Graphics Reference

In-Depth Information

weight of each control point such that the trajectory has higher likelihood.

Intuitively, it can be achieved in a way that states with small variance pull the

trajectory towards them, while states with larger variance allow the trajectory to

stay way from them. Therefore we set the weights to be where

is the trajectory normal vector that also passes is the variance

of the Gaussian distribution in direction. In practice, it is not trivial to

compute Thus, we approximate it by normal vector of line segment

(see in Figure 5.1(a)). Compared to [Brand, 1999], the smooth

trajectory obtained is less optimal in terms of maximum likelihood. But it is

fast and robust, especially when the number of states is small or the assumed

HMM topology fits the data poorly. Ezzat et al. [Ezzat et al., 2002] propose an

trajectory synthesis approach similar in spirit to ours. But our formulation is

easier in a sense that it can viewed as a natural extension of traditional key-frame-

based spline interpolation scheme, given the probability distribution of each key

frame. Figure 5.1 shows a synthetic example comparing conventional NURBS

and our statistically weighted NURBS. The dots are samples of facial shapes.

Figure 5.1.
(a): Conventional NURBS interpolation. (b): Statistically weighted NURBS in-

terpolation.

The dashed line connects centers of the states. The solid line is the generated

facial deformation trajectory. In Figure 5.1(b), the trajectory is pulled towards

the states with smaller variance, thus have higher likelihood than trajectory

in Figure 5.1(a). Ezzat et al. [Ezzat et al., 2002] observed that the generated

trajectory by spline-based method could be “under-articulated”. To reduce

the undesirable effect, they proposed to adjust the center and variance of the

Gaussian model based on training data. We plan to perform similar adjustments

in our framework.

Search WWH ::

Custom Search