FACIAL MOTION ANALYSIS USING FLEXIBLE APPEARANCE MODEL - 3D Face Processing: Modeling, Analysis and Synthesis - page 77

The probability weighted average of the translation vector up to time for
the exemplar of a certain facial motion region is then

where is a normalization factor defined as
and is computed using equation (7.10). Equation (7.11) can be rewritten
in a recursive manner as:

where Equations (7.10) and (7.12) are
applied to updated the translation vectors of each facial motion region.
In an online EM algorithm, the training samples can not be used iteratively for
optimization. Thus, errors made by the initial model may cause the adaptation
method to be unstable. Note that the chosen exemplars are common semantic
symbols which have their intrinsic structure. Therefore, when adapting the ex-
emplar model to a particular person, the translations of the mean vectors should
be constrained rather than random. For the exemplars model of a facial motion
region at time let denote the translation vector constructed by concatenat-
ing the translations of all the mean vectors, i.e., where
K is the number of exemplars. In our algorithm, is a 72-dimension vector.
We impose the constraint that the vector should lie in certain low dimen-
sional subspaces. To learn such a low dimensional subspace, we first leam a
person-independent exemplar model from exemplars of many people. Then
a person-dependent exemplar model is learned for each person. We collect a
training sample set consisting of the translation vectors between the mean
vectors of person-independent model and person-dependent models. Finally,
PCA is applied on the set Principal orthogonal components which account
for major variation in set are chosen to span a low Dimensional subspace.
A translation vector can be projected to the learned subspace by

and the new constrained translation vector can be reconstructed as

where W is the matrix consisting of principal orthogonal components, is the

mean translation vector of the vectors in
In summary, the online EM-based adaptation algorithm is as follows:
E-step: compute exemplar ownership probabilities based on equa-

tions (7.5), (7.6) and (7.8).