# A Novel Computational Imaging Algorithm for Electrical Capacitance Tomography
## 5 Numerical Simulation and Analysis
### Case 1  
To test the performance of the RELM algorithm, seven groups of simulation experiments are carried out in this section, and the imaging quality of various algorithms is compared. The seven kinds of dielectric constant distributions shown in Figure 7 are taken as the phantom objects for numerical simulation, in which the high permittivity is the yellow area and the low permittivity is the blue area. In Figure 7a, the radius of the circle is 10 mm. For Figure 7b, the radius of the circle is 8 mm. The height of laminar flow in Figure 7c is 20 mm. In Figure 7d and Figure 7g, the sides of the square are 16 mm and 15 mm, respectively. For Figure 7e, there are two rectangles with a length of 60 mm and a width of 15 mm. In Figure 7f, the side length of the square is 10 mm, and the radius of the circle is 5 mm. The TR algorithm with the parameter value of 0.01 provides the initial value for the iterative algorithm mentioned in this paper. The parameters of the CG algorithm, the Landweber algorithm, and the SIRT are listed in Tables 3-5, respectively. The number of hidden layer nodes in both ELM and RELM is 1000. The reconstruction results of various algorithms are shown in Figures 8 and 9. To quantitatively evaluate the reconstruction accuracy of the various algorithms, we calculate the REs, shown in Table 6, between the reconstructed and true permittivity. The imaging results of the LBP technique, the TR technique, and the TSVD method are shown in Figure 8. Although they can roughly recognize the contour of the phantom objects, their imaging results are poor when faced with complex imaging targets. In Figure 9, we can see that the reconstruction results of iterative algorithm, i.e., the CG technique, the Landweber technique, and the SIRT, are significantly superior over non-iterative algorithms. However, the imaging results of all three algorithms still have distortion, which cannot reconstruct the phantom objects well. Figure 8: Reconstruction results of the non-iterative algorithms. ((**a1**–**g1**) by LBP algorithm, (**a2**–**g2**) by TRM algorithm, and (**a3**–**g3**) by TSVD algorithm for the phantom objects in Figure 7a–g). Figure 9: Reconstruction results of the iterative algorithms. ((**a1**–**g1**) by the CG algorithm, (**a2**–**g2**) by the Lanweber algorithm, (**a3**–**g3**) by the SIRT algorithm, (**a4**–**g4**) by the GPR algorithm, (**a5**–**g5**) by the ELM algorithm, and (**a6**–**g6**) by the RELM algorithm for the phantom objects in Figure 7a–g). In Figure 9(a4-g4) and Figure 9(a5-g5), the GPR algorithm and the ELM algorithm take full use of the advantages of the machine learning algorithm, which considerably improves the final imaging quality compared to the aforementioned traditional imaging algorithms in this paper. When the imaging target is complex or has a right angle, the imaging quality of the classical imaging algorithms is poor. Due to the learning of a large number of datasets, the GPR algorithm and the ELM algorithm can perform better on imaging tasks. However, from Figure 9(a4-g4) and Figure 9(a5-g5), we see that for seven kinds of imaging targets, the imaging results of the ELM algorithm and the GPR algorithm still have different degrees of artifacts. From Figure 9(a6-g6), we observe that the RELM algorithm has less artifacts compared to the ELM algorithm, and the imaging results are closest to the phantom objects. The reason why the RELM can improve the image reconstruction accuracy in engineering applications is that it not only has strong generalization ability and classification capabilities, but also improves the robustness of the numerical solution. From the above analysis, the RELM algorithm can be divided into two steps: (1) the RELM mapping model between the capacitance correlation coefficient and the imaging target is extracted based on the training samples, and (2) the capacitance correlation coefficient is calculated and then it is used as the input of the RELM model to predict the final imaging.