## 4 Experiments and Evaluation
### Evaluation  
The procedure of the continuous-time fast motion estimation is detailed as follows, shown in Figure 6, where:  
1. Explosion fragments' motion are captured by the HFR cameras with a related frame-rate setting strategy (synchronization or multiplication); the large FOV LWIR cameras monitor the explosion (shown in Figure 6a)
2. With recording images from both HFR cameras, we use the dense optical flow: the Farneback method [27] is used to extract the motion area of the images at each time stamp \(j\). The pixel motion of different fragments are analyzed (shown in Figure 6b). Then we combine the HOG feature extraction and the connected component at every extracted motion area from the optical flow estimation. The image position (area)  
Figure 5: (**a**) Constructed balloons for explosion under long-wavelength infrared (LWIR) view, (**b**) measurement instruments with the monitoring module in the middle. of fragments at each captured frame are extracted, i.e., the dataset of each camera's observed position \(\left\{b_{ij}^{k}\right\}\) of fragment \(k\) is established;
3. The bundle adjustment is optimized based on the HFR cameras' working strategy, which has been detailed in the previous section. Figure 6c demonstrates the variation of three key parameters (damping factor, norm of reprojection error \(\left\|F\right\|\), and \(\left\|g\right\|_{\infty}\), from top to bottom) of the BA along the iteration step. From the BA optimization, the 3D positions \(\left\{PW_{j}^{k}\right\}\) of fragment \(k\) at each captured time stamp \(j\) are established. 4. With optimized 3D positions, the continuous-time motion (position, velocity, acceleration) of each fragment is estimated by cubic B-spline representation. The accuracy of our BA optimization with different frame-rate strategy is presented in this section. We compare synchronization frame-rate bundle adjustment (denoted as SBA) and bundle adjustment with frame-rate magnitude (denoted as BAM) to stereo calculation directly. Euclidean distance is deployed to evaluate the position error, while the norm of the reprojection error from each camera evaluates the accuracy of the proposed methods. We use boxplot to demonstrate the distribution of the position error, shown in Figure 7a, and the reprojection error as shown in Figure 7b (Where -Ci means reprojection in camera i). The related data of the mean error of position (MEP), mean square error of position (MSEP), mean error of reprojection error (MRE) and mean square error of reprojection error (MESRE) in pixel are shown in Table 1.  
Figure 6: Procedure of our method. (**a**) Motion captured by HFR cameras; (**b**) Fragments extraction; (**c**) Bundle adjustment optimization; (**d**) Continuous-time motion estimation. Figure 7 and Table 1 demonstrate that by using bundle adjustment optimization, the accuracy and stability of the trajectory estimation is highly improved, especially using the frame synchronization strategy (SBA). BAM can efficiently increase the motion's recording time. Comparing to SBA, the lack of observed motion from camera one at certain time stamps for BAM causes roughly 5 mm of accuracy degradation of the position, 0.001 pixel in MESRE. This is why the outlier of BAM-C2 is significantly high. When captured motion is around the camera's center area (around 45 pixels), the distortion affection is small, and the position calculated by stereo directly can achieve related high accuracy. Bundle adjustment optimization updates the intrinsic parameters instead of the position around these areas. This is why the minimum position error of stereo calculation directly is lower than BA's optimization. This is the only drawback of trajectory estimation using bundle adjustment. A representative of one fragment's motion estimated by SBA and BAM with uncertainty bounds (3\(\sigma\)) is shown in Figure 8. We plot the whole 419 ms from the explosion to being scattered out of the cameras' FOV. The sample frame-rate for SBA of both HFR cameras is 540 fps, with 1024 \(\times\) 480 resolution.