# Multi-Objective Robust Optimization Reconstruction Algorithm for Electrical Capacitance Tomography
## 8 Conclusions  
This study proposes an innovative multi-objective robust optimization method to alleviate the technical bottleneck caused by low imaging quality. This new approach integrates advanced optimization principles, multimodal learning, and measurement physics to alleviate reconstruction challenges under complex and uncertain conditions, achieving excellent reconstruction performance. It consists of two mutually nested and dependent optimization problems, the ULOP and the LLOP, which can simultaneously achieve the automatic tuning of model parameters and the optimization of reconstruction quality. A new LLOP is derived using the regularization theory to unify the MLPI, sparsity prior, and measurement physics into an optimization framework. To enhance the inference accuracy of the MLPI, we design the MNN, which improves the model's inference ability through the fusion of multimodal information. To mitigate the computational difficulties induced by the ULOP-LLOP interdependencies, an innovative nested algorithm is proposed to solve the proposed multi-objective robust optimization model. The evaluation results indicate that the proposed method exhibits significant advantages over existing mainstream imaging algorithms. It enhances the automation level of the image reconstruction process, improves imaging quality, and demonstrates excellent robustness. For the studied reconstruction objects, the maximum image error is 0.39%, and the minimum correlation coefficient is 1 when capacitance noise is not considered. When the capacitance noise is 15%, the max  
Figure 8: The image errors (**a**) and correlation coefficients (**b**) of different imaging algorithms. (1. L\({}_{1}\)-2R, 2. L1R, 3. FL1R, 4. LRR, 5. PRPCG, 6. L\({}_{1/2}\)R, 7. TL1R, 8. MOROR, 9. L1 TV, 10. L1 SOTV, 11. ELNR, 12. L1 LR, and 13. FOENR). imum image error increases to 1.21%, and the minimum correlation coefficient is 0.9996. Furthermore, the novel algorithm demonstrates remarkable stability in performance, maintaining consistent reconstruction quality without substantial fluctuations across varying reconstruction targets. This research not only provides a comprehensive method to improve the overall performance metrics of image reconstruction but also establishes a new imaging paradigm, which is expected to accelerate the expansion of this measurement technology's application boundaries. Despite numerous advantages and potential of multi-objective optimization, robust optimization, and bilevel optimization, they have not been sufficiently studied and explored in the field of electrical capacitance tomography, limiting the innovative development of imaging algorithms and hindering further improvements in the technology's accuracy, efficiency, and adaptability. Our work aims to overcome this bottleneck and promote the innovative development of reconstruction algorithms. Future research will focus on developing more advanced modeling methods to enhance the automation of the image reconstruction process, strengthen adaptability in complex reconstruction scenarios, and substantially improve the intelligence level of measurement equipment. Data curation, X.Y., and J.L. ; formal analysis, X.Y., J.L., and Q.L. ; funding acquisition, J.L. ; investigation, X.Y., J.L., and Q.L. ; methodology, X.Y., J.L., and Q.L. ; resources, Q.L. ; software, X.Y., and J.L. ; validation, X.Y., J.L., and Q.L. ; visualization, X.Y., J.L., and Q.L. ; writing--original draft, X.Y., and J.L. ; writing--review and editing, X.Y., J.L., and Q.L. All authors have read and agreed to the published version of the manuscript. This study is supported by the Science and Technology Programme Project of the State Administration for Market Regulation (2024 MK081). Not applicable. The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author. Author [PERSON] was employed by the company China Nuclear Power Engineering Co., Ltd.