# A Novel Computational Imaging Algorithm for Electrical Capacitance Tomography
## 1 Introduction  
Electrical capacitance tomography is a promising process imaging technique. It enables visualization of multiphase flow by reconstructing the distribution of multiphase media in the pipe by measuring the capacitance vector between arrays of electrodes outside the pipe. ECT is a growing research topic, with advantages such as its wide range of applications, non-invasiveness, fast response time, high security performance, low cost, etc. It is widely used in the multiphase flow process detection of insulating media [1; 2; 3; 4], such as monitoring flames and combustion conditions in engine combustion chambers, defect detection of thermal protection materials, measuring solid concentration in cyclone separator, pneumatic measurement of high pressure pulverized coal, etc. Image reconstruction is an important process for visualizing the detection results of ECT systems. The imaging quality of the ECT determines whether the ECT technique can be practically implemented in engineering applications. However, ECT image reconstruction process is severely ill-conditioned, so it is necessary to study image reconstruction algorithms to weaken this property. At present, some excellent numerical algorithms have successfully solved the inverse problem for ECT imaging. Two major categories of ECT reconstruction methods exist--non-iterative and iterative image reconstruction algorithms [5]. The advantage of the non-iterative imaging algorithm is that as a \"one-step\" imaging algorithm, it has faster imaging speed and can complete simple imaging tasks. When faced with complex imaging targets, non-iterative imaging algorithms cannot reconstruct the measured objects well and the image quality is poor. Therefore, non-iterative imaging algorithms are mostly used as online imaging algorithms for qualitative analysisor to provide initial values for iterative imaging algorithms. Common iterative algorithms include the gene algorithm (GA) [6], the Landweber algorithm [7], the Newton-Raphson iterative algorithm (NRIA) [8], the Kalman filtering algorithm (KFA) [9], and the extreme learning machine (ELM) algorithm [10]. Compared to non-iterative imaging algorithms, iterative imaging algorithms improve the image reconstruction quality, but this improvement is limited by the high time cost due to many iterations and the difficulty in choosing the parameters of the algorithm. In the past, the aforementioned algorithms have been validated in a number of simple practical applications, which has motivated the development of the ECT techniques. However, as the demand for image reconstruction quality continues to increase, the above algorithms cannot meet the needs of the practical applications of the ECT. To achieve higher image reconstruction quality, this paper proposes a more efficient image reconstruction algorithm. The stability and robustness of the conventional ELM algorithm itself are not very good, and the image quality is low when used directly for image reconstruction. To address this issue, combining the advantages of the ELM algorithm and the Tikhonov regularization (TR) method, a new image reconstruction algorithm aiming to enhance reconstruction accuracy is proposed. In this paper, the RELM method provides several appealing attributes enumerated in the following:  
1. The implementation of the algorithm mainly consists of two stages: at the learning stage, according to the training sample set, the RELM model between capacitance correlation coefficient and imaging target is extracted; at the prediction stage, by taking the calculated capacitance correlation coefficient as the input of the RELM model, the corresponding reconstructed image can be predicted. 2. The solution method of the capacitance correlation coefficient is analyzed in detail.