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Section 12.3 Registering Deformable Objects 381

FIGURE 12.9: Different face intensity masks generated by moving deformation parameters
to different values. Each block shows the effect of a different parameter; the center of that
block shows the parameter at the mean value (where the mean is taken over numerous
example faces), and the left (resp. right) of the block shows the parameter at mean plus
(resp. minus) three standard deviations. Note how a range of expressions is encoded by
these parameter variations. This figure was originally published as Figure 2 of “Active
Appearance Models,” by T. Cootes, G. Edwards, and C. Taylor, IEEE Transactions on
Pattern Analysis and Machine Intelligence, 2001, c IEEE, 2001.

a rotation matrix R, a translation vector t, and a set of parameters θ l , and write

W = R(V + B l θ l )+ t
l
to get a model of the deformations that also incorporates rotation and translation
of the neutral face.
The matrices B l could be obtained by manually aligning a mesh with a de-
formed face, for example (Figure 12.9 shows some deformations encoded by one set
of such matrices). The vertices w are a function of the parameters θ,and so we
must minimize
(k) (k) 2
g(||aI d (p(s j ,t j ; w (θ))) + b −I n (p(s j ,t j ; v ))|| )
k∈triangles j

as a function of R, t, θ l , a,and b.

12.3.2 Active Appearance Models in Practice
We have shown several minimization problems for registering active appearance
models. They are not easy minimization problems at all, though they can be solved
(Figure 12.10). Numerous local minima are likely, and there are several important
strategies that help minimize. First, it is helpful to have an estimate of rotation
and translation before estimating the deformation. We expect that deformations
are relatively small, and that major rotations and translations will be easy to es-
timate. It is natural to first produce a rotation and translation estimate, then fix
that estimate (which is equivalent to working with a new V and a new set of B l )to
estimate the deformations θ l , and then finally polish all estimates simultaneously.