## 4 The Proposed Model
The algorithm stops updating when \(\|\mathcal{A}_{i+1}-\mathcal{W}_{i}\|_{F}^{2}\leq\epsilon\) or \(\|\mathcal{A}_{i}-\mathcal{A}_{i+1}\|_{F}^{2}\leq\epsilon\) or \(\|\mathcal{W}_{i}-\mathcal{W}_{i+1}\|_{F}^{2}\leq\epsilon\). ```
Input:\(\mathcal{G}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\) with mixed noise, outlier and missing value; weight vector \(\mathbf{\omega}\) Initialize:\(\mathcal{C}_{0}=\mathcal{O},\mathcal{C}_{0}=\mathcal{O},Y_{0}=\mathcal{O}\), \(\mu_{0}=1\times 10^{-1}\), \(\mu_{\max}=1\times 10^{10}\), \(\epsilon=1\times 10^{-8}\), \(\rho=1.1\), \(\lambda=\frac{1}{\sqrt{\max(n_{1},n_{2})\times n_{3}}}\)  Reconstruct incompleted \(\mathcal{G}\) by Algorithm 3 while not converged do  Update \(\mathcal{L}_{i+1}=\textbf{WTSVT}(\mathcal{G}-\mathcal{C}_{k}-\frac{\mathcal{Y} _{i}}{\mu_{i}},\omega,\mu_{i})\)  Update \(\mathcal{C}_{i+1}=\text{prox}_{\lim_{i}\frac{\lambda}{\mu_{i}}}(\mathcal{G}- \mathcal{L}_{i+1}-\frac{\mathcal{Y}_{i}}{\mu_{i}})\)  Update \(\mathcal{Y}_{i+1}=\mathcal{Y}_{i}+\mu_{i}(\mathcal{L}_{i+1}+\mathcal{C}_{i+1}- \mathcal{G})\)  Update \(\mu_{i+1}=\text{min}(\rho\mu_{i},\mu_{\max})\)  Check the convergence criteria \(\|\mathcal{L}_{i+1}-\mathcal{L}_{i}\|_{F}\leq\epsilon\) or \(\|\mathcal{G}-\mathcal{L}_{i+1}-\mathcal{C}_{i+1}\|_{F}\leq\epsilon\) endwhile Output:\(\mathcal{L}\) low-rank tensor without missing values, \(\mathcal{C}\) sparse mixture noise, and outlier tensor. ```  
**Algorithm 4** Weighted robust tensor principal component analysis integrated with weighted tensor completion. To reduce multiple noises, outliers in Equation (21) can be solved using WTSVT (Algorithm 2), and the solution of Equation (22) is a proximal function that shrinks all values. The details of the process for reducing mixture noise and outliers and reconstructing missing values are given in Algorithm 4, which describes WRTPCA integrated with WTC and uses the ADMM to extract low-rank data without missing values and sparse noise tensors from the damaged data tensor. This helps reconstruct missing data and reduce multiple noises and outliers. In Algorithm 4, we initialize the components and parameters. An \(n_{1}\times n_{2}\times n_{3}\)-tensor with all its elements set to zero is denoted as \(\mathcal{O}\). \(\mathcal{L}\) and \(\mathcal{C}\) are a low-rank tensor present for normal data without missing values, noise, and outliers and a sparse tensor that indicates outlier and mixed noise, respectively. The ADMM updates the dual variable \(\mathcal{Y}\), which represents the augmented Lagrange penalty parameter. The step size is denoted by \(\mu\), and the maximum step size is \(\mu_{max}\). \(\rho\) is the level of increase in step size \(\mu\) in each iteration. \(\mu\) can be set as fixed, but increasing \(\rho\mu_{i}\) at each iteration helps accelerate convergence. \(\epsilon\), \(\lambda\), and \(\mathbf{\omega}\) are the thresholds of the convergence condition, balance parameter, and weighted vector, respectively. Subsequently, the low-rank tensor \(\mathcal{L}\) is filled with missing values using Algorithm 3. Subsequently, the loop of Algorithm 4 starts reducing the mixed noise and outliers and starts reconstructing the missing values. In the iteration, the tensor \(\mathcal{L}\) is updated by applying the WTSVT operator defined in Algorithm 2 to obtain the low rank of \(\mathcal{L}\). Next, the operator \(\text{prox}_{i_{1},\lambda/\mu_{i}}(. )\) in Algorithm 4 is applied to update \(\mathcal{C}\). For example, with tensor \(\mathcal{G}\), the proximal operator of the \(\text{prox}_{i_{1},\lambda/\mu_{i}}(\mathcal{G})\), given as \(\text{prox}_{i_{1},\lambda/\mu_{i}}(\mathcal{G})=\max\left((|\mathcal{G}_{ijk }|-\lambda/\mu_{i}),0\right)\cdot\text{sign}(\mathcal{G}_{ijk})\), is applicable to all elements \(\mathcal{G}ijk\) within the tensor. The augmented Lagrange penalty parameter \(\mathcal{Y}\) uses a step size denoted as \(\mu_{i}\) when updating.