# Multi-Objective Robust Optimization Reconstruction Algorithm for Electrical Capacitance Tomography
## 3 Multi-Objective Robust Optimization Imaging Model
### Proposed Multi-Objective Robust Optimization Imaging Model
#### 3.4.2 Lower-Level Optimization Problem  
Leveraging the mathematical foundation of regularization theory, we propose a unified framework that synergistically integrates multimodal learning capabilities through the introduction of the MLPI and sparsity-driven prior. This integration culminates in the formalization of the LLOP, which is rigorously defined as the following optimization formulation:  
\[\min\limits_{\mathbf{u}}\{f(\mathbf{A}\mathbf{u},\mathbf{b})+\alpha_{1}\text{Reg}_{1}(\mathbf{u}) +\alpha_{2}\text{Reg}_{2}(\mathbf{u})\} \tag{6}\]  
where \(f(\mathbf{A}\mathbf{u},\mathbf{b})\) defines the data fidelity criterion, and \(\text{Reg}_{1}(\mathbf{u})\) and \(\text{Reg}_{2}(\mathbf{u})\) are regularization terms and model the MLPI and the sparsity prior, respectively.  
Owing to its quadratic structure, the least squares method exhibits convexity. Leveraging this benefit, we use this method as the data fidelity term, which can be mathematically expressed as follows:  
\[f(\mathbf{A}\mathbf{u},\mathbf{b})=||\mathbf{A}\mathbf{u}-\mathbf{b}\mid\mid^{2} \tag{7}\]  
In order to improve the reconstruction quality and promote the application of multimodal learning in image reconstruction, we integrate the MLPI into the reconstruction model in the form of a regularization term. It can be expressed as the following mathematical model:  
\[\text{Reg}_{1}(\mathbf{u})=\left|\left|\mathbf{u}-\mathbf{u}_{p}\right|\right|_{1} \tag{8}\]  
where \(\mathbf{u}_{p}\) defines the MLPI, and it can be predicted by the proposed MNN.  
Integrating the MLPI into the regularized image reconstruction model offers several benefits. Firstly, the MLPI learns complex image features and prior information from collected data, enhancing detail recovery and robustness in noisy conditions by alleviating ill-posedness. This integration allows the algorithm to utilize current measurements and latent data information, optimizing multi-source heterogeneous information and improving imaging quality. Secondly, the MLPI uses a pre-training strategy for near-real-time prior information inference once trained, increasing the reconstruction speed and computational efficiency, reducing the iterations and time compared to traditional methods. Thirdly, the adaptable regularization term in Equation (8) provides a flexible framework for incorporating various priors from multi-sensor measurements and simulation data. This enhances the model's ability to integrate heterogeneous data, offering a more robust reconstruction architecture.  
To further improve the reconstruction performance, this study also introduces the sparsity prior and formulates a regularization framework for image prior fusion. Specifically, the sparsity prior can be integrated into the reconstruction model by the following regularizer:  
\[\text{Reg}_{2}(\mathbf{u})=\left|\left|\mathbf{W}\mathbf{u}\right|\right|_{1} \tag{9}\]where \(\mathbf{W}\) is a nonnegative weighted matrix.  
Based on the developed regularization framework incorporating both the customized data fidelity term and regularization terms, we formulate the novel LLOP as follows:  
\[\min_{\mathbf{u}}\left\{\frac{1}{2}\bigg{|}\bigg{|}\mathbf{A}\mathbf{u}-\mathbf{b}\bigg{|}\bigg{|} ^{2}+\alpha_{1}\bigg{|}\bigg{|}\mathbf{u}-\mathbf{u}_{p}\bigg{|}\bigg{|}_{1}+\alpha_{2} \bigg{|}\bigg{|}\mathbf{W}\mathbf{u}\bigg{|}\bigg{|}_{1}\right\} \tag{10}\]  
The distinguishing aspect of the LLOP, as opposed to conventional compound regularization methods, lies in its incorporation of the MLPI. This inclusion enhances the model's efficacy in handling intricate scenarios while bridging measurement physics and machine learning.