Electrical capacitance tomography (ECT) is used to image cross-sections of industrial processes containing dielectric material. This technique has been under development for more than a decade. The task of image reconstruction for ECT is to determine the permittivity distribution and hence material distribution over the cross-section from capacitance measurements. There are three principal difficulties with image reconstruction for ECT: (1) the relationship between the permittivity distribution and capacitance is non-linear and the electric field is distorted by the material present, the so-called 'soft-field' effect; (2) the number of independent measurements is limited, leading to an under-determined problem and (3) the inverse problem is ill posed and ill conditioned, making the solution sensitive to measurement errors and noise. Regularization methods are needed to treat this ill-posedness. This paper reviews existing image reconstruction algorithms for ECT, including linear back-projection, singular value decomposition, Tikhonov regularization, Newton–Raphson, iterative Tikhonov, the steepest descent method, Landweber iteration, the conjugate gradient method, algebraic reconstruction techniques, simultaneous iterative reconstruction techniques and model-based reconstruction. Some of these algorithms are examined by simulation and experiment for typical permittivity distributions. Future developments in image reconstruction for ECT are discussed.

https://iopscience.iop.org/article/10.1088/0957-0233/14/1/201/pdf

Image reconstruction algorithms for electrical capacitance tomography

Applications of electrical tomography for gas-solids and liquid-solids flows : a review

A new image reconstruction technique for imaging two- and three-phase flows using electrical capacitance tomography (ECT) has been developed based on multi-criteria optimization using an analog neural network, hereafter referred to as Neural Network Multi-criteria Optimization Image Reconstruction (NN-MOIRT)) The reconstruction technique is a combination between multi-criteria optimization image reconstruction technique for linear tomography, and the so-called linear back projection (LBP) technique commonly used for capacitance tomography The multi-criteria optimization image reconstruction problem is solved using Hopfield model dynamic neural-network computing For three-component imaging, the single-step sigmoid function in the Hopfield networks is replaced by a double-step sigmoid function, allowing the neural computation to converge to three-distinct stable regions in the output space corresponding to the three components, enabling the differentiation among the single phases

https://iopscience.iop.org/article/10.1088/0957-0233/12/12/323/pdf

Neural network based multi-criteria optimization image reconstruction technique for imaging two- and three-phase flow systems using electrical capacitance tomography

A multistep inverse solution for two-dimensional electric field distribution is developed to deal with the nonlinear inverse problem of electric field distribution in relation to its boundary condition and the problem of divergence due to errors introduced by the ill-conditioned sensitivity matrix and the noise produced by electrode modelling and instruments. This solution is based on a normalized linear approximation method where the change in mutual impedance is derived from the sensitivity theorem and a method of error vector decomposition. This paper presents an algebraic solution of the linear equations at each inverse step, using a generalized conjugate gradients method. Limiting the number of iterations in the generalized conjugate gradients method controls the artificial errors introduced by the assumption of linearity and the ill-conditioned sensitivity matrix. The solution of the nonlinear problem is approached using a multistep inversion. This paper also reviews the mathematical and physical definitions of the sensitivity back-projection algorithm based on the sensitivity theorem. Simulations and discussion based on the multistep algorithm, the sensitivity coefficient back-projection method and the Newton-Raphson method are given. Examples of imaging gas-liquid mixing and a human hand in brine are presented.

https://iopscience.iop.org/article/10.1088/0957-0233/13/1/314/pdf

Inverse solutions for electrical impedance tomography based on conjugate gradients methods

Electrical capacitance tomography for gas–solids flow measurement for circulating fluidized beds

Due to the 'soft-field' nature of electrical capacitance tomography, it is necessary to employ an iterative approach for image reconstruction in order to obtain good-quality images. In an iterative algorithm it is important to determine the gain factor, i.e., the step length approaching the converging point, because it may either cause divergence or slow down the iterative process. Usually the step length is fixed. In this communication, a method to determine the optimal step length is derived for an iterative algorithm. The efficiency of the method has been demonstrated experimentally.

https://iopscience.iop.org/article/10.1088/0957-0233/10/7/102/pdf

Optimization of an iterative image reconstruction algorithm for electrical capacitance tomography

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Optimization of an iterative image reconstruction algorithm for electrical capacitance tomography