# Containment Control-Guided Boundary Information for Semantic Segmentation
## 1 Introduction  
Semantic segmentation is pivotal in computer vision, facilitating comprehensive scene analysis. Among the inherent challenges faced in semantic segmentation, the precise delineation of boundaries between different categories stands out. These boundaries often imply transition regions between objects or elements with different semantics. Accurately capturing the spatial relationships and transitions between different categories enables a more nuanced and context-aware segmentation, which in turn contributes to improved scene understanding. Thus, in the field of semantic segmentation, the quality of models is intrinsically linked to their ability to recognize complex boundaries [1; 2; 3; 4]. A model's proficiency in accurately delineating boundaries directly affects its ability to interpret complex scenes. Traditional methods may have difficulty in capturing subtle transitions between categories, leading to less accurate segmentation [5; 6]. Effective utilization of these transition cues can significantly improve segmentation accuracy, thereby improving downstream applications such as scene understanding. Therefore, our work aims to address the limitations of traditional semantic segmentation methods by introducing an innovative approach that not only effectively utilizes boundary information but also establishes structured constraints between boundaries and internal features. Containment control is a prevalent issue in the study of multi-agent systems (MASs), aiming to drive a subset of agents, often referred to as followers, to ultimately reside within a convex hull defined by certain specific agents, commonly known as leaders [7; 8]. The development and implementation of containment control strategies integrate control theory and graph theory to address issues such as information diversion, cooperation, and self-organization in MASs [9; 10; 11]. However, the application of containment control should not be limited to the field of control. By transferring the relationship between leaders and followers to semantic segmentation, specifically where leaders represent boundary features and followers represent internal representations, the precise modeling of the interrelationship between boundary features and internal representations can be achieved, guiding more accurate semantic segmentation in images. Our work not only conceptually incorporates containment control but also applies the classical theory of containment control to practically determine the spatial relationship between boundary feature information and internal feature information. This approach helps to quantify the dynamic exchange between boundary features and internal features, which leads to a nuanced control of the model utilizing boundary feature information. This pioneering integration of containment control principles into semantic segmentation models is expected to become a new paradigm in the field of visual scene understanding. Meanwhile, our network achieves high accuracy and real-time requirements as shown in Figure 1.  
In conclusion, our work brings forth significant contributions, including the following:  
* Boundary Feature Integration: We recognize the importance of boundary feature information in semantic segmentation tasks and dig deep into the way it is used. The dynamic fusion process of boundary features and internal representations is quantified by building a features fusion matrix based on the Laplacian matrix, which effectively improves the model's ability to capture and utilize boundary information. * Containment Control-Guided Segmentation: We utilize the containment control criterion to construct a dedicated loss function for updating the features fusion matrix, which achieves finer control over the information fusion between boundary features and internal representations.