LEARNING GEOMETRIC 3D FACIAL MOTION MODEL - 3D Face Processing: Modeling, Analysis and Synthesis - page 23

by summing up the deformation in each part:
where is the deformation of the facial shape. N is the number of
parts. We call this representation parts-based MU, where the j-th part has its
MU set and MUP set
To decompose facial motion into parts, we use NMF together with prior
knowledge. In this method, we randomly initialize the decomposition. Then,
we use NMF to reduce the linear decomposition error to a local minimum. We
impose the non-negativity constraint in the linear combination of the facial mo-
tion energy. We use a matlab implementation of NMF from the web site
http://journalclub.mit.edu (under category “Computational Neuroscience”). The
algorithm is an iterative optimization process. In our experiments, we use 500
iterations. Figure 3.4(a) shows some parts derived by NMF. Adjacent different
parts are shown in different patterns overlayed on the face model. We then
use prior knowledge about facial muscle distribution to refine the learned parts.
The parts can thus be (1) more related to meaningful facial muscle distribution,
(2) less biased by individuality in the motion capture data, and (3) more easily
generalized to different faces. We start with an image of human facial muscle
distribution, illustrated in Figure 3.4(b) [FacialMuscel, 2002]. Next, we align it
with our generic face model via image warping, based on facial feature points
illustrated in Figure 3.7(c). The aligned facial muscle image is shown in Fig-
ure 3.4(c). Then, we overlay the learned parts on facial muscle distribution
(Figure 3.4(d)), and adjust interactively the learned parts such that different
parts correspond different muscles. The final parts are shown in Figure 3.4(e).
The parts are overlapped a bit as learned by NMF. For convenience, the overlap
is not shown Figure 3.4(e).
Figure 3.4. (a): NMF learned parts overlayed on the generic face model. (b): The facial muscle
distribution. (c): The aligned facial muscle distribution. (d): The parts overlayed on muscle
distribution. (e): The final parts decomposition.