# Multi-Objective Robust Optimization Reconstruction Algorithm for Electrical Capacitance Tomography
## 1 Introduction
Our research provides a comprehensive solution to mitigate the challenges in image reconstruction tasks and improve overall reconstruction performance through the extraction and integration of prior information, design and solution of objective functions, adaptive learning of model parameters, and integration of multimodal supervised learning methods and optimization principles. We summarize the main contributions and novelty of this study as follows:  
(1) We model the image reconstruction problem as a multi-objective robust optimization problem. This new model integrates advanced optimization methods, multimodal learning, and measurement physics to ensure reliable reconstruction performance under uncertain conditions. By integrating supervised learning methods and optimization principles, this model not only improves reconstruction accuracy and robustness but also provides a systematic framework that facilitates parameter tuning and optimizes reconstruction quality in practical scenarios, thereby enhancing the automation level of the image reconstruction process. (2) The MLPI, sparsity prior, and measurement principles are integrated into a new LLOP to improve the image reconstruction quality. The MLPI captures the statistical characteristics of objects, serving as both a diverse prior image in the model and an ideal initial value for iterations, thus enhancing algorithm convergence. This integration achieves the fusion of measurement physics with multimodal learning. Additionally, a new optimizer is designed for this LLOP to reduce computational complexity. (3) We design a new MNN to infer the MLPI. This new MNN performs inference tasks with image and capacitance data as input. It fully utilizes information from different modalities, compensating for the limitations of single-modal information, enhancing the model's reasoning ability, reducing potential information loss when using a single modality, and improving the model's overall performance. (4) We design an algorithm capable of effectively solving the proposed multi-objective robust optimization problem with a hierarchical optimization structure. By employing an alternating optimization strategy, this nested algorithm decomposes the complex bilevel optimization problem into two relatively independent but interrelated single-level optimization problems for solving. This approach not only reduces the complexity of the solution but also effectively handles the interdependencies between the ULOP and the LLOP. (5) Performance evaluation results indicate that our proposed multi-objective robust optimization imaging algorithm significantly enhances the overall image reconstruction performance compared to currently popular algorithms. It maintains stable performance across different targets, with improvements in various imaging performance metrics and increased model automation. This approach revolutionizes image reconstruction by integrating optimization principles with advanced multimodal learning techniques, filling research gaps and advancing measurement technology for greater practical value. To address the challenges of image reconstruction, this study proposes a systematic solution. The structure of this paper is summarized as follows: Section 2 reviews and analyzes major image reconstruction algorithms, laying the theoretical foundation for subsequent research. Section 3 models the imaging problem as a multi-objective robust optimization problem, introducing a new perspective for alleviating imaging challenges in complex scenarios. On this basis, Section 4 proposes an innovative optimization technique to solve the multi-objective robust optimization model. In Section 5, a novel MNN for inferring MLPIs is designed. The overall reconstruction methodology and its computational workflow are summarized in Section 6. Section 7 fully demonstrates the superiority of the proposed method through performance evaluation. Finally, Section 8 concludes the research findings and highlights this study's significant value in both theoretical and applied domains.