# Multi-Objective Robust Optimization Reconstruction Algorithm for Electrical Capacitance Tomography
## 1 Introduction  
Accurate multiphase flow measurement remains a key challenge. Electrical capacitance tomography has emerged as a promising solution to this problem through an inverse problem method that reconstructs permittivity distribution from capacitance data. Recent advances in sensor design, reconstruction algorithms, and hardware/software systems have significantly improved its sensitivity, resolution, and practicality. The technology has found valuable applications in the energy, petroleum, chemical, and storage industries, with potential for broader adoption as it matures further. Reconstruction algorithms have developed along two distinct paths: physics-based and learning-based methods. Learning-based methods automate the imaging process and reduce reliance on specific physical mechanisms, thus minimizing the impact of manualparameter settings on the results and enhancing imaging quality. However, they also face challenges such as high data quality and quantity requirements and potential over-fitting, which may limit performance in new or complex scenarios. These methods also demand significant computational resources, increasing costs and complicating application in resource-limited settings. These models' task-specific nature requires retraining for new tasks, adding complexity, and their lack of physical interpretability can cause controversy in scientific and engineering fields. Despite these hurdles, learning-based methods have great potential. Future research could focus on efficient data techniques, improved generalization, physics-guided learning, and creating interpretable models, bridging the gap between traditional and learning methods to advance the measurement technology. Physics-based reconstruction methods, especially regularization methods, have been dominant due to their solid theoretical foundation, strong interpretability, and effective use of prior information. However, as imaging targets become more complex, these methods face different challenges. Selecting regularization parameters is also difficult. To alleviate these challenges, new methodologies that fuse physics-based and learning-based approaches are needed. Hybrid frameworks that combine the strengths of both methods are promising, aiming to balance physical principles with data-driven insights. Developing such frameworks requires addressing the formulation of effective objective functions and creating efficient solvers. Regularization approaches are vital for solving inverse problems and can integrate measurement-physics-based methods with learning-based algorithms. This study will use this method as a theoretical foundation to design a novel image reconstruction model that integrates measurement physics, optimization principles, and advanced machine learning methods to adapt to different imaging scenarios and reconstruction requirements. To achieve this goal, we need to address several critical challenges. How can we design an effective reconstruction model that minimizes the impact of measurement noise and enables adaptive model parameter selection? How can we mine and integrate more valuable prior information to improve reconstruction quality? How can we solve the reconstruction model? To address the first challenge, we formulate the image reconstruction problem as a multi-objective robust optimization model. The framework is structured with two hierarchical components: an upper-level optimization problem (ULOP) and a lower-level optimization problem (LLOP). The ULOP, framed as a multi-objective optimization problem, focuses on estimating the optimal values of model parameters, using the mean of and variance in performance metrics as objective functions to ensure robust performance under uncertainty. The LLOP, framed as a single-objective optimization problem, utilizes the solution from the ULOP as input parameters to execute the image reconstruction task.