# Multi-Objective Robust Optimization Reconstruction Algorithm for Electrical Capacitance Tomography
## 7 Validation and Discussion
### Results and Discussion
#### 7.5.1 The Results of the Iterative Reconstruction Algorithms
Notably, our proposed model can integrate any prior images, including inference information from any machine learning and deep learning methods, sensing information from other sensors, simulation information based on physical models, and more, without being limited to a specific type of algorithm. This enhances the model's flexibility and adaptability. Moreover, we develop an efficient algorithm to solve the hierarchical multi-objective robust optimization problem. Both the visual results illustrated in Figure 4 and the quantitative data presented in Tables 3 and 4 validate the efficacy of the introduced MNN and the newly designed optimizer. Sparsity serves as a crucial prior in imaging, extensively utilized across numerous algorithms such as TL1R, L1R, FL1R, L\({}_{1-2}\)R, and L\({}_{1/2}\)R. This prior holds significant theoretical importance, helping algorithms to process image data more effectively and enhance image quality. However, extensive computational results indicate that in practical applications of electrical capacitance tomography, this prior information does not always perform well. For the reconstruction targets in Figure 3, we conduct a detailed comparative analysis of these reconstruction algorithms with different sparsity characteristics. The results show that the maximum image errors for these algorithms are 12.14%, 16.13%, 15.82%, 17.55%, and 13.75%, while the minimum correlation coefficients are 0.9029, 0.8984, 0.9306, 0.9229, and 0.9316, respectively. These data indicate that the quantitative image errors are substantial,rendering them incapable of precisely representing the spatial distribution of dielectric constants. This not only compromises the image quality but also leads to erroneous conclusions. In scenarios requiring high-precision reconstruction, these algorithms obviously cannot meet the demands. Although the sparsity prior has demonstrated certain advantages, there are still several issues in practical applications that need continuous research and improvement to meet the requirements of applications with high reconstruction precision. The low-rank prior is another widely employed form of prior knowledge. The LRR algorithm incorporates the low-rank characteristics of the reconstruction target by utilizing the nuclear norm. It is difficult to fully characterize complex objects with the LRR method, and it tends to produce overly smooth solutions, making edges and subtle structures unclear. From the visual results in Figure 5d and the quantitative results provided in Tables 3 and 4, it can be seen that the LRR algorithm cannot achieve high-quality reconstruction. For the reconstruction objects in Figure 3a-e, the image errors are 11.78%, 15.16%, 9.45%, 8.30%, and 10.93%, with correlation coefficients of 0.8948, 0.8970, 0.9751, 0.9825, and 0.9577, respectively. This result further underscores that relying exclusively on low-rank priors is inadequate for addressing intricate reconstruction challenges. Incorporating additional and more robust prior knowledge and enhancing the automation capabilities of models during the image reconstruction process can significantly enhance imaging quality. The effectiveness of the MOROR algorithm further validates this insight. Figure 5: Results reconstructed by the compound regularization methods: (**a**) L1 TV, (**b**) L1 SOTV, (**c**) ELNR, (**d**) L1 LR, and (**e**) FOENR. We also demonstrate the reconstruction results of the PRPCG algorithm. As seen in Figure 4e, this algorithm has certain limitations, posing challenges for achieving high-quality reconstruction. For the reconstruction objects in Figure 3, the PRPCG algorithm's maximum image error reaches 16.88%, and the minimum correlation coefficient is 0.8693. In electrical capacitance tomography, leveraging prior information to improve reconstruction quality is essential. However, the PRPCG algorithm, being a gradient-based optimization algorithm, fails to incorporate such prior knowledge. Consequently, it falls short of reconstructing high-quality images.