LEARNING GEOMETRIC 3D FACIAL MOTION MODEL - 3D Face Processing: Modeling, Analysis and Synthesis - page 27

displacement of a certain vertex is of the form

where is the coordinate of a marker, and is its displace-
ment. is a radial basis kernel function such as Gaussian function, and
are the weights. and need to be carefully designed to ensure the inter-
polation quality. For facial deformation, the muscle influence region is local.
Thus, we choose a cut-off region for each vertex. We set the weights to be zero
for markers that are outside of the cut-off region, i.e., they are too far away
to influence the vertex. In our current system, the local influence region for

the i-th vertex is heuristically assigned as a circle, with the radius as the
average of the distances to its two nearest neighbors. Similar to [Marschner

et al., 2000], we choose the radial basis kernel to be
where We choose to be the normalization factor such
that The lips and eye lids are two special cases
for this RBF interpolation, because the motions of upper parts of them are not
correlated with the motions of the lower parts. To address this problem, we
add “upper” or “lower” tags to vertices and markers near mouth and eyes. If a
marker M and a vertex V have different tags, M has no influence on V. Thus, the
weight of the marker M is set to be zero in the RBF interpolation (equation 3.3)
of the vertex V. These RBF weights need to be computed only once for one set
of marker positions. The weights are stored in a matrix. The matrix is sparse
because marker influence is local. During synthesis, the movement of mesh
vertices can be computed by one multiplication of the sparse RBF matrix based
on equation 3.3. Thus the interpolation is fast.
After the MU fitting, MUs can be used to animate any 3D face models
or analyze facial motion in image sequences. Deformed face image examples
presented in this chapter, such as faces deformed by first four MUs in Figure 3.3,
are produced after fitting MU to iFACE models. MU fitting is also used in facial
motion analysis in Chapter 4.

6. Temporal Facial Motion Model
In this section, we describe the temporal facial deformation trajectory mod-
eling in the 3D face processing framework. The model describes temporal
variation of facial deformation given constraints (e.g. key shapes) at certain
time instances. For compactness and usability, we propose to use an spline-
based model similar to the spirit of the approach of Ezzat et al. [Ezzat et al.,
2002]. The advantage of this approach is that it employs only a few key facial
shapes estimated from training data. In order to get a smooth trajectory, we
use NURBS (Nonuniform Rational B-splines) interpolation, which is easy to