A new hierarchical reconstruction algorithm for electrical capacitance tomography using a relaxation region-based approach
Highlights
► We developed an electrical capacitance tomography image reconstruction algorithm. ► The algorithm consists to confine progressively the regions of interest. ► We compared the performance of this algorithm over a regular reconstruction method. ► The image reconstruction is significantly accelerated and the resolution is enhanced.
Introduction
Process flow tomography is a technique for extracting spatial and temporal information about the flow properties by mounting multiple sensors around the process of interest. It has provided a step forward in online flow estimation for improved industrial process monitoring and control [1], [2].
Among the flow properties that are widely used in tomography is the permittivity which varies function of its composition. Different components of a multiphase flow have different permittivities (e.g. gas, oil and water composition). Hence, the transition of the flow regime and the change of the amount of each material will result in a variation of the capacitances between different points of the pipe or the vessel. Thus, it is possible to use these values to determine the material permittivity distribution, and consequently one can deduce the corresponding flow regime and the quantity of each phase.
Compared to other tomography techniques such as X-ray and γ-ray, ECT is relatively new and has the advantage of being safe (i.e. non-radioactive), lower cost, non-invasive, non-intrusive, portable, and much faster. Thus, it can be adopted for applications requiring real-time performance such as visualization of oil pipeline flows and gas/solids flows [3], [4].
Many different approaches to solve the ECT image reconstruction problem have been proposed in literature. One of the most common consists to divide the imaging area into a finite number of elements or pixels where the permittivity of each pixel is supposed constant; and then minimize the square norm of the difference between the measured boundary capacitances and the calculated boundary capacitances using an optimization algorithm [5]. However, because of the ill-posed nature of the problem, some regularization techniques are needed. Regularization is obtained by introducing an additional term into the square norm objective function. The most popular regularization technique is the Tikhonov regularization which incorporates a smoothness assumption on the solution [6], [7]. This latest is very simple for implementation, however, it performs weakly in case a sharp change of the permittivity occurs near the object boundary. Total variation (TV) regularization allows reconstructing sharp edges and it takes into account jumps in permittivity profile [8], [9]. However, the TV regularization is non-quadratic [10] and involves the minimization of a non-differentiable objective function. Thus, special attention is required to use it in a non-linear inverse problem [11]. One possibility, adopted in the present work, is to stabilize the inverse problem using a Tikhonov-type regularization with finer elements or pixels near the boundary of the inhomogeneities.
The size of the elements or the pixels in the mesh is very important. Finer sizes usually improve the spatial resolution of the reconstructed image and lead to higher accuracy of the reconstruction algorithm. However, this increases the computational time and resources and makes the inverse problem more underdetermined since the number of unknowns is increased. To remedy, some researchers proposed an adaptive mesh refinement strategy during the reconstruction process (see e.g., [9], [12]). Most of these techniques proceed by refining the mesh in areas with sharp permittivity gradients.
In this paper, a new hierarchical reconstruction algorithm is proposed to solve the ECT image reconstruction problem. The algorithm is hierarchical and consists to confine progressively the regions of interest which hold the inhomogeneous phase by refining the finite element mesh size around their boundaries. This is done by localizing, at each stage of the hierarchy, outer and inner boundaries within which the real boundaries of regions shall exist. At each stage of the hierarchy, the permittivity distribution (the imaging area) is obtained by minimizing the objective function that consists of squared errors between the measured and the calculated capacitances plus a Tikhonov regularization term. This optimization problem is solved accordingly using an iterative non-linear GN algorithm. One of the main features of this algorithm is to speed up the reconstruction procedure, while keeping the image quality high.
Section snippets
Previous work
Among the non-iterative or direct ECT image reconstruction algorithms, the D-bar algorithm and the Calderon’s algorithm have been introduced just recently [13], [14], [15]. Numerical experiments reported in [15] show that the D-bar method and Calderon’s linearization method produce quite similar reconstructions, while Calderon’s method is computationally faster. Calderon’s algorithm can be directly implemented in terms of the scattering transform [13] which requires an infinite number of
Image reconstruction method
In this paper, an ECT device with N = 8 electrodes is considered. By exciting one electrode at a time and measuring the charges induced at all the other electrodes, one obtains M = N(N − 1)/2 = 28 independent mutual capacitance values. For this sensor, the Laplace’s equation can be used to describe the governing equation:where ε(x, y) is the permittivity distribution and Φ(x, y) is the electric field distribution.
On an incentive electrode Ei, one has the boundary condition
Simulation results
The quality of the reconstructed images using the proposed hierarchical algorithm was compared with a stand-alone GN reconstruction algorithm using extensive simulations. In this paper, the results corresponding to five typical permittivity distributions are presented. The ECT system used consists of a circular sensor composed of eight electrodes evenly surrounding the sensing area. The forward problem and the reconstruction algorithms were implemented using Matlab on a PC with a Core 2 Duo 2.8
Conclusions
In this paper, a new hierarchical mesh algorithm was proposed to solve the ECT image reconstruction problem. It consists of locating progressively the inhomogeneity boundaries by refining the mesh inside an outer and an inner boundary. At each step of the hierarchy the image was obtained using a GN algorithm. The proposed algorithm has been compared with a stand-alone GN algorithm. The algorithm gives two advantages: the speed of image reconstruction is significantly accelerated and the spatial
Acknowledgment
The authors would like to thank the Petroleum Institute for supporting the actual work.
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