Learning 3D Face Deformation Model: Methods and Applications - 3D Modeling and Animation: Synthesis and Analysis Techniques for the Human Body - page 325

visualizing MUs is described in the subsection “MU adaptation.” Any arbitrary
facial deformation can be approximated by a linear combination of the MUs,
weighted by the MUPs.

Learning Parts-Based Linear Subspace

It is well known that the facial motion is localized, which makes it possible to
decompose the complex facial motion into parts. The decomposition helps
reduce the complexity in deformation modeling and improves the analysis'
robustness and the synthesis' flexibility. The decomposition can be done
manually based on the prior knowledge of facial muscle distribution (Tao, 1998).
However, it may not be optimal for the linear model used because of the high
nonlinearity of facial motion. Parts-based learning techniques provide a way to
help design parts-based facial deformation models , which can better approxi-
mate real, local facial motion. Recently, Non-negative Matrix Factorization
(NMF) (Lee & Seung, 1999), a technique for learning localized representation
of data samples, has been shown to be able to learn basis images that resemble
parts of faces. In learning the basis of subspace, NMF imposes non-negativity
constraints, which is compatible to the intuitive notion of combining parts to form
a whole in a non-subtractive way.
In our framework, we present a parts-based face deformation model. In the
model, each part corresponds to a facial region where facial motion is mostly
generated by local muscles. The motion of each part is modeled by PCA. Then,
the overall facial deformation is approximated by summing up the deformation
in each part:

∑

N
j ∑∑

N
j

M
i

∆ s = ∆ s = (

j

c e + e )

,

j ij ij 0 j

= 1 = 1 = 1

∆ s = s −

s 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 {

e c 1 , …, M c }.

To decompose facial motion into parts, we propose an NMF-based method. 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 motion energy. Figure
5(a) shows some parts derived by NMF. Adjacent different parts are shown in
different colors that are overlayed on the face model. We then use prior
knowledge about facial muscle distribution to refine the learned parts. The parts

e 0 e 1 c 0

, , ..., }, and MUP set { , j j Mj j j