  Trajectory analysis is essential in many applications. In this paper, we
address the problem of representing motion trajectories in a highly informative
way, and consequently utilize it for analyzing trajectories. Our approach first
leverages the complete information from given trajectories to construct a
thermal transfer field which provides a context-rich way to describe the global
motion pattern in a scene. Then, a 3D tube is derived which depicts an input
trajectory by integrating its surrounding motion patterns contained in the
thermal transfer field. The 3D tube effectively: 1) maintains the movement
information of a trajectory, 2) embeds the complete contextual motion pattern
around a trajectory, 3) visualizes information about a trajectory in a clear
and unified way. We further introduce a droplet-based process. It derives a
droplet vector from a 3D tube, so as to characterize the high-dimensional 3D
tube information in a simple but effective way. Finally, we apply our
tube-and-droplet representation to trajectory analysis applications including
trajectory clustering, trajectory classification & abnormality detection, and
3D action recognition. Experimental comparisons with state-of-the-art
algorithms demonstrate the effectiveness of our approach.
