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some new data samples. Then the Expectation-Maximization (EM) frame-

work [Dempster et al., 1977] can be applied to update the model parameters.

At time

the E-step provides exemplar ownership probabilities defined as

where is the index of the exemplars. In the M-step, the model is adapted

by computing new maximum likelihood estimates of its parameters. Note that

we only adapt the texture part of the model because shape features are less

person-dependent and not sensitive to changes of lighting.

The idea of Maximum Likelihood Linear Regression (MLLR) can be gener-

alized to this adaptation problem, where we estimate a linear transformation of

the GMM mean vectors to maximize the likelihood of new observations. How-

ever, conventional MLLR is not an online method which requires multiple data

samples for maximum likelihood optimization. In the M-step of our online EM

algorithm, only one data sample is available at a time. Thus we constrain the

transformation of the GMM mean vectors to be translation only. The M-step

of our algorithm is then to estimate which denotes the translation of

the GMM mean vectors from initial model, for the facial motion region, the

exemplar at time To weight the current data sample appropriately against

history, we consider the data samples under an exponential envelope located

at the current time as in [Jepson et al., 2001],

Here,

For the GMM model of certain value, suppose the GMM has
M
com-

ponents denoted by Here the subscripts

are dropped for simplicity. Given an adaptation data sample the

ML estimate of the translation can be computed by solving equation (7.9)

according to [Gales and Woodland, 1996]:

where is GMM component occupancy probability defined as the probability

that draws from the component of the GMM given draws from this

GMM. is the texture feature of the current adaptation data. A closed-form

solution for equation (7.9) is feasible when

is diagonal. The

element of

can be computed as

where

is the

diagonal element of

and

is the

element of

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