Electrical Impedance Tomography Inverse Problem Research Articles

Detection of indentation damage in carbon fiber/epoxy composites via EIT during the application of bending loads

Electrical impedance tomography (EIT) is a method of spatially mapping the conductivity distribution of a domain and has been studied as a potential embedded sensing or nondestructive evaluation (NDE) tool. An often touted advantage of EIT is that it can be used in-situ; that is, because the method only requires the application of unobtrusive electrodes, it can conceivably be used while the component or structure is in operation. This material-as-the-sensor philosophy strongly aligns with key components of the NDE 4.0 vision such as the realization of intelligent cyber–physical systems (CPS) and digital twins. To date, however, the claim of in-situ sensing via EIT has not been significantly substantiated. This is problematic because operational loads induce strains that often change the conductivity of the material. Establishing that EIT can detect damage-induced conductivity changes through the presence of unrelated strain-induced conductivity changes is therefore important. To that end, we herein study the application of EIT for detecting indentation damage in a carbon fiber/epoxy composite as the composite is loaded in a four-point bend. It was found that the bending load changes the contact impedance of the electrodes, which resulted in poor EIT images when solving the EIT inverse problem with the ℓ2-norm on the error term. Using the ℓ1-norm on the error term, solved via the primal–dual interior point method (PDIPM), significantly improved image quality. Image quality was even further improved through the use of a mixed prior for regularization, and EIT images were compared to thermography with good agreement. These results show that EIT can indeed detect damage through the presence of an applied load, but care must be taken to account for factors such as outlier data arising from electrode degradation and changing contact impedance. Use of the ℓ1-norm on the error term is therefore highly recommended for in-situ imaging via EIT.

Validation of a multilayer perceptron for rapid, direct solution of the electrical impedance tomography inverse problem

Validation of a multilayer perceptron for rapid, direct solution of the electrical impedance tomography inverse problem

Advances in Electrical Impedance Tomography Inverse Problem Solution Methods: From Traditional Regularization to Deep Learning

Advances in Electrical Impedance Tomography Inverse Problem Solution Methods: From Traditional Regularization to Deep Learning

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.

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.

Damage mapping via electrical impedance tomography in complex AM shapes using mixed smoothness and Bayesian regularization

Additive manufacturing (AM) holds immense potential for rapidly producing complexly-shaped composite components and structures. However, AM is also well known for producing composites of lesser quality than more traditional manufacturing methods. Hence, nondestructive evaluation (NDE) and/or embedded sensing (i.e. integrated NDE) is important for mitigating failures in these materials. Unfortunately, many prevailing NDE modalities are ill-suited for complex AM-produced shapes or are not amenable to embedded sensing. To overcome this limitation, we herein study electrical impedance tomography (EIT) as a means of detecting and localizing damage in a complexly-shaped carbon fiber composite truss structure produced by Impossible Objects. Unlike the current state of the art which is overly fixated on applying EIT to flat composite plates not representative of real components and structures, this work shows that EIT can indeed adeptly find damages in complex shapes. We also present a novel mixed smoothness + conditionally Gaussian regularization formulation for the EIT inverse problem that shows marked improvement over the traditionally used smoothness-only prior. Combined, these two contributions are an important step in translating EIT into practice for AM composites.

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.

Implicit Solutions of the Electrical Impedance Tomography Inverse Problem in the Continuous Domain with Deep Neural Networks.

Electrical impedance tomography (EIT) is a non-invasive imaging modality used for estimating the conductivity of an object Ω from boundary electrode measurements. In recent years, researchers achieved substantial progress in analytical and numerical methods for the EIT inverse problem. Despite the success, numerical instability is still a major hurdle due to many factors, including the discretization error of the problem. Furthermore, most algorithms with good performance are relatively time consuming and do not allow real-time applications. In our approach, the goal is to separate the unknown conductivity into two regions, namely the region of homogeneous background conductivity and the region of non-homogeneous conductivity. Therefore, we pose and solve the problem of shape reconstruction using machine learning. We propose a novel and simple jet intriguing neural network architecture capable of solving the EIT inverse problem. It addresses previous difficulties, including instability, and is easily adaptable to other ill-posed coefficient inverse problems. That is, the proposed model estimates the probability for a point of whether the conductivity belongs to the background region or to the non-homogeneous region on the continuous space Rd∩Ω with d∈{2,3}. The proposed model does not make assumptions about the forward model and allows for solving the inverse problem in real time. The proposed machine learning approach for shape reconstruction is also used to improve gradient-based methods for estimating the unknown conductivity. In this paper, we propose a piece-wise constant reconstruction method that is novel in the inverse problem setting but inspired by recent approaches from the 3D vision community. We also extend this method into a novel constrained reconstruction method. We present extensive numerical experiments to show the performance of the architecture and compare the proposed method with previous analytic algorithms, mainly the monotonicity-based shape reconstruction algorithm and iteratively regularized Gauss-Newton method.

Enhanced damage imaging in three-dimensional composite structures via electrical impedance tomography with mixed and level set regularization

Electrical impedance tomography (EIT) is a promising tool for nondestructive evaluation (NDE) of materials that exhibit stimulus-responsive electrical conductivity. To date, materials practitioners of EIT have used standard regularization methods such as smoothness-promoting operators to solve the ill-posed EIT inverse problem. However, more sophisticated and better performing regularization methods exist. It is essential that these other regularization methods be explored in order for EIT to see more widespread adoption in NDE. To that end, we present two novel regularization techniques – a mixed smoothness/focal method and a level set method – for application to damage imaging via EIT. These new techniques are experimentally validated on two three-dimensional composite components, a carbon black-modified glass fiber/epoxy tube and a carbon fiber/epoxy laminate airfoil subject to multiple damaging impacts. It is shown that both of the new regularization techniques markedly outperform traditional smoothness-only methods. These results are an important step towards adoption of more sophisticated EIT formulations in NDE and transition of EIT from the proof-of-concept phase and into practice for engineered materials.

A path integral Monte Carlo (PIMC) method based on Feynman-Kac formula for electrical impedance tomography

A path integral Monte Carlo method (PIMC) based on a Feynman-Kac formula for the Laplace equation with mixed boundary conditions is proposed to solve the forward problem of the electrical impedance tomography (EIT). The forward problem is an important part of iterative algorithms of the inverse EIT problem, and the proposed PIMC provides a local solution to find the potentials and currents on individual electrodes. Improved techniques are proposed to compute with better accuracy both the local time of reflecting Brownian motions (RBMs) and the Feynman-Kac formula for mixed boundary problems of the Laplace equation. Accurate voltage-to-current maps on the electrodes of a model 3-D EIT problem with eight electrodes are obtained by solving a mixed boundary problem with the proposed PIMC method.

An Image Reconstruction Algorithm for Electrical Impedance Tomography Using Measurement Estimation of Virtual Electrodes

For biomedical imaging in mini-scale, the inverse problem of electrical impedance tomography (EIT) is severely ill-conditioned due to the number of electrodes is very limited. In this paper, a novel difference imaging algorithm for 2D-EIT using measurement estimation of virtual electrodes is proposed. The proposed reconstruction algorithm breaks though the limitation of micro EIT sensor’s structure and tackles the problem of low spatial resolution in mini-scale by introducing virtual electrodes. The inverse problem of EIT is decomposed into two separately tasks: estimation potentials of virtual electrodes with proper priors, determination of the conductivity distribution using both virtual and real electrodes. It is formulated as a possibility model using virtual electrodes’ potentials as latent variables. Real electrodes’ potentials are regarded as the observable variables. Conductivity distribution is predicted by a maximum likelihood of the possibility model by using EM algorithm. The proposed algorithm is verified by both simulation and experiment. In experiment, medaka fish embryo which is about 1mm in diameter is reconstructed. Comparing with Tikhonov and NOSER regularization method, average ICC is improved 11.54% from 0.7620 to 0.8499 in simulation and improved 5.79% from 0.6978 to 0.7382 in experiment. The algorithm not only performs well in conductivity reconstruction, but also in shape reconstruction. Average position error is decreased 47.57%, average shape error is decreased 24.44% in experiment. It is very suitable for image reconstruction in mini-scale.

Pressure Mapping Using Nanocomposite-Enhanced Foam and Machine Learning

Pressure mapping has garnered considerable interest in the healthcare and robotic industries. Low-cost and large-area compliant devices, as well as fast and effective computational algorithms, have been proposed in the last few years to facilitate distributed pressure sensing. One approach is to use electrical impedance tomography (EIT) to reconstruct the contact pressure distribution of piezoresistive materials. While tremendous success has been demonstrated, conventional algorithms may be unsuitable for real-time monitoring due to its computational demand and runtime. Moreover, the low resolution of reconstructed images is a well-known issue related to the regularization strategies typically employed for traditional EIT methods. Therefore, in this study, two different supervised machine learning (ML) approaches, namely, radial basis function networks and deep neural networks, were employed to efficiently solve the inverse EIT problem and improve the resolution of reconstructed pressure maps. The demonstration of high-resolution pressure mapping, specifically, for identifying pressure hotspots, was achieved using a carbon nanotube-based thin film integrated with foam.

Improvement of performance and sensitivity of 2D and 3D image reconstruction in EIT using EFG forward model

This paper presents a pure element-free Galerkin method (EFGM) forward model for image reconstruction in 2D and 3D electrical impedance tomography (EIT) using an adaptive current injection method. In EIT systems with the adapting current injection method, both static and dynamic images can be reconstructed; however, determination of electrode contact impedances in the complete electrode model is difficult and the Gap model is used. In this paper, in the EIT forward problem a weak form functional based on the Gap model and a pure EFGM approach are developed, and in the EIT inverse problem, Jacobian matrix is computed by the EFGM, and a fast integration technique is introduced to calculate the entries of the Jacobian matrix within an adequate computation time. The influence of increasing the density of nodes at and near the electrodes with steep electric potential gradients on the accuracy of FEM and EFGM forward solutions is investigated, and the performance of the image reconstruction algorithm with the proposed fast integration technique is examined. The numerical results reveal that the proposed EFGM forward model with the fast integration technique has an efficient performance both in terms of mean relative imaging errors and computational time.

A Rotational Invariant Neural Network for Electrical Impedance Tomography Imaging without Reference Voltage: RF-REIM-NET.

Background: Electrical Impedance Tomography (EIT) is a radiation-free technique for image reconstruction. However, as the inverse problem of EIT is non-linear and ill-posed, the reconstruction of sharp conductivity images poses a major problem. With the emergence of artificial neural networks (ANN), their application in EIT has recently gained interest. Methodology: We propose an ANN that can solve the inverse problem without the presence of a reference voltage. At the end of the ANN, we reused the dense layers multiple times, considering that the EIT exhibits rotational symmetries in a circular domain. To avoid bias in training data, the conductivity range used in the simulations was greater than expected in measurements. We also propose a new method that creates new data samples from existing training data. Results: We show that our ANN is more robust with respect to noise compared with the analytical Gauss–Newton approach. The reconstruction results for EIT phantom tank measurements are also clearer, as ringing artefacts are less pronounced. To evaluate the performance of the ANN under real-world conditions, we perform reconstructions on an experimental pig study with computed tomography for comparison. Conclusions: Our proposed ANN can reconstruct EIT images without the need of a reference voltage.

R-UNet Deep Learning-Based Damage Detection of CFRP With Electrical Impedance Tomography

Carbon fiber reinforced polymer (CFRP) with excellent properties is widely used in many fields. During the production process and service life of CFRP-related products, CFRP may suffer some damages which are difficult to be detected. Due to the electrical conductivity of CFRP, electrical impedance tomography (EIT) can be used to detect the damage in CFRP. However, the inverse problem of EIT is nonlinear and ill-conditioned and the reconstructed images based on EIT have low accuracy. In order to solve these problems, a R-UNet network with four modules (preliminary imaging module, backbone feature extraction module, enhanced feature extraction module, and prediction module) is proposed in the paper. Firstly, a shallow residual network is introduced to reconstruct the image. Then, the initial image is input into U-Net for backbone feature extraction and enhanced feature extraction. Finally, the image with fewer artifacts and clear edges is output. The network is trained with 12,600 simulated data. Four algorithms were compared with R-UNet. The simulation results showed that the R-UNet algorithm performed better than other algorithms in imaging quality. CC and SSIM were used as the indicators to evaluate the simulation results. The average CC and SSIM of R-UNet reached 0.93 and 0.95, respectively. In addition, an EIT experimental platform was established to verify R-UNet. The comparison of infrared thermal imaging results and experimental results of EIT indicated that R-UNet realized reliable detection results of CFRP damage.

Sorted L1 regularization method for damage detection based on electrical impedance tomography.

Carbon fiber reinforced polymers (CFRPs) are composite materials in which carbon provides strength and stiffness, whereas polymers provide cohesiveness and toughness. The electrical impedance of CFRP laminates is changed due to different kinds of damages. Electrical impedance tomography (EIT) has significant advantages such as non-intrusion, portability, low cost, and quick response and has widely been used as a nondestructive testing method. Therefore, EIT has great potential in structural health monitoring of CFRPs. Regularization can solve the ill-posed inverse problem of EIT. However, conventional regularization algorithms have their own limitations, such as over-smoothness of reconstructed edges and unstable solution caused by measurement noise. In addition, the anisotropic property of CFRPs also affects the image quality based on traditional methods. In this paper, the sorted L1-norm regularization is proposed. It promotes grouping highly correlated variables while encouraging sparsity by using more effective penalty terms. The sharp edges between different materials can be obtained, and the obtained solution is more stable. The image quality of different objects, especially the image quality of multi-targets, can be significantly improved with this new method. In addition, the sorted L1 norm can generate adaptive regularization parameters without empirical selection. The new regularization problem is solved by the alternating direction method of multipliers. Both experimental and simulation results demonstrate that the sorted L1 norm improves the quality of reconstructed images under various noise levels. The proposed method is comprehensively evaluated with three image quality criteria by numerical simulation quantitatively.

Tactile Sensing Using Machine Learning-Driven Electrical Impedance Tomography

Electrical Impedance Tomography (EIT) tactile sensors have limited success in equipping robots with tactile sensing capabilities due to the low spatial resolution of the resulting tactile images and the presence of image artifacts. To address these limitations, we propose a modular framework for invariant recognition of objects, within the context of an EIT artificial skin sensor. Three interconnected problems, i.e., EIT image reconstruction, segmentation and object recognition, are tackled in this work with the aid of machine learning. A novel conductivity surface decomposition approach, based on low order bivariate polynomials and RBF networks is introduced for the efficient solution of the EIT inverse problem. Next, segmentation of the reconstructed images is performed using a convolutional neural network and transfer learning. Finally, a subspace KNN ensemble classifier is trained on the set of object descriptors extracted from the segmented inhomogeneities to classify the objects. The proposed framework provides an accuracy of 97.5% on unseen data.

EIT-4LDNN: A Novel Neural Network For Electrical Impedance Tomography

EIT (Electrical Impedance Tomography) is a non-invasive dynamic image detection technology that can well reflect the distribution of objects in a uniform field. But the inverse problem of EIT is a non-linear, highly ill-conditioned, and ill-posed problem. Images reconstructed by traditional iterative algorithms and non-iterative algorithms have too many uneliminated artifacts and low spatial resolution. This paper proposes a deep neural network containing a multi-layer fully connected network so that predicting the conductivity distribution of different samples through the excellent nonlinear fitting ability of the neural network. Compared with the CG(Conjugate Gradient) algorithm and TR ( Tikhonov Regularization) algorithm, the quality of images is improved and the noise is reduced, but the generalization ability and prediction accuracy of the network need to be further improved.

Imaging Conductivity Changes in Monolayer Graphene Using Electrical Impedance Tomography.

Recently, graphene has gained a lot of attention in the electronic industry due to its unique properties and has paved the way for realizing novel devices in the field of electronics. For the development of new device applications, it is necessary to grow large wafer-sized monolayer graphene samples. Among the methods to synthesize large graphene films, chemical vapor deposition (CVD) is one of the promising and common techniques. However, during the growth and transfer of the CVD graphene monolayer, defects such as wrinkles, cracks, and holes appear on the graphene surface. These defects can influence the electrical properties and it is of interest to know the quality of graphene samples non-destructively. Electrical impedance tomography (EIT) can be applied as an alternate method to determine conductivity distribution non-destructively. The EIT inverse problem of reconstructing conductivity is highly non-linear and is heavily dependent on measurement accuracy and modeling errors related to an accurate knowledge of electrode location, contact resistances, the exact outer boundary of the graphene wafer, etc. In practical situations, it is difficult to eliminate these modeling errors as complete knowledge of the electrode contact impedance and outer domain boundary is not fully available, and this leads to an undesirable solution. In this paper, a difference imaging approach is proposed to estimate the conductivity change of graphene with respect to the reference distribution from the data sets collected before and after the change. The estimated conductivity change can be used to locate the defects on the graphene surface caused due to the CVD transfer process or environment interaction. Numerical and experimental results with graphene sample of size 2.5 × 2.5 cm are performed to determine the change in conductivity distribution and the results show that the proposed difference imaging approach handles the modeling errors and estimates the conductivity distribution with good accuracy.