## 1 Introduction  
A continuous-time motion analysis has great engineering support for the damage assessment and cause analysis of an explosion [1]. Reconstructing a continuous-time trajectory of scattered fragments has great value. In this paper, we present a method of fast motion estimation of the explosion fragments. The difficulty of the fragment's motion estimation is randomly scattered. The fragments created by an explosion follow these four stages:  
1. Radial expansion of the shell;
2. A crack appears somewhere in the shell;
3. Material leakage, resulting in pressure difference between the inside and outside of the shell;
4. The shell is broken randomly, forming irregular fragments which are scattered with a related high initial velocity.  
To capture this fast, diffuse motion, the high frame-rate (HFR) cameras are deployed in this research. We used two HFR cameras to build a high-speed vision system capturing the high-speed motion.  
The HFR camera has overcome the restrictions posed by conventional video signals. Many offline HFR cameras can be operated at over 1000 FPS (Frame Per Second) during a very short time [2, 3, 4]. HFR cameras can help us recognize the high-speed phenomena in the real world, which cannot be captured by conventional cameras (e.g., 30 FPS). Many phenomena have been analysed by the HFR camera, such as crash testing, sports motion estimation, industrial robot control, chemical reaction, etc. For 3D measurement, 3D shape inspection using HFR cameras, cooperated with the related projected and coded pattern, can reconstruct the dynamic phenomena of the shape [5; 6; 7; 8]. Deformation and vibration can also be analyzed by only the HFR system or a similar camera array [9; 10]. For motion estimation, unlike shape inspection and vibration analysis, the measured object is not a continuous plane or continuous rigid body. A high frequency projector does not work well. A pure vision-based system is widely deployed for activity sensing. To capture the detail of the motion, plenty of HFR vision systems have been developed [11; 12; 13]. Vision-based motion capture systems are either using passive markers [14; 15] or marker-less systems where motion estimation employs object kinematic models. Image analyzing methods, such as pixel level temporal frequency analyzing, are also presented [16; 17]. Our high-speed vision system uses an infrared camera which senses the heating information which can control the HFR cameras and evaluate our method.  
HFR cameras can only capture some samples at a certain time stamp at the moment of the explosion. The triangulation measurement of the stereo vision has accuracy drift problems, etc. To avoid this problem, we inherent the bundle adjustment (BA) method from a static object's 3D reconstruction. Bundle adjustment has played an essential role in the 3D reconstruction problem. It estimates six degrees of freedom (DOF), (three for position and three for orientation) of the camera's moving trajectory and three DOF of point clouds (e.g., 3D map) [18; 19; 20; 21; 22].  
We explore the bundle adjustment to reconstruct the motion along the time axis, i.e., the moving trajectory of each fragment created by the explosion is a 3D structure along continuous time. Furthermore, the derivation of these 3D positions have their own physical meaning, i.e., velocity and acceleration. We give the analytical result of the continuous-time position, velocity and acceleration estimation by using the cubic B-spline representation. The key idea of this study is shown in Figure 1.