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Electrical Capacitance Tomography (ECT) image reconstruction has been largely applied for industrial applications. However, there is still a crucial need to develop a new framework to enhance the quality of reconstructed images and make it faster. Deep learning has recently boomed and applied in many fields since it is good at mapping complicated nonlinear functions based on series of artificial neural networks. In this paper, a novel image reconstruction method based on a deep neural network is proposed. The proposed image reconstruction algorithm mainly uses Long Short-Term Memory (LSTM) deep neural network, which is abbreviated as LSTM-IR algorithm. A big simulation dataset containing 160k pairs of instances is created to train and test the performance of the proposed LSTM-IR algorithm. Each pair of the sample has a predefined permittivity distribution vector and corresponding capacitance vector. The generalization ability and feasibility of the LSTM-IR network are measured using contaminated data, data not included in the training dataset, and experimental data. The preliminary results show that the proposed LSTM-IR method can create fast and more accurate ECT images than traditional and deep learning image reconstruction algorithms.

Image reconstruction of electrical capacitance tomography (ECT) is a nonlinear and ill-posed inverse problem. Therefore, how to introduce an effective algorithm to reduce the ill conditioned degree of ECT imaging, thereby improving the imaging accuracy is an important subject of ECT algorithm research. In order to further study the subject, a novel ECT image reconstruction algorithm based on an adaptive support driven Bayesian reweighted (ASDBR) algorithm was proposed in this paper. The great advantage of this algorithm is that it can accurately extract the main features of the flow pattern and remove redundant information. This algorithm transforms the original problem into a series of subproblems with iteratively reweighted weights, and solves these subproblems by the iterative shrinkage-thresholding algorithm (ISTA). Comparisons are made among the ASDBR algorithm, the Landweber iterative algorithm, the sparse Bayesian learning (SBL) algorithm, and Lasso. Both simulation and experiment results show that the proposed new method considerably enhances the quality of the reconstructed image.

To improve image quality generated from the electrical capacitance tomography measurement system, the use of entropic thresholding techniques is investigated in this article. Based on the analysis of the principle of Electrical Capacitance Tomography (ECT) image reconstruction and entropic thresholding, various algorithms have been proposed for easy extraction of quantitative information from tomograms generated from the ECT system. Experiments indicate that proposed algorithms can provide high-quality images at no or minimum computational cost. It is easier to implement and integrate with classical algorithms such as Linear Back Projection, Singular value decomposition, Tikhonov regularization, and Landweber. Entropic thresholding techniques present a feasible and effective way toward the industrial utilization of ECT measurement systems. Keywords: Electrical Capacitance Tomography; Inverse Problem; Image Reconstruction; Entropic Thresholding

For great achievements in recent decades, image reconstruction for electrical capacitance tomography (ECT) has been considered in this study. ECT has demonstrated impressive potentials in multiprocess measurement, and the obtained images are of high resolution, which are suitable for advanced procedures in industrial and medical applications and across different tasks and domains. But the ECT system still requires improvements in the quality of image reconstruction given its importance of great significance to obtain the reliability and usefulness of measurement results. The deep neural network is used in this study to extract new features and to update the number of nodes and hidden layers in the system. Recently, deep learning exhibits suitable solutions in many flourishing fields based on different series of artificial neural networks for mapping nonlinear functions. To address the obstacles, this paper proposes an imaging method using an optimizer reconstruction model. An optimization model for imaging is generated as a powerful optimizer for building a computational model to ameliorate the reconstruction accuracy. Based on the deep learning methodology, the previous images reconstructed by using one of the imaging techniques to the required images are abstracted and stored in the deep learning machine, resulting in an error rate of 8.9%, and this is considered good on ECT. Therefore, an artificial neural network of the capacitance (ANNoC) system is introduced to estimate capacitance measurements.

Electrical capacitance tomography (ECT) has been applied in many fields for process monitoring. Knowing the permittivity values of target dielectric objects is essential in lots of application scenarios. Conventional methods have difficulty in predicting permittivity values of dielectrics and cannot solve the dilemmas between image quality and time efficiency. A deep learning-based method is proposed to refine the images reconstructed by conventional methods, with a multi-level fusion layer added to derive the permittivity values of multiple (>2) different dielectrics in the sensing field. A hybrid training strategy is implemented by taking images of varied qualities as inputs to the deep neural network during training, which are reconstructed by non-iterative and iterative methods such as Linear back projection (LBP) and Landweber iteration. With this strategy, low-quality input images reconstructed by LBP can be refined to the equivalent level of quality as those by iterative methods such as Landweber iteration, with the time consumption significantly reduced. Simulation and experimental results confirm the effectiveness of the proposed method by comparing with the conventional reconstruction methods as well as other deep learning-based methods without taking images reconstructed by conventional methods as network inputs or without using the hybrid training strategy.

Purpose Crystallization is the process widely used for components separation and solids purification. The systems for crystallization process evaluation applied so far, involve numerous non-invasive tomographic measurement techniques which suffers from some reported problems. The purpose of this paper is to show the abilities of three-dimensional Electrical Capacitance Tomography (3D ECT) in the context of non-invasive and non-intrusive visualization of crystallization processes. Multiple aspects and problems of ECT imaging, as well as the computer model design to work with the high relative permittivity liquids, have been pointed out. Design/methodology/approach To design the most efficient (from a mechanical and electrical point of view) 3D ECT sensor structure, the high-precise impedance meter was applied. The three types of sensor were designed, built, and tested. To meet the new concept requirements, the dedicated ECT device has been constructed. Findings It has been shown that the ECT technique can be applied to the diagnosis of crystallization. The crystals distribution can be identified using this technique. The achieved measurement resolution allows detecting the localization of crystals. The usage of stabilized electrodes improves the sensitivity of the sensor and provides the images better suitable for further analysis. Originality/value The dedicated 3D ECT sensor construction has been proposed to increase its sensitivity in the border area, where the crystals grow. Regarding this feature, some new algorithms for the potential field distribution and the sensitivity matrix calculation have been developed. The adaptation of the iterative 3D image reconstruction process has also been described.

Electrical capacitance tomography (ECT) utilizes measured mutual capacitances across a region of interest to visualize distributions inside. As typical two-phase flows can be roughly treated as binary-valued material distributions, in this article, a fuzzy PID-controlled iterative algorithm is proposed for image reconstruction in cases of binary distributions. A closed-loop control system includes a fuzzy PID controller, Calderon’s method, and fast calculation of the Dirichlet-to-Neumann map. Capacitances measured in an electrode array of the ECT sensor are compared with the feedback, and the difference is input to the controller. Fuzzy rules are used to automatically adjust the three parameters of the controller, i.e., $K_{P}$ , $K_{I}$ , and $K_{D}$ . The controller passes the difference to Calderon’s method for reconstructing permittivity distribution. Reconstructed distribution is used to calculate a boundary map for feedback, by fast calculation of the Dirichlet-to-Neumann map, and serves as an updated reference for measured capacitances. A smooth segmentation method is also introduced to deal with the binary distribution and release the fluctuation in the tuning of the PID controller. Numerical simulations were done to verify the performance of the proposed iterative Calderon’s method for binary distributions. Experiments on real phantoms were also carried out using an ECT system to evaluate the proposed method. Several distributions were set up with solid particles and air. The results show that the proposed method can produce images with clear edges and shapes of binary distributions.

In this article, an annular electrical capacitance tomography (ECT) sensor was designed, and a revised Calderon image reconstruction method was proposed for monitoring the flashback flame of a bluff-body burner. The central support of the bluff-body was used as the internal electrode of the sensor, and the external electrode array was fixed on the outer surface of the burner. The capacitance values of all pairs of electrodes, including the internal and external, composed a boundary capacitance matrix of the annular sensing field. The revised Calderon method was compared with the commonly used sensitivity matrix-based methods including the linear back projection (LBP) and Landweber methods for imaging the cross-sectional distribution of permittivity which is continuous in value. Simulation results show that the revised Calderon method is of higher image reconstruction quality than the LBP and Landweber methods. In the experiment, the designed annular ECT sensor and revised Calderon method were used to monitor the flame flashback phenomenon of the bluff-body burner. The experiment results not only monitored the occurrence of flame flashback, revealed the variations of flame flashback intensity and time with the fuel:air equivalent ratio, but also determined the boundary condition of flame flashback of the bluff-body burner. The proposed method provides a new approach for experimental study on the influencing factors of flame flashback, and thus can be applied to research on combustion stability and high-performance burner design.

Image reconstruction for industrial applications based on Electrical Capacitance Tomography (ECT) has been broadly applied. The goal of image reconstruction based ECT is to locate the distribution of permittivity for the dielectric substances along the cross-section based on the collected capacitance data. In the ECT-based image reconstruction process: (1) the relationship between capacitance measurements and permittivity distribution is nonlinear, (2) the capacitance measurements collected during image reconstruction are inadequate due to the limited number of electrodes, and (3) the reconstruction process is subject to noise leading to an ill-posed problem. Thence, constructing an accurate algorithm for real images is critical to overcoming such restrictions. This paper presents novel image reconstruction methods using Deep Learning for solving the forward and inverse problems of the ECT system for generating high-quality images of conductive materials in the Lost Foam Casting (LFC) process. Here, Long Short-Term Memory Recurrent Neural Network (LSTM-RNN) models were implemented to predict the distribution of metal filling for the LFC process-based ECT. The recurrent connection and the gating mechanism of the LSTM is capable of extracting the contextual information that is repeatedly passing through the neural network while filtering out the noise caused by adverse factors. Experimental results showed that the presented ECT-LSTM-RNN model is highly reliable for industrial applications and can be utilized for other manufacturing processes.

In this article, a fuzzy adaptive Kalman filter (FAKaF)-based method was proposed for image reconstruction in electrical capacitance tomography (ECT). When the Kalman filter (KF) is applied for image reconstruction in ECT, two key parameters need to be predetermined, i.e., the observation noise covariance ( ${R}$ ) and the initial estimation error covariance ( ${P}_{0}$ ). These two parameters play significant roles in image reconstruction. For instance, a larger ${R}$ may lead to a blurrier image. A larger ${P}_{0}$ can cause increasing artifacts or even heavier distortion of the reconstructed image. In this work, a FAKaF was established to adjust ${P}_{0}$ using ${R}$ calculated from the measured capacitances so as to improve the quality of the reconstructed image. The implementation of the FAKaF-based reconstruction method was divided into offline and online parts. In the offline part, the Kalman gain and the corresponding fuzzy control table were precalculated, aiming to save resource consumption and improve imaging speed. Simulations and experiments were carried out to evaluate the image quality and computational cost of the proposed method. Comparisons were made with three widely-used algorithms. Results show that the proposed FAKaF-based method yields good quality images and few artifacts, needs few iterations and consumes less computational cost.

This article presents a method for real-time 3-D imaging and velocity measurement of two-phase flows using a twin-plane electrical capacitance tomography (ECT) sensor with 24 electrodes. A dedicated digital ECT system with a data acquisition rate of 241 frames/s is used for capacitance measurement. In this article, the sensitivity distribution of the ECT sensor is analyzed, and the two most commonly used algorithms, linear back projection (LBP) and Tikhonov, are used to achieve high-speed 3-D image reconstruction. The cross correlation technique is used to calculate the velocity of a two-phase flow. Experiments were carried out to evaluate the performance of the proposed method. Several typical distributions were constructed and used to analyze the axial imaging resolution. A free-falling sphere was used to test the ability of the system in velocity measurement. Experimental results show that the proposed method can provide high axial resolution in the central zone of the 3-D sensor and high temporal resolution in velocity measurement.

In this study, we propose a new sensor structure with driving electrodes and a new excitation mode in electrical capacitance tomography (ECT) to improve: 1) the low sensitivity distribution in the central area caused by low potential distribution in the central area and 2) the nonuniform sensitivity distribution caused by the nonuniform potential distribution. An imaging method based on sensitivity map optimization is derived through electromagnetic field analysis and numerical simulation. In addition, the change rule of the optimal driving voltage is studied when the excitation voltage of the excitation electrodes and the length of the driving electrodes change. The numerical simulation and experimental results show that the imaging method proposed in this article can distinctively alter the potential distribution and sensitivity distribution in the central area by selecting the optimal driving voltage, which brings significant improvement in image reconstruction.

As a fast and nonintrusive measurement and visualization technique, electrical capacitance tomography (ECT) is rapidly expanding its applications in the research on multiphase flow, fluidization, drying, combustion, and so on. However, the marked unevenness of the sensitivity maps sometimes causes unexpected effects in imaging reconstruction, particularly in 3-D cases. To exploit the positive potential of this phenomenon, the authors proposed an image fusion method using the data from two units of ECT sensors in this study. This method is used in image fusion on the reconstructed images for a planar sensor and a cylindrical sensor. In contrast to the conventional fusion models that use fixed weight factors for two sources of data, our model forges weight functions that are set preference the strength of the sensitivity maps. The new algorithm is implemented by first extracting the characteristic information out of the ECT images using the latent low-rank representation and then performing a fusion algorithm with linear weight functions in preference to the significance of the sensitivity maps. The simulation results show that the algorithm effectively retains the advantages of the two units of sensors and mutually compensates the weak points of theirs, and significantly improves the reconstruction quality. The fusion image quality by the new method can response the real situation better in different heights. The results imply that this data fusion method can amend the weakness of ECT cause by the uneven sensitivity maps to a significant extend.

Benefiting from the ability to image the permittivity distribution of dielectric materials, electrical capacitance tomography (ECT) has been applied for multiphase flow metering for decades as a contactless method. However, the water-continuous flow brings challenges for ECT since the conductivity in water makes ECT fail to reconstruct the distribution. Therefore, complex-valued ECT (CV-ECT) is introduced to image both permittivity and conductivity distribution based on complex-valued capacitance measurements using the same sensor head of ECT. Different from conventional ECT, the investigation of excitation frequency and linearization point selection is vital for CV-ECT, as the conductivity information is coupled with permittivity and frequency. An 8-electrode CV-ECT system was set up to obtain measurements both in simulations and experiments. The measurements on different phantoms over different excitation frequencies were conducted and the images were reconstructed to elaborate the selection of the linearization point and excitation frequency range.

Electrical capacitance tomography is a widely used sensor modality for flow imaging in many industrial settings. Adaptive electrical capacitance volume tomography (AECVT) extends the capabilities of traditional ECT by enabling direct volumetric imaging and an improved resolution. Construction of the sensitivity matrix is a necessary step to obtain flow images. This step requires the computation of the electric field inside the sensing domain, which is done via a typical field solver, such as the finite-element method. In this work, we present an efficient and flexible method to construct the sensitivity matrix for AECVT based on individual electrode segment excitations and their judicious combination to form desired matrix elements. We illustrate how the proposed method yields the same sensitivity matrix as the traditional method but at a much lower computational cost. Once all segment contributions are obtained, we also indicate how the proposed method, unlike the traditional approach, can generate the sensitivity matrix on demand for an arbitrary combination of synthetic electrodes and obviating the need for any additional field computations. Finally, we present image reconstruction results for two different experimental scenarios where the mutual capacitance data and the corresponding sensitivity vectors are obtained through the proposed measurement combination scheme.

Electrical capacitance tomography (ECT) is a technique to visualize the permittivity distribution inside a domain from the interelectrode mutual capacitances on the domain boundary. The potential applications of ECT in nondestructive testing (NDT) depend on accurate and stable image reconstruction. To improve image quality, this article presents a reconstruction framework with the sparse representation of phase boundaries in two-phase distributions. By adopting a sparsity-promoting basis, i.e., radial basis functions, the reconstruction problem is transformed into searching for the optimal phase boundaries and real-permittivity values of inclusions by formulating an optimization problem accordingly. This can reduce the number of unknowns significantly and has a strong ability to accommodate to the topology changes of inclusion geometries, which can improve the reconstruction accuracy and robustness. Both simulation and phantom experiments are performed to investigate the reconstruction performance, especially regarding the antinoise ability and robustness against choices of model parameters. Compared with typical conventional methods, our method can not only provide significantly improved image quality but also accurately estimate the real-permittivity values of inclusions. This would enable ECT to be applied in challenging NDT applications.

One of the vital processes that should be monitored and analyzed continuously in the oil-gas and petroleum-related industries is the multi-phase flow inside pipes. Multi-phase flow means flowing two or more phases of gas, liquid, or solid inside a pipe. Electrical Capacitance Tomography (ECT) is a feasible and economical solution for monitoring dynamic applications. The ECT system offers the benefits of no radiation, non-intrusive, and non-invasive. Despite its potential, ECT systems deployment's major limitation is the crucial need to develop rapid image reconstruction algorithms. In this paper, a Local Ensemble Transform Kalman Filter (LETKF) is developed as a non-linear system estimator for reconstructing images in the ECT system. This method manages each node of the model independently by assimilating only the observations at a predefined distance. The localized approach of the LETKF gives it high computational efficiency allowing it to be applied to large dynamic systems. A quantitative analysis using Image Error (IE) and Coefficient Correlation (CC) measures has been applied to prove the effectiveness of the proposed algorithm. Indeed, the IE has been significantly decreased (around 62%), and the CC greatly increased (around 58%). Then, the influence of the noise was discussed. The results are promising and prove the algorithm feasibility.

In this paper, an algorithm using electrical capacitance tomography has been proposed to image and continuously monitor individual phases of a multiphase flow components. Instead of directly reconstructing the permittivity of the imaging domain, the proposed method presents successful reconstruction of fractional area/volume of each phase of a multiphase flow using only single frequency measurements. The permittivity image can then be calculated from the reconstructed fraction parameters. For the fraction reconstruction, the conventional cost function is reformulated and solved employing the well-known Landweber method typically used in ECT. Several three and two phase flow scenarios, including gas-oil-water process, have been analyzed and successful fraction parameter reconstructions are demonstrated. Apart from yielding successful reconstruction of individual phases, the proposed method also increases the overall accuracy of permittivity recovery consistently for all the explored cases. Hence, the proposed fraction imaging algorithm can potentially serve as a tool to decompose individual phases of a multiphase flow along with increased image accuracy.

Electrical capacitance tomography (ECT) has been developed for imaging permittivity distribution of two-phase flows in a pipe or vessel with high temporal resolution. The spatial resolution of ECT is relatively low due to its soft-field, ill-posed, and ill-conditioned nature. So far, there is no comprehensive analysis of the spatial resolution of ECT for complex two-phase flows. To assess the merits of ECT systematically, 2-D modeling is carried out based on fluid-field and electrostatic-field coupling simulation for liquid–solids two-phase flows. The real permittivity distribution and the corresponding capacitance measurements can be obtained simultaneously. The effect of the number of electrodes, the mesh used for reconstruction, and the initialization method for the Landweber iteration algorithm on image quality are investigated. In terms of the correlation coefficient between the reconstructed permittivity distribution and the real permittivity distribution obtained from the coupling simulation model, the accuracy and the stability of reconstructed images are quantitatively evaluated. Furthermore, the sensitivity matrices and point spread functions calculated without capacitance measurements are used to compare the performance of ECT sensors and validate the results with capacitance measurements from the coupling simulation. The two methods, with and without capacitance measurements, are used to evaluate the change in spatial resolution of ECT with a different number of electrodes.

The characteristics of flow configuration in pipes are very important in the oil industry due to its role in governing equipment design. In vertical risers, many flow configurations could be observed such as bubbly, slug, churn, and annular flow. In this project, two tomographic techniques have been applied simultaneously to the flow in a vertical riser: the Electrical Capacitance Tomography (ECT) technique and the Capacitance Wire Mesh Sensor (WMS) technique. The employed pipe diameter was 50mm and the superficial studied velocities were 0.06-3.0m/s for gas and 0.06-0.4m/s for oil. Several techniques have been used to analyze the output data of the two tomography techniques such as time series of cross-sectional averaged void fraction, Probability Density Function (PDF), image reconstruction, and liquid hold-up profile. The averaged void fractions were calculated from the output signal of the two measurement techniques and plotted as functions of the superficial velocity of the gas. The flow patterns were identified from the PDF of the averaged void fraction. In addition, it was found that both tomographic techniques are reliable in identifying the flow regimes in pipes.

Electrical Capacitance Tomography (ECT) has been developed for many years and made great progresses. Successful applications of ECT depend on the accuracy and speed of image reconstruction. In this paper, we propose a new image reconstruction method based on deep neural network. The proposed neural network mainly consists of three parts, i.e. a forward problem network, an inverse problem network and a permittivity prediction network. Compared to previous reconstruction algorithms, the benefit of our method is that it can not only reconstruct the object shape, but also predict its permittivity value with the preset resolution which is desired in many multiphase flow applications. In experiment, 10000 frames of simulation data and additional measured capacitance data were used. The results show that the proposed method can reconstruct images more accurately than typical iterative methods with real permittivity values of objects predicted correctly, and reduce the computational cost to about 2.24 seconds per frame for an image resolution of 200*200.

The linear back projection (LBP) algorithm is often used for real-time online image reconstruction of electrical capacitance tomography (ECT) due to its high speed. However, due to the fact that the image reconstruction of ECT is a nonlinear ill-posed inverse problem, reconstructed images obtained by the LBP algorithm that simplifies ECT image reconstruction as a linear problem, tend to have distortion and can only be used for qualitative observation. In this paper, a deep fully-connected neural network, which improves the imaging quality of the LBP algorithm by compensating its imaging results is proposed. Instead of simplifying the ECT image reconstruction as a linear problem, our proposed compensated LBP algorithm uses a deep neural network to map the nonlinear relationship from capacitance to permittivity distribution. Furthermore, the difference between the capacitance regarding the permittivity distribution reconstructed by the LBP algorithm and the actual capacitance is used as the input of the network while the difference between the reconstructed permittivity distribution and the actual permittivity distribution is used as the output of the network. The results of the network can be used to compensate the image reconstruction results of the LBP. This strategy makes the ECT image reconstruction need only to deal with a support interval significantly smaller than that of the original ECT image reconstruction problem and is helpful to suppress the nonlinearity to be trained. Both the training and testing results based on simulation data instances and experimental data show that the proposed compensation network has a great improvement on image reconstruction results of the LBP algorithm. In addition, the computation load is comparable to the original LBP algorithm.

In order to improve the quality of reconstructed images in electrical capacitance tomography (ECT), the image reconstruction method based on compressed sensing for ECT is studied. First, the traditional discrete Fourier transform and discrete cosine transform are used as a sparsity basis to make the grey vectors of the typical two-phase flow distributions sparse. The energy loss of the sparse signals under different sparsity degrees is calculated, and the effect of energy loss on the quality of reconstructed images is studied. Then, using the natural sparsity of the original signal, an improved orthogonal matching pursuit algorithm for ECT image reconstruction is proposed. There are two main improvements in the proposed algorithm. First, multiple columns instead of one column in each iteration are selected for improving the reconstruction speed. Second, a regularisation solution instead of the least-squares solution is used for improving the adaptability to ill-posed inverse problems. Simulation and experimental tests are carried out and the results show that the proposed method can effectively improve the reconstructed images quality, and on the whole, obtain better reconstruction results than the Landweber iteration algorithm, the Tikhonov regularisation algorithm, and the gradient projection for sparse reconstruction algorithm.

We present a Relevance Vector Machine (RVM) based algorithm for electrical capacitance tomography (ECT) applications that can concurrently provide image reconstruction results and uncertainty estimates about the reconstruction. To illustrate the RVM operation in ECT, we simulate typical ECT scenarios, making explicit the connection between the reconstructed pixel values and the corresponding uncertainty estimates in each case. We compare the RVM reconstruction performance with that of the Iterative Landweber Method (ILM) and the least absolute shrinkage and selection operator (LASSO) in all the considered scenarios. The results show that, in addition to the key advantage of providing uncertainty measures, RVM can achieve similar reconstruction results with either lower or similar computational complexity.

The electrical capacitance tomography (ECT) technique has been extensively studied for real-time tomographic imaging of various industrial processes. Due to the ‘soft-field’ and under-determined problems in the image reconstruction, the tomographic images of ECT are usually low in the spatial resolution when compared with its radioactive counterparts, i.e. $\gamma $ -ray tomography and computed tomography. This paper presents an ECT sensor that consists of 8 external electrodes and 8 internal electrodes, with enhanced image quality at the center of imaging region and better accuracy on phase fraction measurement. The internal electrodes are placed on cross planes inside the investigated vessel. In this way, the signal-to-noise ratio of measurement channels as well as image quality are improved. Numerical simulations show that the proposed sensor can achieve better reconstructed images than a conventional 12-electrode sensor. In addition, the phase fraction calculation using ECT data can also benefit from the internal electrodes and corresponding measurements, which is an important parameter for multiphase flow processes.

This paper presents a sectional water fraction measurement method using electrical capacitance tomography (ECT) image. To achieve a desired measurement accuracy, an optimal threshold-search method based on fast simulated annealing (FSA) algorithm is applied for image binarization; and to guarantee the imaging time consumption, fast fixed point continuation (FFPC) iterative algorithm cooperating with the compressed sensing (CS) theory was applied to the image reconstruction. In this study, the imaging time consumption and the measurement accuracy of water fraction are used to compare the proposed method and other methods. A numerical model is established with COMSOL Multiphysics to conduct the simulative validation of the proposed method firstly and physical experiments are conducted then. The experiment results show the average relative error of water fraction is lower than 15%, where the even as low as 3.68%.

Electrical Capacitance Tomography (ECT) is a well-established industrial process tomography technique. Image reconstruction for the ECT is a nonlinear problem, and the inverse problem is usually ill-posed and ill-conditioned. Hence, the solutions for the ECT are not unique and highly sensitive to the measurement noise. In this paper, a novel tuned fuzzy algorithm is proposed for reconstructing accurate images to monitor the distribution of the multi-phase flow in the industrial process. The proposed algorithm utilizes a Tuned Fuzzy Inference System (TFIS) to overcome the nonlinear characteristics of the ECT system. The optimal parameters of the fuzzy membership functions are obtained using the Particle Swarm Optimization (PSO) technique. In the past few decades, the naturally inspired intelligent swarm algorithms got more attention due to their wide spectrum of research for real-world complex problems optimization. The proposed PSO-tuned fuzzy algorithm is fast since it does not require solving the forward problem to update the sensitivity matrix. Comparing the results with traditional reconstruction algorithms, the proposed algorithm performs better in visual effects and imaging quality, since the image edges and details are better preserved.

Image reconstruction of electrical capacitance tomography (ECT) is an ill-posed inverse problem. The simulated annealing (SA) algorithm is suitable for the solution of the ECT inverse problem. However, selection of related parameters of the SA algorithm will influence the reconstruction performance of ECT. We present an ECT image reconstruction method based on the adaptive simulated annealing (ASA) algorithm. We adopt the bat algorithm in the new solution generation strategy of the ASA algorithm. Moreover, the definition of the energy function introduces the sparsity of the reconstructed image. As a result, an adaptive annealing strategy is proposed to select the appropriate annealing rate. We made reconstructed image comparisons among linear back-projection, Landweber iteration, SA, and ASA algorithms via simulation and static experiment. Results show that improved reconstructed images can be obtained using the ASA algorithm.

Electrical capacitance tomography is an industrial process tomography technology, mainly used to measure and plot two-phase flow and multi-phase flow. The technology is based on the properties of various dielectric constants between the phases of the measured substance. This work focuses on the improvement of particle swarm optimization and intelligent algorithms, especially for the parameter control of particle swarm optimization and intelligent algorithms. The efficiency of group optimization algorithms and intelligent algorithms that solve optimization problems. This paper mainly aims at the image reconstruction process in electrical capacitance tomography system, and proposes an image reconstruction algorithm based on intelligent algorithm and particle swarm algorithm. Combined with the experimental environment of the self-made electrical capacitance tomography system, the actual imaging effect of the algorithm was compared and analyzed with the traditional imaging algorithm, and the verification of the algorithm improvement effect was theoretically completed. According to the finite element analysis method, the internal area of the sensor is subdivided on the entire network, and three flow modes are modeled, which provides conditions for the construction of the following experimental environment. Based on the theory, the principle of the classic Landweber imaging algorithm is discussed in detail. Using the experimental environment with built-in electrical capacitance tomography system, experiments were conducted to visualize the gas–liquid two-phase flow. The traditional Landweber algorithm and imaging algorithm proposed in this paper are used to reconstruct the image using the obtained volume data. Through comparative analysis of the resulting images, the results show that the imaging algorithm proposed in this paper improves the accuracy of flow pattern recognition and image accuracy, which proves the improvement. The feasibility of algorithm and particle swarm algorithm.

For the problems of low sensitivity, weak signal of high and low frequency and low signal-to-noise ratio in ECT, the mathematical model of the sensor is established. From the aspects of electrostatic field distribution and soft field effect, the influence of the structural parameters of the sensor on the sensor performance is analyzed. According to the influence of the components of the sensor on the sensitivity, the principle of optimal design is put forward. Based on the optimized Landweber image reconstruction algorithm, an ART image reconstruction algorithm with iterative correction is proposed, and the mathematical model of the algorithm is designed. According to constructing the target functional regularization term in the negative problems of electrical capacitance tomography, the iterative process of the modified art algorithm is deduced, and with adaptive step size, the convergence is speeded and accuracy of image reconstruction is improved. The experimental results show that the semi-convergence in the improved algorithm is obviously weakened, and the reconstructed image quality is better than that of the traditional art algorithm.

In the image reconstruction of the electrical capacitance tomography (ECT) system, the application of the total least squares theory transforms the ill-posed problem into a nonlinear unconstrained minimization problem, which avoids calculating the matrix inversion. But in the iterative process of the coefficient matrix, the ill-posed problem is also produced. For the effect on the final image reconstruction accuracy of this problem, combined with the principle of the ECT system, the coefficient matrix is targeted and updated in the overall least squares iteration process. The new coefficient matrix is calculated, and then, the regularization matrix is corrected according to the adaptive targeting singular value, which can reduce the ill-posed effect. In this study, the total least squares iterative method is improved by introducing the mathematical model of EIV to deal with the errors in the measured capacitance data and coefficient matrix. The effect of noise interference on the measurement capacitance data is reduced, and finally, the high-quality reconstructed images are calculated iteratively.

While Landweber iteration is a popular image reconstruction algorithm for electrical capacitance tomography (ECT), it suffers from semi-convergence. In process control, PID controllers are commonly used, where P stands for proportional, I stands for integral, and D stands for derivative. Proper setting of the PID parameters can reduce the response time and enhance the accuracy and stability of control systems. In this paper, an extended PI controller is integrated with the Landweber method to relieve its semi-convergence, which is named the Landweber-PI method. To stabilize the iteration process, constraints are imposed on the integral term of the PI controller. In comparison with the conventional Landweber method and its modified variants, simulation was conducted to verify the effectiveness of the proposed method by imaging typical distributions using ECT sensors with different number of electrodes, 8 or 12 or 16 electrodes, and capacitance data without and with noise. The simulation results were validated by static experiment, showing that the proposed method converges within 50 iterations for the specified core, annular, stratified, and other two complex distributions and remains stable after 1000 iterations, while the other methods converge slowly and become divergent eventually. Regarding the reconstruction accuracy, the derived void fractions by the proposed method have smaller absolute deviations than the other methods in most cases, with a minimum of 0.3% and a maximum of 4.3%. The gas–solid flows in a real fluidized bed were imaged, with more image details reconstructed after five iterations. The improved performance of the Landweber-PI method was justified by singular value decomposition analysis.

Electrical capacitance tomography (ECT) is a widely used imaging modality to image two-dimensional cross sections of multiphase flows. Recent developments in electrical capacitance volume tomography (ECVT) have made it possible to directly obtain volumetric images from measured data. An ECVT is instrumental for obtaining accurate phase hold up information and velocity information that are needed for the optimization of certain flow processes. However, compared to ECT, the high correlation between the measurements in ECVT exacerbates the ill-conditioning of the associated image reconstruction problem. Previous studies have suggested that neglecting mutual capacitance data between the ECVT electrodes located at cross-planes that are well separated along the sensor axis can be done without significantly affecting the reconstructed image. In addition, this may help constrain the ill-conditioning of the reconstruction problem. In this paper, we examine in detail and quantify the effect of reduced cross-plane acquisition strategies for optimizing the image reconstruction and constraining the ill-conditioning of typical ECVT settings.

Electrical capacitance tomography (ECT) is one kind of two/three-phase flow parameter visualization measurement technique. Image reconstruction of ECT is an ill-posed problem. Among the two-phase flow regimes, there are such distribution situations as objects being in the centre, objects being rather small or many objects being close to each other, the accuracy of reconstructed images is relatively poor. Based on the fact that the permittivity distribution of these flow regimes satisfies the priori condition of sparsity, the ECT image reconstruction algorithm based on Barzilai-Borwein gradient projection for sparse reconstruction (GPSR-BB) was presented in this paper. Simulation and experimental tests were carried out for the sparse distribution phantoms and experimental results show that the quality of reconstructed images can be enhanced obviously.

In this paper, the shortcut D-bar method is introduced to electrical capacitance tomography to directly reconstruct the permittivity distribution within the region of interest. The method reveals the analytic relationship between the permittivity distribution and the Dirichlet-to-Neumann (DN) map. The DN map can be directly estimated with measured capacitances. However, the direct estimate of the DN map is inaccurate in the cases of small electrode numbers. To better adapt the method to real applications of small electrode numbers, the original shortcut D-bar method is modified by splitting estimation of the DN map into the analytical expression of a homogeneous background and the estimation of the other variation part. Comparisons are made among the original, modified shortcut D-bar methods, and the classical Landweber iteration method through simulation and experiment, respectively. Results confirm the improved tomographic imaging quality by using the modified shortcut D-bar method.

The image reconstruction is an important step in the electrical capacitance tomography (ECT) technology, and its performance directly impacts the reconstruction precision (RP). Beyond existing optimization-based imaging techniques, in this study the data-dependent reference image abstracted by the regularized random vector functional link network (RVFLN) and the domain expertise about imaging targets (ITs) are simultaneously encapsulated as regularizers to form a more effective imaging model. The three-operator splitting (TOS) technique is developed to solve the proposed imaging model more effectively, which extends the flexibility of the TOS method with the improvement in the utilization of image priors. Numerical validation results indicate that the proposed imaging technique achieves better reconstructions as compared with the state-of-the-art methods.

In the electrical capacitance tomography (ECT) systems, the electrodes, and cables may fail to function properly, which will cause several measurements missing. In these cases, image reconstruction can only use the remaining effective measurements. In order to make the reconstructed images close to the image results of the complete measurement set, it is necessary to use the incomplete measurements reasonably. The measurement/data recovery method and image reconstruction can be conducted to obtain the results, which meet the imaging needs under these circumstances. The measurement/data recovery method by using the sensitivity matrix and the regression model of least square support vector machine (LS-SVM) are proposed. The image recovery result is reconstructed by the method of total variation (TV) minimization. The simulations and experiments of gas-solids two-phase measurement are conducted to validate the method.

Imaging of multiphase flows holding water as continuous phase (i.e., water-dominated flows) is very challenging for conventional electrical capacitance tomography (ECT) due to the high permittivity of water. In this paper, we introduce a new approach, based on the multi-frequency excitation of ECT sensors, for imaging and real-time monitoring of water-dominated columnar or slug vertical flows. The proposed method exploits differences between measurements obtained at distinct frequencies caused by the Maxwell–Wagner–Sillars effect, which is present in multiphase flows with at least one conducting phase. To illustrate this new approach, several numerical simulations are carried out for two-phase and three-phase mixtures containing air, methylamine, and/or oil as dispersed phases and with water as the continuous phase. Experimental results are also provided to validate the findings.

Electrical capacitance tomography (ECT) is a non-invasive and non-intrusive imaging modality that utilizes mutual capacitance measurements between electrode plates to reconstruct the cross-sectional spatial electrical permittivity distribution inside a region of interest (RoI). Increasing attention has been given in recent years to the extension of ECT to volumetric imaging, also known as electrical capacitance volume tomography (ECVT). Most of the desirable properties of ECT carry over to ECVT. However, the speed of reconstruction is reduced in ECVT due to an increase in the number of measurements and in the number of voxels used to represent the permittivity distribution in the RoI. In this paper, we adopt a Fourier basis representation as a means to accelerate the speed of image reconstruction in ECVT. By considering only spatial frequency components below certain threshold, we show that the number of unknowns can be greatly reduced and the overall reconstruction process significantly accelerated. We present both numerical and experimental results to corroborate our findings.

The electrical capacitance tomography (ECT) technology is a promising tomography technology that can image the distribution information of permittivity in a measurement region. The imaging algorithms play crucial roles in practical ECT image reconstruction problems. Different from previous numerical methods, this study proposes a novel cost function to model the ECT inverse problem, in which the L1-norm is used as the data fidelity to weaken the influence of the outliers contained in the capacitance data, the L1 regularization is introduced to enhance the sparseness of reconstructed objects, and the second order total variation (STV) regularization is used to weaken the staircasing effects caused by the first order total variation (FTV) regularization. The split Bregman iteration (SBI) algorithm that splits a complicated ECT imaging problem into several simpler sub-problems is developed to solve the proposed cost function effectively. The numerical simulation results show that the imaging algorithm proposed in this paper can reconstruct satisfactory results.

As a powerful tool to get a recursive solution of least squares estimation, the Kalman filter has been used for image reconstruction in Electrical Capacitance Tomography (ECT). In the Kalman-filter-based image reconstruction method, some key parameters, e.g., initial guess, observation noise covariance and initial estimate error covariance, greatly influence the performance of the method. Inappropriate values of these parameters may cause a series of problems, such as lower convergence rate, artifacts, or filter divergence. This paper aims to analyze the influence of the parameters on the image quality for ECT and guide the selection of the parameters. Numerical simulation and experiment were carried out and the results show that with an initial guess obtained by linear back projection (LBP) method and a good match of observation noise covariance and initial estimate error covariance, the performance of the Kalman-filter-based method can be improved.

Electrical capacitance tomography (ECT) is a technique that uses image reconstruction algorithms to reconstruct permittivity distributions in the region of interest (ROI) with capacitances measured at the boundary of ROI. In this paper, a closed-loop iterative reconstruction algorithm is proposed, and it aims to eliminate the error between the measured capacitances and recuperative capacitances, and in this way, the quality of the reconstructed images is improved. The proposed algorithm includes three parts, namely, Fractional-order Proportional Integral Derivative (FPID) controller, Calderon's method, and a fast-solving forward method for numerical simulated capacitances in ECT. The error to be eliminated is the difference between measured capacitances and the simulated capacitances and is input to the FPID controller. When the coefficients in the FPID controller are determined, the output of controller is used to reconstruct permittivity distribution in the Calderon's method. Reconstructed image provides simulated capacitances as an updating reference for measured capacitances for the iterative reconstructions. Both noise-free and noisy data are used and verified the feasibility and effectiveness of the proposed algorithm for ECT.

Electrostatic tomography (EST) is a passive tomographic image method, which determines that the number of independent measurements and the signal-to-noise ratio (SNR) of its signal are smaller than those of active excited electrical tomography (ET) such as electrical capacitance tomography (ECT). The nonlinearity and ill-posed property of EST are more prominent due to the nature of passive measurement. The traditional iterative approximation method is insufficient to express the nonlinear nature of image reconstruction, so the accuracy of the reconstructed image is low. To solve this problem, a residual network (ResNet) model is proposed in this paper. A new loss function based on the prior knowledge of the boundary measurement of EST is proposed to make the model fit the imaging principle better. In order to build the dataset, 3500 samples are generated through simulation and divided into training set and validation set. Several typical flow patterns are simulated separately as test set. The reconstructed results of the model on the test set have high accuracy compared with some conventional algorithms. When Gaussian white noise with SNR of 10 dB, 15 dB and 20dB is added to the test set, the reconstructed results of the model can still represent the approximate position of the sample, which proves the robustness of the model.

An important part of ECT technology is image reconstruction in Electrical Capacitance Tomography (ECT).In order to study the influence of different sensors on the reconstructed image quality, sensors of different shapes are designed for specific ring objects in this paper, and LBP and Landweber imaging algorithms are used to analyze the influence of different sensors on image reconstruction results. Experiments show that the designed ring sensor has better imaging effect than the traditional square or cylindrical sensor.

Image reconstruction algorithm and its controller constitute the main modules of the electrical capacitance tomography (ECT) system; in order to achieve the trade-off between the attainable performance and the flexibility of the image reconstruction and control design of the ECT system, hardware-software codesign of a digital processing unit (DPU) targeting FPGA system-on-chip (SoC) is presented. Design and implementation of software and hardware components of the ECT-DPU and their integration and verification based on the model-based design (MBD) paradigm are proposed. The inner-product of large vectors constitutes the core of the majority of these ECT image reconstruction algorithms. Full parallel implementation of large vector multiplication on FPGA consumes a huge number of resources and incurs long combinational path delay. The proposed MBD of the ECT-DPU tackles this problem by crafting a parametric segmented parallel inner-product architecture so as to work as the shared hardware core unit for the parallel matrix multiplication in the image reconstruction and control of the ECT system. This allowed the parameterized core unit to be configured at system-level to tackle large matrices with the segment length working as a design degree of freedom. It allows the trade-off between performance and resource usage and determines the level of computation parallelism. Using MBD with the proposed segmented architecture, the system design can be flexibly tailored to the designer specifications to fulfill the required performance while meeting the resources constraint. In the linear-back projection image reconstruction algorithm, the segmentation scheme has exhibited high resource saving of 43% and 71% for a small degradation in a frame rate of 3% and 14%, respectively.

Complex-valued, multifrequency electrical capacitance tomography (CVMF-ECT) is a recently developed tomographic concept which is capable of simultaneously reconstructing spectral permittivity and conductivity properties of target objects within the region of interest. To date, this concept has been limited to simulation and another key issue restricting its wide adoption lies in its poor image quality. This letter reports a CVMF-ECT system to verify its practical feasibility and further proposes a novel image reconstruction framework to effectively and efficiently reconstruct multifrequency images using complex-valued capacitance data. The image reconstruction framework utilizes the inherent spatial correlations of the multifrequency images as a priori information and encodes it by using multiple measurement vector (MMV) model. Alternating direction method of multipliers was introduced to solve the MMV problem. Real-world experiments validate the feasibility of CVMF-ECT, and MMV-based CVMF-ECT method demonstrates superior performance compared with conventional ECT approaches.

Solid contaminants have been found to be a major obstacle in assuring the quality of flow in pipelines. The presence of these substances leads to clogging of pipes that causes serious issues downstream when transporting oil or gas. This paper suggests an Electrical Capacitance Tomography (ECT) system for realtime measurement of solid contaminants in gas pipelines. It consists of a ring of eight electrodes evenly distributed in the circular cross section of the probe. The speed-up enhancement is achieved using a Field Programmable Gate Array (FPGA) for the post-processing part of the system to accelerate the intensive matrix multiplications which are required in the image reconstruction algorithm. Experimental results on field-collected solid contaminants demonstrate the capability of the system to build in real-time two dimensional cross section images of the contaminants while giving an estimated measurement of their concentration. This helps identify the flow regime of the contaminants in the pipeline, which is required to know their flow characteristics which helps mitigating their formation. Experimental results indicate that using Altera’s Stratix V FPGA, 305 KLEs are required to achieve image reconstruction throughput of up to 3,233 frames/s for image size of 64 x 64 pixels. Simulation results were also conducted using finite element method solver to assess the ECT probe for various image reconstruction algorithms (i.e Linear back projection, Landweber, and modified Landweber algorithms). The results indicate a good matching with the experimental results.

This paper presents the feasibility study of dynamic flow measurements using the concept of a rotatable electrical capacitance tomography (ECT) sensor. The experiment considered horizontal flow in a pneumatic conveying flow loop in the case of dense phase flow. Slugs and settled layers were imaged and a comparison was made between no rotation or rotation of the sensor for two image reconstruction schemas: linear back projection (LBP) and non-linear iterative back projection. Data were evaluated both qualitatively and quantitatively by estimating the solids concentration level for different hue levels.

In this paper, Ansoft Maxwell software was used to build the model, and sensitivity field analysis and reconstruction imaging were carried out with Matlab. Firstly, Ansoft Maxwell finite element analysis software was used to simulate the sensor model, and different capacitance values were obtained by exciting different electrode plates in the sensor. Secondly, the imaging of the sensitivity field is realized based on Matlab software, and the influence of different electrode plates on the sensitivity field is analyzed. Finally, 2d ECT image reconstruction of the cylinder model is carried out, and good results are obtained.