# Containment Control-Guided Boundary Information for Semantic Segmentation
## 3 Method
### How Containment Control Guides Semantic Segmentation
\(\otimes\) denotes the Kronnecker product, and \(I_{M}\) denotes a \(M\)-dimensional identity matrix. Based on the conclusion provided in the preceding paragraph,  
\[x_{f}+(\mathcal{L}_{1}^{-1}\mathcal{L}_{2}\otimes I_{n})x_{l}\to 0 \tag{4}\]  
means all followers in the MAS will ultimately reside within the convex hull formed by the leaders, thereby indicating the successful resolution of the containment control. Equation (4) is just like the containment control criterion. In the realm of semantic segmentation, conventional feature fusion methods often grapple with fully exploiting boundary information, motivating us to introduce the ground-breaking concept of containment control to intricately model the interplay between boundary and internal information. Our novel methodology strategically incorporates containment control as a guiding principle, seamlessly integrating it into the training process of the segmentation model. We establish a robust mapping between containment control, as encapsulated by Equation (4), and the intricate feature landscape within the model. Within the context of our bespoke semantic segmentation task, a dedicated network branch adeptly extracts boundary information, aligning with \(x_{f}\) in Equation (4). Concurrently, other branches of the network extract specific segmentation results, corresponding to \(x_{l}\) in Equation (4). Drawing inspiration from neural architecture search principles, we introduce a set of learnable feature fusion matrices \(\tilde{\mathcal{L}}\) (i.e., \(\mathcal{L}_{1}^{-1}\mathcal{L}_{2}\)) to orchestrate seamless communication between internal and external feature information within the model. Utilizing the features fusion matrix-derived specific weights, each channel of the internal feature map will undergo a sophisticated weighting process involving all external feature channels. Based on the above approach, we introduce a relevant loss function. Through linear neural mapping, different features extracted by our network are collapsed into one dimension. Then, the internal information (\(\mathcal{I}\)) and boundary information (\(\mathcal{B}\)) are stacked as column  
Figure 3: The leaders and followers communicate based on the Laplacian matrix \(\mathcal{L}\) to eventually achieve containment. The suitable communication weights of the features fusion matrix \(\tilde{\mathcal{L}}\) that guide the feature fusion process are derived through \(Loss_{cc}\) that infers the relationship between internal representation and boundary information. vectors. Using the containment control criterion (4) and the features fusion matrix \(\bar{\mathcal{L}}\), the loss function is formulated as follows:  
\[Loss_{cc}=\frac{1}{\mathcal{C}}\sum_{i=1}^{\mathcal{C}}(\mathcal{I}+\bar{ \mathcal{L}}\mathcal{B})_{i}, \tag{5}\]  
in which \(C\) denotes the number of feature channels, and \((\mathcal{I}+\bar{\mathcal{L}}\mathcal{B})_{i}\) denotes the \(i\)-th element of the vector \((\mathcal{I}+\bar{\mathcal{L}}\mathcal{B})\). Here, we employ the term \"internal information\" as a comprehensive representation of detailed features and contextual features. Subsequently, we will provide a detailed exposition on this matter. The loss calculation based on the condensed information reduces the computational complexity while preserving feature information. When we consider image boundary features and internal features as \"leaders\" and \"followers\" in containment control, it should be noted that \(L_{1}\) in Equation (2) reflects the connection between the internal information of the image, while \(L_{2}\) reflects the connection between the image boundary information and the internal information. Since the internal information of the image has already been interpreted by the detail features and the contextual feature branches, the spatial constraints between the internal information of the image are not taken into account when parsing the relationship between the image boundary and the internal information by using the containment control criterion.