# A Novel Computational Imaging Algorithm for Electrical Capacitance Tomography
## 4 Regularized Extreme Learning Machine
### Regularized Extreme Learning Machine  
As a powerful machine learning algorithm, the ELM algorithm has been successfully applied to the ECT imaging, but the stability and robustness of the ELM algorithm itself are not very good. When the conventional ELM algorithm is directly used for image reconstruction, it leads to the following two problems: (1) when directly calculating the least square solution of the cost function, the user cannot change the parameters of the ELM algorithm according to the characteristics of the training sample set, which makes the controllability of the conventional ELM algorithm worse; and (2) when the data set contains some disturbances the conventional ELM algorithm will be seriously affected, making the imaging quality worse. To solve these intractable problems, this paper proposes the RELM algorithm based on the TR technique. The TR theory usually transforms ill-posed problems into optimization problems to solve. Its objective function consists of two parts: the data fidelity term and the regularizer [18]. The generalized expression is as follows:  
\[\min_{V}\{P(V)+\sum_{i=1}^{n}\lambda_{i}Q_{i}(V)\} \tag{10}\]  
where \(P(V)\) is the data fidelity term, which measures the difference between \(HV\) and \(Y\), \(Q_{i}(V)\) is the regularizer, and \(\lambda_{i}\) is the regularization parameter, which adjusts the proportion of the data fidelity term and the regularizer, and \(\lambda_{i}>0\)[19]. When designing the objective function based on the TR technology, introducing the prior information and the characteristics of the imaging targets into the inverse problem-solving process is helpful to improve the accuracy and stability of the numerical solution. Theoretically, a variety of estimation functions can be used as the data fidelity term and the regularizer, and different choices determine the different stability and accuracy of the numerical solution, so their selection is particularly important. Because reconstructed images in ECT generally have sparse features, the L\({}_{1}\) norm has the advantage of generating sparse solutions compared to other norms. Therefore, we deploy the L\({}_{1}\)-norm as the data fidelity term to enhance the accuracy of numerical solution [20], whose mathematical expression is as follows:  
\[P(V)=\left\|HV-Y\right\|_{1} \tag{11}\]  
In addition to the data fidelity term of the solution, the design of the regularizer is also very crucial. The first-order total variation (FTV) regularization has the ability to preserve the discontinuity characteristics of the measured region [21], but this method commonly leads to the ladder effect, so we deploy the STV regularization as the regularizer to get a more stable numerical solution. The STV norm has incomparable advantages in enhancing the sparse characteristics and improving imaging ability for the edge of the imaging target compared to the FTV norm [22]. However, when the STV regularization is employed alone, the imaging results may still have the ladder effect. Therefore, a new penalty function must be added to eliminate the imaging artifacts caused by the STV regularization. Because most imaging targets have sparse characteristics, we add additional sparse constraints through the L\({}_{1}\) regularization. The mathematical expression of the regularization term is as follows:  
\[\text{Q}(V)=\lambda_{1}\left\|V\right\|_{1}+\lambda_{2}\left(\left\|D_{xx}V \right\|+\left\|D_{xy}V\right\|+\left\|D_{yx}V\right\|+\left\|D_{yy}V\right\|\right) \tag{12}\]  
where \(D_{xx}\), \(D_{xy}\), \(D_{yx}\), and \(D_{yy}\) are different difference operators, and \(\lambda_{1}\) and \(\lambda_{2}\) are regularization parameters.