Consider a motion sequence of multiple objects viewed by a static camera. Assuming no particular 3D model of motion, the problem is restricted to 2D projection of real 3D motion. Let , , be the th frame of the sequence, where is the total number of the frames. Assume feature points have been detected in each prior to tracking. The number of points in the th frame, , may vary from frame to frame. Denote the total number of distinct points that appear in the sequence by . This number is equal to the total number of distinct trajectories . An occluded trajectory counts as one although it consists of 2 or more pieces.
When a point enters or leaves the view field in any frame , the trajectory is called partial. A trajectory is broken if the point temporarily disappears within the view field, and later reappears again. In this case, we speak of (temporary) occlusion. If a trajectory is broken, partial, or both, it is called incomplete. Entries, exits and temporal occlusions are called events.
The feature point tracking problem is a motion correspondence problem under the general assumptions of:
Assumption 2 means limited accelerations, i.e., limited changes in motion directions and speeds. The speeds themselves are also limited so that a point is observed a sufficient number of times as it crosses the view field. However, small inter-frame displacements are not assumed. Assumption 4 implies directional continuity of broken trajectories, which makes smoothness applicable to occluded paths as well.
In addition to the general assumptions 1-4, most algorithms use specific assumptions concerning the admissible events. These assumptions are discussed in section 3.