## 2 Suggested Techniques
### Bounding Box
#### 2.1.2 Merging and Expanding Bounding Boxes  
BB merging is a two-step process. First, the merge process combines two nested BBs to consolidate them. Secondly, there is an expansion process that increases the size of an independent BB that does not overlap with other BBs, allowing it to be merged with other BBs. The ratio variable for scaling BB in this process is m_factor, let us call it \(\alpha\). Hyperparameters are shown in Table 1. BB size is \(W_{init}\times H_{init}\), which is the initial size of the BB. The maximum size of the BB, max_factor, is \(S_{max}=(W_{max},H_{max})\), the IoU Threshold is \(\tau_{threshold}\), the information of each BB is \(bb_{j}=(l_{j},m_{j},w_{j},h_{j})\), \(l\) is the x-axis position, \(m\) is the y-axis position, \(w\) is the width, \(h\) is the height, and \(D^{\prime}\) is the dataset index included in the BB. Accordingly, BB information is represented as \(BB=\{bb_{1},\ldots,bb_{l}\}\). The initial BB setup is as shown in Equation (3) below. \[BB =\{bb_{1},\ldots,bb_{k}\},forallbb_{i}\in BB,w(bb_{i})=W_{init},h( bb_{i}) \tag{3}\] \[=H_{init},l(bb_{i})=y_{i,l}-\frac{W_{init}}{2},m(bb_{i})=y_{i,m}- \frac{H_{init}}{2}\]  
Next, calculate the IoU value between the input BBs using Equation (4). \[getIoU(BB)= \begin{bmatrix}-1&\tau_{1,2}&\cdots&\tau_{1,k}\\ \tau_{2,1}&-1&\cdots&\tau_{2,k}\\ \vdots&\vdots&\ddots&\vdots\\ \tau_{k,1}&\tau_{k,2}&\cdots&-1\end{bmatrix}  \tag{4}\]  
To save the status information of the merging process, we set up a BB List to store and manage the BBs created in the current stage, and checked whether the merging or expanding process was performed. At this point, if any merging or expanding process is performed in the current step, it will proceed to the next step. If all BBs are larger than max_factor, the process will terminate immediately without proceeding with the merging and expanding process of the next step. Therefore, if the size of all BBs does not yet meet the termination condition, the process of combining and enlarging BBs will be repeated for the number of initial BBs. In this iteration, the IoU value of the current BB and other BBs are calculated and the BBs that are greater than the IoU Threshold value are merged. This checks to see if the current BB is already a combined BB, and at every iteration it checks for combining or expanding information per BB. It then calculates the minimum BB size of the current BB and the other selected BB to calculate the minimum BB that contains both BBs, creating a new combined BB and adding it to the BB List. The BB List is then used to update the status information of the BBs used in the union, specifying that they are the BBs used in the union. If BBs have an IoU value that is less than the IoU Threshold required for joining, they do not have overlapping regions and can be judged to be independently located BBs. Thus, independent BBs expand their size to merge with other BBs around them. It sets the expanded BB information to the BB List and updates the status of the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Hyper Parameter** & **BBi Size** & **m\_factor** & **max\_factor** & **IoU Threshold** \\ \hline value & 0.5 m \(\times\) 0.5 m & \(\times\)1.5 & 5 m & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Hyperparameters required for merging BBs. current BB. The expansion of a BB is conducted by multiplying the current size of the BB by m_factor. In our previous paper, we set a threshold in the form of a constant to prove neighboring data points [12]. If this is applied to the BB of this paper, the proximity of independent BBs can be determined using GIoU. However, when using GIoU, the same issue as using a threshold constant value arises. We solved this problem by extending the nested BBs using IoU. BBs that were not used for merging and expansion are processed in the next step. Checks for BBs that have not been merged or expanded in the current staff are saved to the BB List.