LEARNING GEOMETRIC 3D FACIAL MOTION MODEL - 3D Face Processing: Modeling, Analysis and Synthesis - page 19

trajectory which minimizes an objective function consisting of a target term and
a smoothness term. The target term is a distance function between the trajectory
and the given key shapes. The optimization of the objective function yields mul-
tivariate additive quintic splines [Wahba, 1990]. The results produced by this
approach could look under-articulated. To solve this problem, gradient descent
learning [Bishop, 1995] is employed to adjust the mean and covariances. In the
learning process, the goal is to reduce the difference between the synthesized
trajectories and the trajectories in the training data. Experimental results show
that the learning improves the articulation.
1.3 Machine learning techniques for facial deformation
modeling
In recent years, more available facial motion capture data enables researchers
to learn models which capture the characteristics of real facial deformation.
Artificial Neural Network (ANN) is a powerful tool to approximate func-
tions. It has been used to approximate the functional relationship between
motion capture data and the parameters of pre-defined facial deformation mod-
els. Morishima et al. [Morishima et al., 1998] used ANN to learn a function,
which maps 2D marker movements to the parameters or a physics-based 3D
face deformation model. This helped to automate the construction of physics-
based face muscle model, and to improve the animation produced. Moreover,
ANN has been used to learn the correlation between facial deformation and
other related signals. For example, ANN is used to map speech to face ani-
mation [Lavagetto, 1995, Morishima and Yotsukura, 1999, Massaro and et al.,
1999].
Principal Component Analysis (PCA) [Jolliffe, 1986] learns orthogonal com-
ponents that explain the maximum amount of variance in a given data set. Be-
cause facial deformation is complex yet structured, PCA has been applied to
learn a compact low dimensional linear subspace representation of 3D face de-
formation [Hong et al., 2001b, Kshirsagar et al., 2001, Reveret and Essa, 2001].
Then, arbitrary complex face deformation can be approximated by a linear com-
bination of just a few basis vectors. Besides animation, the low dimensional
linear subspace can be used to constrain noisy low-level motion estimation to
achieve more robust 3D facial motion analysis [Hong et al., 2001b, Reveret
and Essa, 2001]. Furthermore, facial deformation is known to be localized. To
learn a localized subspace representation of facial deformation, Non-negative
Matrix Factorization (NMF) [Lee and Seung, 1999] could be used. It has been
shown that NMF and its variants are effective to learn parts-based face image
components, which outperform PCA in face recognition when there are occlu-
sions [Li and et al., 2001]. In this chapter, we describe how NMF may help to
learn a parts-based facial deformation model. The advantage of a parts-based
model is its flexibility in local facial motion analysis and synthesis.