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To get the hand trajectory.

We took several opening and closing action sequences from different view points, although the trajectories look very different, we can detect three instants .

The instants capture the real physical changes happening during actions. So this instants and interval representation can reflect the action characteristics

We will show the sign of instant is view invariant in the later slides.

In this research, we use affine camera model, which assumes the variation in depth of the points is small compared to the depth of the centroid of the point set.

Affine camera model is a generalization of orthographic,

weak perspective, and paraperspective projections.

This property is important because it can define the action categories without using any model.

If we take 3d trajectory as 3D points, we can use rank theorem to determine if two action trajectories are from the same action

The rank of the observation matrix of a 3D action trajectory is at most 3.

S is a set of 3-D points and Πs are projection matrices

for different view points, then we can arrange image

coordinates in matrix M

Based on the rank theorem and the singular value decomposition, we can get the the similarity measurement of two actions by this equation.

This system can start without any model and learn the actions from the sequences. And the representation system make the system get pretty good result by using simple matching algorithm.