## 2 Methodology
### Relationship-Based Weight Analysis
\[A_{norm}=\frac{A_{R}-A_{min}}{A_{max}-A_{min}} \tag{6}\]  
In addition to distance and attribute relationships, the spatial relationship association value \(S_{R}\) is a factor that quantitatively evaluates the relative positional information between the pouring boom and the detected concrete object. It plays a role in improving detection accuracy by analyzing whether the relationship between objects follows a specific pattern. Generally, concrete discharged from a pouring boom exists directly below the boom, and this relationship can serve as a structural constraint in the object detection process. Therefore, it is important to check whether the detected object satisfies a specific spatial relationship with the pouring boom. The spatial relationship association value \(S_{R}\) is calculated according to Equation (7), by evaluating whether the detected object is positioned at the vertical bottom of the boom bounding box. For this, the bottom of the boom bounding box (\(y_{Boom,max}\)) and the top of the concrete bounding box (\(y_{Concrete,min}\)) are compared to analyze whether the object is positioned below the boom. Here, \(S_{R}\) is the spatial relationship association value, which is a binary value (0 or 1) before normalization. \[S_{R}= \begin{cases}\textbf{1},\textbf{if}\ y_{Concrete,min}>y_{Boom,max}( \textbf{Concrete is below boom})\\ \textbf{0},\ \textbf{Otherwise}\end{cases}  \tag{7}\]  
According to this condition, if the detected concrete object is positioned at the vertical bottom of the boom object, \(S_{R}\) = 1, otherwise \(S_{R}\) = 0. This allows us to determine whether the detected object has the correct spatial relationship with the boom. Having established the three relationship factors, the final detection is determined by evaluating the normalized association values through mathematical formula-based analysis. The final weight between the two detected objects is defined by Equation (8). \[W(\textbf{Boom,Concrete})=\alpha\cdot\textbf{D}_{norm}+\beta\cdot\textbf{A}_ {norm}+\gamma\cdot S_{R} \tag{8}\]  
where \(\alpha\), \(\beta\), and \(\gamma\) are coefficients adjusting the importance of distance, attributes, and spatial relationships, respectively, starting with initial values set equally at 0.33. This initially sets the contribution of each factor equally to evaluate the reliability of relationship-based detection, and then adjusts based on training data. During the learning process, optimal importance coefficient values are determined experimentally, considering the balance between detection precision (Precision) and recall (Recall). Following Equation (9), the computed value is normalized through a sigmoid function and compared with a pre-set threshold to determine whether the detected object is finally classified as concrete being poured. \[\hat{\mathbf{W}}=\mathbf{\sigma}(\mathbf{W})=\frac{\mathbf{1}}{\mathbf{1}+\mathbf{e}^{-\mathbf{W}}} \tag{9}\]  
Upon comparison with the threshold, if the calculated weight value exceeds the pre-set threshold, the object is confirmed as concrete being poured; otherwise, the detection is invalidated. In this study, the threshold was set based on the top 80% of average weight values based on training data, and optimized considering the balance between precision and recall. The detailed process by which the threshold was calculated will be elaborated on in Section 2.6. Through this module, the limitations of YOLOv11 are complemented, enabling more reliable concrete pouring monitoring through the application of a detection technique that reflects inter-object relationships. The final decision, based on normalized weight, improves detection reliability compared to the existing individual object detection method. It maintains high accuracy even in environments with high boundary ambiguity by reflecting the structural relationship between the pouring boom and concrete objects. Consequently, the Ambient Detection module proposed in this study is designed to go beyond simple object detection.