## 3 Proposed Method
### Overview  
Semi-supervised semantic segmentation aims to efficiently utilize the information of both unlabeled data and labeled data. Therefore, we have a small amount of labeled data \(D_{L}=\{(x_{i}^{u},y_{i}^{l})\}_{i=1}^{|D_{L}|}\) and a large amount of unlabeled data \(D_{U}=\{x_{i}^{u}\}_{i=1}^{|D_{U}|}\). Specifically, \(x_{i}^{l}\), \(x_{i}^{u}\in\mathbb{R}^{H\times W\times 3}\) and \(y_{i}^{l}\in\mathbb{R}^{H\times W\times C}\), where \(H\) and \(W\) are the height and width of the image and \(C\) is the number of classes. This dataset is used to train students and all teachers. Additionally, \(y_{i}^{l}\) is converted and redefined for students and teachers. Figure 2 shows the transformation of the semantic ground truth to train all our models when the number of classes is 6. First, we construct \(C\)-1 teacher networks and 1 student network for our approach. The student and teachers output \(\hat{y}_{s}\in\{0,1\}^{H\times W}\) and \(\hat{y}_{t}\in\{0,1,2\}^{H\times W}\), respectively. Therefore, labels for students and teachers have 2 classes and 3 classes, respectively That is, we re-define the labels for students and teachers using Equations (1) and (2) below for the semantic ground truth:  
\[y_{i}^{l(n)}= \begin{cases}0,&\text{if }y_{i}^{l}=0\\ 1,&\text{if }y_{i}^{l}=n\\ 2,&\text{otherwise},\end{cases}  \tag{1}\]  
\[y_{i}^{l(s)}= \begin{cases}0,&\text{if }y_{i}^{l}=0\\ 1,&\text{otherwise},\end{cases}  \tag{2}\]  
where \(y_{i}^{l}\) is the \(i\)-th semantic ground truth, and \(y_{i}^{l(n)}\) and \(y_{i}^{l(s)}\) are the re-defined ground truths for the \(n\)-th teacher and student, respectively. The student is trained based on input perturbation, and teachers are trained based on new output perturbation. Figure 3 shows an overview of MVIE. In the MVIE architecture, the model named teacher consists of a CNN-based encoder \(h\) and decoder \(g\) with a segmentation head. The teacher model is decomposed into encoder \(h_{g_{h}^{l}}:\mathcal{X}\rightarrow\mathcal{Z}\) and decoder \(g_{g_{g}^{l}}:\mathcal{Z}\rightarrow\mathcal{Y}\), where \(\mathcal{Z}\subset\mathbb{R}^{\mathcal{Z}}\) represents the feature space of dimension \(\mathcal{Z}\). All teacher models consist of the number of classes minus 1, and have the same structure. Hence, teachers are denoted \(f_{\theta^{l}}=\{f_{\theta^{l(n)}}\}_{i=1}^{C-1}\), where \(C\) is the number of classes. The student consists of one model with only a decoder \(g_{g_{g}^{l}}:\mathcal{Z}\rightarrow\mathcal{Y}\), where \(\mathcal{Z}\subset\mathbb{R}^{\mathcal{Z}}\) encoded by a fixed teacher represents a \(Z\)-dimensional feature space. Therefore, one of the teacher models, named fixed teacher, is responsible for encoding the input and passing the features to the student. All configured teacher models and the student model have different initial weights. For all labeled images, the goal of the student and all teachers is to minimize the standard cross-entropy of Equations (4) and (7). For unlabeled images, each teacher model receives an image with weak augmentation and outputs \(\hat{y}_{t}\in\{0,1,2\}^{H\times W}\). After that, each teacher obtains pseudo-labels with a new ensemble method using the predictions of all other teachers and computes the teacher's unsupervised loss in Equation (6). This part is introduced in detail in Section 3.2. The student decodes the features encoded by the fixed teacher from the image with strong augmentation and outputs \(\hat{y}_{s}\in\{0,1\}^{H\times W}\), and the results of applying a hard voting ensemble from predictions of all teachers are used as pseudo-labels to calculate the student's unsupervised loss in Equation (9). This part is introduced in detail in Section 3.3. The optimization target for our students and teachers is to minimize the overall loss, which can be formulated as follows:  
\[ \begin{split}\mathcal{L}_{T}&=\mathcal{L}_{sup}^{t}+ \lambda_{t}\mathcal{L}_{unsup}^{t},\\ \mathcal{L}_{S}&=\mathcal{L}_{sup}^{s}+\lambda_{s} \mathcal{L}_{unsup}^{s},\end{split}