# A Deep Learning-Based Fragment Detection Approach for the Arena Fragmentation Test
## 2 Deep Learning-Based Fragment Detection
### Temporal Filtering  
In this subsection, we briefly review the temporal filtering scheme [22]. The deep learning-based warhead fragment detector described in Section 2.2 detected all fragments in each frame of the test image. This includes fragment-like flames in addition to the first penetrating fragments, as mentioned in Section 2.1. To solve this problem, we used an efficient temporal filtering technique that detected only the fragments penetrating the steel plate first. As a warhead fragment penetrates the steel plate, a flame is created at that location. These flames are characterized by expanding with time and dispersing finely after a certain amount of time. Unfortunately, these flames are similar to warhead fragments and so they are also detected at every frame after the first penetration. Therefore, our proposed method employed temporal filtering with information of the fragments and their following flames: (1) Size diversity of the first penetrating fragment, and (2) the expandability of the warhead fragment size (flame size). Based on the aforementioned features, the size of each initial penetrating warhead fragment was normalized, and the size change (flame) over time was statistically modeled. The proposed method used the Levenberg-Marquardt non-linear squares algorithm to perform the following non-linear logarithmic fitting on the size of warhead fragments as follows:  
\[r^{t}=a\text{log}\left(t\right)+1. \tag{1}\]  
\(r^{t}\) is the result of modeling the relative size of the warhead fragment detected in the \(t\)th frame when the size of the first penetrating warhead fragment is normalized to 1, as shown in Figure 9. \(a\) is a parameter required for non-linear log fitting of the collected fragment data. The bounding box in the \(t\)th frame after the initial penetration of each warhead fragment can be expressed as follows using \(\left(x_{i}^{1},y_{i}^{1},w_{i}^{1},h_{i}^{1}\right)\), the size of the initial penetrating warhead fragment (\(i\) indicates the index):  
\[ \begin{array}{ll}x_{i}^{t}=x_{i}^{1}+\frac{w_{i}^{1}\left(1-r^{t}\right)}{2},&w_{i}^{t}=w_{i}^{1}r^{t}.\\ y_{i}^{t}=y_{i}^{1}+\frac{h_{i}^{1}\left(1-r^{t}\right)}{2},&h_{i}^{t}=h_{i}^{1 }r^{t}.\end{array}  \tag{2}\]  
In this case, \(x\) and \(y\) are the upper left coordinates of the bounding box, respectively, and \(w\) and \(h\) are the horizontal and vertical lengths, respectively. Equation (2) can be used to estimate the size of warhead fragments at the \(t\)th frame after the initial penetration. At this time, the error between the size of the warhead fragments predicted by non-linear log fitting and the one for the actual warhead  
Figure 8: An illustration of the proposed anchor boxes for fragment detection. fragments is less than about 20% on average. By using this, the mask for the first penetrating fragment is defined as follows:  
\[mask_{i}^{t}= \begin{cases}1,&mask_{i}^{t-1}\cup Rect\left(x_{i}^{t},y_{i}^{t},w_ {i}^{t},h_{i}^{t}\right),\\ 0,&otherwise\end{cases}  \tag{3}\]  
where \(\cup\) represents a union operator. \(Rect\left(\cdot\right)\) is a function that creates a rectangular mask from the input \(x,y,w,h\). Equation (3) is used to obtain \(mask_{i}^{t}\) in the \(t\)th frame. Therefore, the proposed method could detect the first penetrating fragments using a deep learning-based warhead fragment detector over every frame, and could then use the filter generated by Equation (3) to eliminate the duplicated detection results after the first penetration (see Figure 10). However, the aforementioned temporal filtering method does not use the tracking information of the detected warhead fragments but uses only the detection result of the warhead fragment at every frame and the change of the warhead fragment size estimated by the non-linear log fitting. The disadvantage is that it is difficult to create appropriate masks.