A non-convex regularization method combined with Landweber method for image reconstruction in electrical resistance tomography

Electrical resistance tomography (ERT) is a promising technique with which the conductivity distribution in the detected region can be visualized. Mathematically, the reconstruction of conductivity distribution is a seriously ill-posed inverse problem which poses a great challenge for the ERT sensing technique. The regularization method has been found to be an effective approach in coping with the inverse problem. In this work, a novel reconstruction strategy which combines the non-convex regularization method with Landweber method is proposed for the image reconstruction in ERT. At each iteration, the non-convex regularization is used to constrain the conductivity calculated with the Landweber method. A simple and efficient generalized iterated shrinkage algorithm is developed to solve the proposed method. To validate the performance of the proposed method, a series of numerical simulation is conducted and comparative analysis with other methods is performed. From the results, it can be observed that images with high quality are obtained when reconstructing with the proposed method. The impact of noise on the reconstruction is also investigated which shows that the images reconstructed by the proposed method are the least sensitive to the noise. The performance of the proposed method in the image reconstruction is also verified by experimental data. The results demonstrate that the inclusion is accurately reconstructed and the background is clear when the proposed method is adopted for the image reconstruction.     

A hybrid regularization method combining Tikhonov with total variation for electrical resistance tomography

Electrical resistance tomography (ERT) is a promising measurement technique in industrial process imaging. However, image reconstruction in ERT is an ill-posed inverse problem. Regularization methods have been developed to solve the ill-posed inverse problem. Since the penalty term is a form of L2-norm, Tikhonov regularization method guarantees the stability of the solution, but it always makes the image edge oversmoothed. Total variation (TV) regularization method has good ability of preserving image edges. A hybrid regularization method, which combines Tikhonov with TV regularization method, is proposed to get better reconstructed images. The choice of the adaptive weighted parameter between TV and Tikhonov penalty term has been discussed in detail. In the proposed hybrid regularization method, the function of conductivity gradients is used as the adaptive weighted parameter to control automatically the weighting between the penalty terms from TV and Tikhonov regularization. For the model with sharp edges, the proportion of the penalty term from TV regularization is increased to preserve the edges, while for the model with smooth edges, the proportion of penalty term from Tikhonov regularization is increased to make the solution stable and robust to noise. Both simulation and experimental results of Tikhonov, TV and hybrid regularization method are shown respectively, which indicates that the hybrid regularization method can improve the reconstruction quality with sharp edges and is more robust to noise, and it is applicable for models with different edge characteristic.     

An adaptive Tikhonov regularization parameter choice method for electrical resistance tomography

Electrical resistance tomography (ERT) reconstructs the conductivity distribution from the boundary changes of electrical measurements. The inverse problem of ERT is seriously ill-posed where regularization methods are needed to treat this ill-posedness. A proper choice of regularization parameter which controls the degree of smoothing is very important for these regularization methods. Although have been a variety of methods, such as L-curve method, to choose a reasonable parameter for the problem, these methods usually result in a scalar parameter which cannot distinctly express the spatial characteristic of the conductivity distribution. So a spatially adaptive regularization parameter choice method is proposed for regularizing the inverse problem of ERT based on Tikhonov regularization. Since large regularization parameters can stabilize and smoothen the solution, while small regularization parameters can approximate and sharpen the solution, the proposed method adaptively updates the regularization parameters during the iteration process and provides spatially varying parameter for each pixel of the reconstructed image. When the iteration is stopped, large regularization parameters for the smooth background region and small regularization parameters for the object region can be obtained. The method is discussed using simulated data for some typical conductivity distributions, and further applied to the analysis of real measurement data acquiring from the practical system. The results demonstrate that flexible regularization parameter vectors can be achieved for different distributions and the strength of regularization is adaptively provided for different regions in a specific distribution. The adaptive method achieves an efficient and reliable regularization solution and has outstanding performance in noise immunity especially in smooth background regions.     

Window function-based regularization for electrical capacitance tomography image reconstruction

Image reconstruction for electrical capacitance tomography (ECT) is to retrieve the permittivity distribution of materials inside the sensor from the capacitance measurements outside. It is a typical inverse problem and has long been a challenge for its nonlinearity and ill-posedness. This paper discusses the application of Tikhonov regularization, widely used for ill-posed problems, to the image reconstruction for electrical capacitance tomography. Two methods using different regularizations are investigated, which are the standard Tikhonov regularization and the Tikhonov regularization based on the second order derivative operator. Particularly, a combined method using the linear back projection (LBP) result as the prior constraint for the Tikhonov regularization with the second order derivative operator is suggested. Simulation and experiment results show that this combined method takes advantages from both the linear back projection and the Tikhonov regularization and provides reconstructions better than those from the LBP and the Tikhonov regularization. In addition, considering the essence that the Tikhonov regularization can be described as a spectral filter characterized by its corresponding window function, we propose the possibility of applying other window functions to the ECT image reconstruction, which include the Gauss window, the Hanning window, the Blackman window, and the cosine window. Results also show the feasibility of using window functions as regularization, which presents a new strategy for the regularization of ECT image reconstruction.     

Total fractional-order variation regularization based image reconstruction method for capacitively coupled electrical resistance tomography

Compared with electrical resistance tomography, capacitively coupled electrical resistance tomography (CCERT) is preferred since it avoids problems of electrode corrosion and electrode polarization. However, reconstruction of conductivity distribution is still a great challenge for CCERT. To improve reconstruction quality, this work proposes a novel image reconstruction method based on total fractional-order variation regularization. Simulation work is conducted and reconstruction of several typical models is studied. Robustness of the proposed method to noise is also conducted. Additionally, the performance of the proposed reconstruction method is quantitatively evaluated. We have also carried out phantom experiment to further verify the effectiveness of the proposed method. The results demonstrate that the quality of reconstruction has been largely improved when compared with the images reconstructed by Landweber, Newton-Raphson and Tikhonov methods. The inclusion is more accurately reconstructed and the background is much clearer even under the impact of noise.     

An image reconstruction algorithm based on total variation with adaptive mesh refinement for ECT

In this paper, a robust image reconstruction algorithm for electrical capacitance tomography (ECT) is proposed. The key feature of the algorithm is the use of adaptive mesh refinement based on total variation (TV) in solving the inverse problem. It keeps the edge preserving and scale-dependent properties of total variation regularization, and enhances the distinguishability by using adaptive mesh refinement. This strategy improves the spatial resolution efficiently with less calculation and is less underdetermined than uniform refinement. Simulation and experimental results show that the algorithm performs better than both standard Tikhonov regularization and the conventional total variation method.     

基于两步迭代TV正则化的电阻抗图像重建算法

针对电阻抗层析成像(electrical impedance tomography,EIT)逆问题求解的欠定性和病态性,克服传统基于L2范数的Tikhonov正则化对介质边界的模糊效应,提出一种基于两步迭代的正则化图像重建算法.该算法采用具有良好保边性的总变差(total variation,TV)正则化函数,利用两步迭代法引入TV去噪算子,达到解的双重正则化效果.与传统最小二乘迭代算法、TV相关迭代算法相比,不仅保证了逆问题求解的稳定性,而且进一步提高了非连续分布介质区域成像的分辨能力,具有较好的成像精度.     

Split Bregman iteration based image reconstruction algorithm for electrical capacitance tomography

Practical applications of the electrical capacitance tomography (ECT) rely mainly on the effectiveness of reconstruction algorithms. In this paper the solution of the inverse problem with the focus on the ECT imaging is reformulated to be an optimization problem by introducing a new loss function with regularizes encoding multiple features of solution. An iterative scheme that decomposes a complex optimization problem into several simpler sub-problems is developed to solve the proposed loss function, in which the linearization approximation and the acceleration strategy are introduced to improve numerical performances. Numerical experiments validate the effectiveness of the proposed imaging method in tackling the ECT inverse problem.     

一种基于U2-Net模型的电阻抗成像方法

电阻抗成像(EIT)是一种实现场域内电导率分布情况图像重建的成像技术.传统的电阻抗成像算法成像精度较低,为解决此问题,提出一种基于U2-Net深度学习模型的新型电阻抗图像重建方法.首先,以U2-Net模型为基础,创新地提出了拼接层(CAT)的概念用于数据扩展,使得U2-Net的输入层结构简单,运算速度快;其次,使用仿真...     

Coping with electrode polarization for development of DC-Driven electrical impedance tomography

An electrical impedance tomography (EIT) system design is proposed for imaging of phase distribution in gas-water two-phase flow from boundary measurement of electrical potentials in response to direct current (DC) injection. DC injection simplifies substantially the system design, but introduces problems due to polarization of injection electrodes. Electrode polarization means charge accumulation on the electrode-water interface causing a drift in the interfacial potential difference. The polarization problems are coped with by using dedicated electrodes for injection and potential measurement, and using a current source unaffected by the polarization of current-carrying electrodes (CCEs). Furthermore, the polarization of CCEs is controlled, to lessen the possible influence on the sensing electrodes (SEs), by using a short (milliseconds in width) pulse for injection with a charge balanced injection strategy. The impact of electrode polarization and the effectiveness of countermeasures introduced in the present design are discussed through comparisons of measured boundary potentials and of images reconstructed for a simple object simulating large bubbles in water.     

Sparsity-inspired image reconstruction for electrical capacitance tomography

We present a new image reconstruction method for Electrical Capacitance Tomography (ECT). ECT image reconstruction is generally ill-posed because the number of measurements is small whereas the image dimensions are large. Here, we present a sparsity-inspired approach to achieve better ECT image reconstruction from the small number of measurements. Our approach for ECT image reconstruction is based on Total Variation (TV) regularization. We apply an efficient Split-Bregman Iteration (SBI) approach to solve the problem. We also propose three metrics to evaluate image reconstruction performance, i.e., a joint metric of positive reconstruction rate (PRR) and false reconstruction rate (FRR), correlation coefficient, and a shape and location metric. The results on both synthetic and real data show that the proposed TV-SBI method can better preserve the edges of images and better resolve different objects within reconstructed images, as compared to a representative state-of-the-art ECT image reconstruction algorithm, Projected Landweber Iteration with Linear Back Projection initialization (LBP-PLI).     

Estimation of conductivity distribution based on fast inversion using eigenvalue and eigenvector in electrical impedance tomography

In order to solve the inverse solution for conductivity distribution in electrical impedance tomography, the one-step Gauss–Newton method is usually employed. Major computational time is involved in the calculation of inverse term of the Jacobian matrix and the complexity increases with the number of electrodes and finite elements. Therefore, to reduce the computational time, the inverse term is replaced with a summation term based on the eigenvalue and eigenvector in the inverse solver. In this paper, a fast inversion method using eigenvalue and eigenvector is developed to monitor the conductivity distribution. Therefore, using the proposed method the computation of inverse matrix is avoided resulting in decrease of the on-line computational time. Numerical simulations and experiments have been carried out to evaluate the performance of the proposed method.     

Image reconstruction algorithm for capacitively coupled electrical resistance tomography

Capacitively Coupled Electrical Resistance Tomography (CCERT), which is on the basis of Capacitively Coupled Contactless Conductivity Detection (C⁴D), is a novel electrical tomography technique. As a developing technique, more research work should be undertaken. This work focuses on the study of image reconstruction algorithm of CCERT. Combining Tikhonov regularization principle and Simultaneous Iterative Reconstruction Technique (SIRT), a new hybrid image reconstruction algorithm is proposed. Tikhonov regularization is introduced to obtain the initial reconstructed image. SIRT is used to obtain the final reconstructed image. With a 12-electrode CCERT prototype, image reconstruction experiments are carried out. Experimental results show that the images reconstructed by the proposed image reconstruction algorithm are satisfactory and are in accord with the actual distributions of two-phase flows. The research work also indicates that the proposed image reconstruction algorithm is more suitable for image reconstruction of CCERT, comparing with the conventional image reconstruction algorithms of Electrical Capacitance Tomography (ECT) and Electrical Resistance Tomography (ERT).     

基于Tikhonov和TV混合正则化的接地网电阻抗成像算法研究

电阻抗成像技术(EIT)是解决接地网腐蚀定位问题的方法之一,为了改善接地网EIT逆问题的病态性,提高求解稳定性以及重建图像清晰度,提出了经典吉洪诺夫正则化(Tikhonov)和全变差正则化(TV)相结合的混合正则化(Tikhonov-TV)接地网成像算法.首先,在循环测量原理的基础上,创新地借助COMSOL与MATLA...     

Dynamic imaging method using the low n-rank tensor for electrical capacitance tomography

Imaging objects in electrical capacitance tomography (ECT) measurement are often in a dynamic evolution process, and exploiting the spatial–temporal properties of the dynamic reconstruction objects is crucial for the improvement of the reconstruction quality. Based on the multiple measurement vectors, in this paper a robust dynamic reconstruction model that incorporates the ECT measurement information and the dynamic evolution information of a dynamic object, in which a series of dynamic images is cast as a third-order tensor that the first two dimensions are space and the third is time, is proposed. Under the considerations of the two-dimensional spatial structure property of a difference image and the spatial–temporal property of a third-order image tensor, a new objective functional that fuses the ECT measurement information, the dynamic evolution information, the temporal constraint, the spatial constraint, the low rank constraint of a difference image and the low n-rank constraint of a third-order tensor is proposed, where the images are reconstructed by a batching pattern. The split Bregman iteration (SBI) algorithm is developed for solving the proposed objective functional. Numerical simulations are implemented to demonstrate the advantages of the proposed algorithm on improving the reconstruction quality and the robustness.     

Design of wearable and wireless electrical impedance tomography system

Electrical impedance tomography (EIT) is a non-invasive approach to reconstruct the cross-section impedance image of the body. Many EIT systems and impedance image reconstruction algorithms have been proposed in previous studies. However, most of these EIT systems are bulky to cause the limitation of applications. In this study, a wearable and wireless EIT system is proposed to reconstruct impedance images non-invasively and wirelessly. By microminiaturizing the conventional EIT system, the proposed system can provide the advantages of small volume and wireless transmission to reduce the application limitation of conventional EIT systems. Finally, the phantom experiment is tested to validate the performance of the proposed EIT system. The experimental results show the average BR value of the reconstructed image obtained by the proposed system being 1.3 ± 0.2 and the averaged location error ratio being about 6.27 ± 3.14%. Therefore, the proposed wearable and wireless EIT system can be viewed as a good system prototype and may be applied to more clinical applications in the future.     

动态电容层析成像图像重建算法

提出了融合ECT测量信息和被测对象动态演化信息的新型图像重建模型;基于Tikhonov正则化方法,建立一个同时考虑了ECT测量信息、被测对象动态演化信息、时间与空间约束的新型图像重建目标泛涵,将图像重建问题转化为最优化问题;提出了集成分裂Bregman迭代法优势的新型算法求解该目标泛涵。数值仿真结果表明,所提出的图像重建算法其图像重建质量均优于OIOR算法、STR算法及PLI算法;同时由于所提出的图像重建算法同时考虑了测量数据和重建模型的不精确性,其抵抗测量噪声的能力得以提高。     

Nonlinear image reconstruction using a GA-ECT technique in electrical capacitance tomography

Electrical capacitance tomography (ECT) is a non-invasive measurement technique that estimates the dielectric permittivity distribution of an inhomogeneous object from the boundary potentials at floating electrodes or mutual capacitances. In this paper, a stochastic inverse technique based on genetic algorithm (GA-ECT) is developed, which is adapted to the two different methods, i.e. potential measurement and capacitance measurement. Numerical simulation results are presented to evaluate the inverse technique both for noise free and noisy data and the results show that quantitative image can be reconstructed not only with the low permittivity contrast but also with the high contrast. Furthermore, the influence of a priori knowledge to image reconstruction is discussed.     

A hybrid deep learning model for ECT image reconstruction of cryogenic fluids

Compared to general room-temperature fluids, the characteristics of cryogenic fluids, as well as the complexity of the cryogenic environment, pose greater challenges for reconstruction algorithms for Electrical Capacitance Tomography (ECT). Based on deep learning, a hybrid model is proposed for cryogenic fluid ECT image reconstruction in this study. The multi-head self-attention mechanism is employed to initially establish the mapping of capacitance to the image, and then an improved U-net-like convolution neural network is presented to perform deep feature extraction and image reconstruction. The ConvNeXt block is adopted for multi-level feature extraction, and a separate downsampling layer is used to replace the pooling layer. A dataset covering a variety of two-phase typical flow patterns and irregular flow patterns is built for training. A capacitance vector and an image of phase distribution are included in each sample. Extensive numerical experiments are carried out on the trained model. The results show that the model can accurately predict phase distribution and produce a clear interface. Finally, the model was successfully applied in cryogenic experiment to obtain the phase distribution image of liquid nitrogen stratified flow.     

梯度引导电学成像自适应网格生成方法

电学层析成像是一种观测场域内电导率分布的无损检测技术。有限元法是求解电学层析成像问题的常用方法。其作为线性化的近似方法,剖分单元的大小会影响有限元法求解的精度。更密的尺寸可以提高重建图像的空间分辨率,但会增加计算成本,同时未知量个数的增加会加剧逆问题的欠定性。针对上述问题,提出一种基于图像梯度的自适应网格生成方法。根据初始重建图像的梯度,自适应地提高内含物区域的网格密度,降低其他区域的网格密度,并对场域边界进行精确拟合来优化被测场域的网格剖分。通过仿真与实验研究对比分析了所提方法与常用网格剖分方法。结果表明,所提方法的重建结果图像误差平均降低15%,相关系数平均提高7%,因此所提方法在不显著增加或减少网格数的情况下,可以有效提高内含物的重建精度和图像重建质量。