchet derivative, instead, uses an approximate form of it and at the same time, no additional assumptions are required to establish the convergence of the scheme. We illustrate the theoretical result through numerical examples. In order to ensure that the proposed scheme can be applied to practical problems, we have applied the scheme to EIT problem and the reconstruction process clearly demonstrates that the method can be successfully applied to practical problems.

Series reversion for electrical impedance tomography with modeling errors *

This work extends the results of Garde and Hyvönen (2022 Math. Comput. 91 1925–1953) on series reversion for Calderón’s problem to the case of realistic electrode measurements, with both the internal admittivity of the investigated body and the contact admittivity at the electrode-object interfaces treated as unknowns. The forward operator, sending the internal and contact admittivities to the linear electrode current-to-potential map, is first proven to be analytic. A reversion of the corresponding Taylor series yields a family of numerical methods of different orders for solving the inverse problem of electrical impedance tomography, with the possibility to employ different parametrizations for the unknown internal and boundary admittivities. The functionality and convergence of the methods is established only if the employed finite-dimensional parametrization of the unknowns allows the Fréchet derivative of the forward map to be injective, but we also heuristically extend the methods to more general settings by resorting to regularization motivated by Bayesian inversion. The performance of this regularized approach is tested via three-dimensional numerical examples based on simulated data. The effect of modeling errors related to electrode shapes and contact admittances is a focal point of the numerical studies.

Lumped Element Method Based Conductivity Reconstruction Algorithm for Localization Using Symmetric Discrete Operators on Coarse Meshes

The inverse conductivity problem in electrical impedance tomography involves the solving of a nonlinear and under-determined system of equations. This paper presents a new approach, which leads to a quadratic and overdetermined system of equations. The aim of the paper is to establish new research directions in handling of the inverse conductivity problem. The basis of the proposed method is that the material, which can be considered as an isotropic continuum, is modeled as a linear network with concentrated parameters. The weights of the obtained graph represent the properties of the discretized continuum. Further, the application of the developed procedure allows for the dielectric constant to be used in the multi-frequency approach, as a result of which the optimized system of equations always remains overdetermined. Through case studies, the efficacy of the reconstruction method by changing the mesh resolution applied for discretizing is presented and evaluated. The presented results show, that, due to the application of discrete, symmetric mathematical structures, the new approach even at coarse mesh resolution is capable of localizing the inhomogeneities of the material.

Crack identification using smart paint and machine learning

Information on the presence and location of cracks in civil structures can be precious to support operators in making decisions related to structural management and scheduling informed maintenance. This paper investigates the efficacy of supervised machine learning to solve the inverse electrical impedance tomography problem and to reconstruct the conductivity distribution of a piezoresistive sensing film. This film consists of a conductive paint applied onto structural components, and operators can use its conductivity distribution to identify crack sizes and locations in the underlying structure. A deep neural network is employed to reconstruct a dense conductivity distribution within the painted area by using only voltage measurements collected at sparse boundary locations. Since one of the most challenging aspects of using supervised learning tools for real-world applications is generating a representative training dataset, this paper presents a new approach to test the suitability of synthetic datasets built using a finite element model of the sensing film. Results are reported for four sensing specimens fabricated with two different techniques (i.e., using carbon nanotubes and graphene nanosheets, respectively). Crack-like damage is induced to the substrate of the sensing film and identified using the proposed machine learning technique. Promising results are obtained as compared to conventional methods.

An INN-Based Solution for the Underdetermined Image Reconstruction Problem in EIT

Electrical impedance tomography (EIT) has been widely concerned in online nondestructive testing of carbon fiber-reinforced polymers (CFRPs) due to its significant advantages of high real-time performance, zero radiation, simple structure, and low cost. However, its inverse problem is highly underdetermined and nonlinear, thus limiting its imaging quality and application scope. Based on the architecture of an invertible neural network (INN), an INN image reconstruction algorithm with latent output variables is proposed in this article. With latent output variables, the information lost during training is reused, and a unique input can be determined by sampling latent output variables and predicted outputs. In addition, the forward and reverse processes are combined together to weaken the dependency of effective output and latent output variables, reduce the errors between input and output domains, and solve the underdetermination and nonlinearity problems of EIT. The simulation and experimental results showed that compared with the traditional algorithms and the convolution neural network (CNN) network algorithm, the INN can accurately reconstruct the damage location, yield a clearer damaged edge, and reduce artifacts. Noise-added simulations and prototype experiments showed that the INN achieves both good imaging quality and enhanced noise immunity, to improve the performance of the EIT system in CFRP laminate inspection.

An equilibrium finite element method for electrical impedance tomography

We present a new numerical method for the forward problem of electrical impedance tomography with the shunt model. The new method builds on a discretization of the total Joule heat consumed over the region, in terms of the current density. It gives an approximation of the distribution of current density before using it to reconstruct the potential distribution. Simulations on a 2D and a 3D EIT model indicate the new method can produce more accurate approximations than the traditional FEM, based on the same mesh. The new method has the potential to be a competitive alternative of the traditional FEMs and incoporated into some reconstruction algorithms of EIT.