A COMPARISON AND EVALUATION OF MOTION INDEXING ...

Once the boundary is found, the algorithm is restarted with motion containing
only the frames starting at the boundary frame till the end. Again, all the steps
starting from the calculation of r are followed. The algorithm finds all the possible
boundaries in the motion starting from the first frame to the last frame and outputs
the boundaries of the segments in terms of frame numbers.
2.1.4 PPCA Technique
This algorithm starts with defining the mean ¯x and the covariance C for the
Gaussian distribution model using the first k frames of the motion. The intrinsic
dimensionality calculation is done for these k frames and the value of r is calculated
as in the SVD-PCA technique. For the PPCA technique, the value of τ is 0.95. This
approach provides an accurate model of a particular behavior because it captures the
correlation in the motion of different joint angles as well as the variance of all joint
angles. The SVD calculation for the first k frames is done with the value of k set to
150 yielding matrices U, V and Σ.
The mean ¯x of the distribution is equal to the center of motion, which is defined
in equation (2.2) for the SVD calculation. The covariance C is based on the estimated
intrinsic dimensionality of the motion. To determine C, first the averaged square of
discarded singular values σ is calculated:
σ 2 = 1
D − r ·
D
i=r+1
where D = 4J and J is the number of joints in the body.
The covariance matrix C is calculated by the following equations:
and
22
σ 2 i , (2.5)
W = Vr · Σr 2 − σ 2 1
2 I (2.6)
C = 1
n − 1 · W W T + σ 2 I = 1
n − 1 · V ˜ Σ 2 V T , (2.7)