# Multi-Objective Robust Optimization Reconstruction Algorithm for Electrical Capacitance Tomography
## 6 Proposed Multi-Objective Bilevel Optimization Imaging Method  
By integrating multi-objective optimization, bilevel optimization, robust optimization, measurement theory, and multimodal learning, our study proposes an innovative multi-objective robust optimization reconstruction (MOROR) algorithm. Its computational workflow is summarized in Figure 2.  
Figure 2: Proposed MOROR method.  
The MOROR algorithm provides a comprehensive and systematic approach for image reconstruction. It integrates advanced machine learning techniques and multi-objective optimization methods to achieve high-quality reconstruction results. As shown in Figure 2, the implementation of the MOROR algorithm consists of six key stages, each providing crucial support for robust and efficient image reconstruction:  
(1) Training sample collection. The collected training samples are used to train the MNN, ensuring that the algorithm has good generalization capabilities in various scenarios.  
(2) Training the MNN using Algorithm 3. Based on the collected training samples, Algorithm 3 is executed to train the MNN. This step establishes a predictive model that serves as the core foundation for image reconstruction. The MNN can capture complex patterns and relationships in the data, thus providing accurate prior image predictions for subsequent stages.  
(3) Determining the optimal configuration of model parameters using Algorithm 2. At this stage, Algorithm 2 is executed to determine the optimal configuration of model parameters. This step involves solving a multi-objective robust optimization problem to ensure the robustness and accuracy of image reconstruction under uncertain conditions.  
(4) Initial image reconstruction using the newly acquired capacitance data. After determining the optimal model parameters, the initial image reconstruction is implemented using the newly acquired capacitance data. This step generates a preliminary image that serves as the input of the trained MNN.  
(5) Inferring the MLPI using the trained MNN. The trained MNN is used for inferring the MLPI, a step that plays a pivotal role in enhancing reconstruction quality. This is because the MLPI serves a dual purpose: it not only acts as valuable prior knowledge that can be integrated into the reconstruction model but also functions as an effective initialization for iterative optimization processes. By providing a high-quality starting point, the MLPI significantly accelerates the convergence rate of optimization algorithms while simultaneously improving the accuracy and reliability of the final solution. Furthermore, the incorporation of the MLPI helps mitigate common reconstruction challenges such as noise amplification and artifact generation, particularly in ill-posed or underdetermined problems.  
(6) Conducting final image reconstruction using Algorithm 1. In the final stage, the algorithm combines the newly acquired capacitance data with the inferred MLPI and the determined model parameters and completes the final image reconstruction by executing Algorithm 1.  
In the MOROR algorithm, the first three steps (i.e., training sample collection, MNN training, and model parameter optimization) can be performed offline, meaning that they do not occupy time during the image reconstruction process. This offline processing enhances the algorithm's efficiency, especially in applications with high real-time requirements. Once the MNN training is completed, its inference process is efficient, allowing the MLPI to be completed in a very short time. Since the MLPI inference time is extremely brief, its impact on the overall reconstruction time of the algorithm is negligible. The majority of the algorithm's execution time is consumed by the operation of Algorithm 1, which is responsible for the final image reconstruction.