Problem Statement

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 $F_{k}$, $k=1,2,\ldots,M$, be the $k$th frame of the sequence, where $M$ is the total number of the frames. Assume feature points $P_{k,i}$ have been detected in each $F_{k}$ prior to tracking. The number of points in the $k$th frame, $N_{k}$, may vary from frame to frame. Denote the total number of distinct points that appear in the sequence by $N$. This number is equal to the total number of distinct trajectories $T$. 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 $k \neq 1,M$, 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:

1.

Indistinguishable points

2.

Smooth motion

3.

Limited speeds

4.

Short occlusions

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.