# Multi-Objective Robust Optimization Reconstruction Algorithm for Electrical Capacitance Tomography
## 7 Validation and Discussion
### Results and Discussion
#### 7.5.2 The Results of the Compound Regularization Algorithms  
We compare the MOROR algorithm with five well-known compound regularization algorithms, including the L1 TV method with the regularization parameters of 0.01 and 0.0003 for the \(L_{1}\) norm and the total variation term, the L1 SOTV method with the regularization parameters of 0.01 and 0.0008 for the \(L_{1}\) norm and the second-order total variation term, the ELNR method with the regularization parameters of 0.02 and 0.003 for the \(L_{1}\) norm and the \(L_{2}\) norm, the L1 LR method with the regularization parameters of 0.08 and 0.08 for the \(L_{1}\) norm and the nuclear norm, and the FOENR algorithm with the regularization parameters of 0.0001 and 0.05 for the \(L_{2}\) norm and the \(L_{1}\) norm. Figure 5 displays quantitative visualization outcomes, offering an intuitive visual representation of algorithmic performance comparisons. For rigorous and unbiased assessment, Tables 5 and 6 summarize key numerical indicators, reconstruction errors and correlation metrics, which are critical for appraising the result precision and reliability across diverse methods. This dual analytical framework, merging qualitative and quantitative evaluations, allows the systematic identification of algorithmic advantages and limitations. We compare the performance of several advanced compound regularization methods with the newly proposed MOROR algorithm. The results reveal the limitations of traditional compound regularization methods while highlighting the excellent performance of the MOROR algorithm in high-fidelity image reconstruction. By analyzing the data from Figure 5 and Tables 5 and 6, we find that the five commonly used compound regularization methods struggle to achieve high-precision reconstruction for the targets shown in Figure 3a-e. The image errors for these methods range from 5.96% to 16.485%, while the correlation coefficients range from 0.9014 to 0.9860. These data indicate that despite these methods performing well in certain scenarios, they fall short when facing complex reconstruction tasks. In contrast, the MOROR algorithm demonstrates remarkable perfor \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Algorithms** & **Object 1** & **Object 2** & **Object 3** & **Object 4** & **Object 5** \\ \hline L1 TV & 0.9576 & 0.9418 & 0.9352 & 0.9282 & 0.9454 \\ L1 SOTV & 0.9028 & 0.9061 & 0.9726 & 0.9745 & 0.9689 \\ ELNR & 0.9729 & 0.9576 & 0.9536 & 0.9491 & 0.9555 \\ L1 LR & 0.9014 & 0.9196 & 0.9822 & 0.9860 & 0.9632 \\ FOENR & 0.9291 & 0.9409 & 0.9785 & 0.9796 & 0.9672 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Correlation coefficients of the compound regularization methods. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Algorithms** & **Object 1** & **Object 2** & **Object 3** & **Object 4** & **Object 5** \\ \hline L1 TV & 7.70 & 11.82 & 15.04 & 16.48 & 12.35 \\ L1 SOTV & 11.59 & 14.86 & 9.90 & 10.01 & 9.42 \\ ELNR & 5.96 & 9.71 & 12.76 & 13.95 & 11.11 \\ L1 LR & 12.29 & 13.94 & 7.95 & 7.39 & 10.17 \\ FOENR & 10.19 & 11.95 & 8.73 & 8.86 & 9.70 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Image errors of the compound regularization methods (%). mance advantages. When handling the same reconstruction targets, the MOROR algorithm keeps the image error below 0.39% while maintaining a correlation coefficient of 1. This significant performance improvement is evident not only in quantitative metrics but also visibly through comparing the visual images in Figures 4 and 5, where the superiority of the MOROR algorithm's image quality is apparent. Traditional compound regularization methods attempt to incorporate different priors by introducing regularization terms, but these methods rely on domain expert knowledge, making it difficult to fully and accurately capture the features of the reconstruction target. This has been confirmed by the visualization results in Figure 5 and the quantitative results in Tables 5 and 6.