# Multi-Objective Robust Optimization Reconstruction Algorithm for Electrical Capacitance Tomography
## 1 Introduction
This new model not only effectively handles uncertainties but also enhances the automation of the image reconstruction process by adaptively learning and adjusting model parameters. To ensure the robustness of the image reconstruction process against uncertainties and varying conditions, in the new image reconstruction model, the ULOP is modeled as a multi-objective optimization problem. The objective function consists of the mean of and variance in the performance function. The core principle of this design is to achieve optimal performance while minimizing performance fluctuations when faced with various possible disturbances and uncertainties. The mean value represents the system's overall performance level across specified parameter conditions. The significance of this objective lies in ensuring that the system can achieve optimal performance in volatile environments or under varying input conditions. In the measurement process, when there are different levels of interference, noise, or other influencing factors, the mean represents the system's average performance under these conditions. The variance measures the volatility or instability of system performance. A lower variance indicates lower sensitivity of the system to input changes or noise, thus demonstrating stronger robustness. When there are fluctuations in the input data, a system with lower variance is able to resist these fluctuations and maintain stable output. By minimizing variance as an objective, it is possible to reduce performance volatility, thereby enhancing the stability of image reconstruction under different conditions. This multi-objective optimization approach provides a more comprehensive and effective solution for image reconstruction, allowing the process to maintain good performance in complex and changing environments. To address the second key challenge, we develop a multimodal neural network (MNN) to infer prior images, defined as the multimodal learning prior image (MLPI). This method improves reconstruction quality by enhancing the quality of the MLPI. The MNN employs a dual-channel input strategy (image and capacitance data) to capture cross-modal associations, enabling more accurate predictions. By integrating multimodal information, the MNN overcomes the limitations of a single data source, enhancing the model's inference capability and robustness while minimizing information loss. In this new reconstruction model, we propose an innovative LLOP. The uniqueness of the LLOP lies in its integration of measurement physics principles, the MLPI, and sparsity priors. To maximize the effectiveness of the LLOP, we design an efficient solver that decreases reconstruction time without compromising the quality of the reconstructed images. We propose a new imaging model presented in the form of a multi-objective bilevel optimization problem. Although this innovative modeling approach opens new avenues for improving imaging accuracy and robustness, it also introduces new computational challenges, which constitute the third challenge in our study. Our research introduces an innovative nested algorithm architecture that implements an alternating iteration mechanism to separate and independently solve the ULOP and the LLOP. This hybrid method not only alleviates the inherent computational difficulties of the bilevel optimization structure but also handles the complexity arising from multi-objective optimization. Additionally, the proposed algorithm exhibits good scalability and adaptability. It is not only suitable for solving problems of varying scale and complexity but can also integrate existing optimization algorithms to directly solve bilevel optimization problems without the need to specially design specific bilevel optimization solvers. This flexibility implies that our approach has the potential for wide application across various real-world scenarios.