Graphics Reference
In-Depth Information
.
.
.
Curves of index
i
Curves of index
j
Curves of index
k
(a)
(b)
(c)
Figure 3.18
Radial curves collections for training step
information available for this prediction are (1) the current (partially observed) curve and (2)
several (complete) training curves at the same angle that are extracted from full faces. The
basic idea is to develop a sparse model for the curve from the training curves and use that
to complete the observed curve. To keep the model simple, we use the PCA of the training
data (in an appropriate vector space) to form an orthogonal basis representing training shapes.
Then, this basis is used to estimate coefficients of the observed curve, and the coefficients help
us to reconstruct the full curve. Because the shape space of curve
S
is a nonlinear space, we
use the tangent space
T
μ
(
is the mean of the training shapes, to perform PCA.
Some examples of curves used during training step are shown in Figure 3.18.
Let
S
), where
μ
denote the angular index of the observed curve, and let
q
1
q
2
q
k
α
be the SRVFs
of the curves taken from training faces at that angle. As described earlier, we can compute the
sample Karcher mean of their shapes
α
α
,
α
,...,
[
q
i
α
{
]
∈
S
}
, denoted by
μ
α
. Then, using the geometry of
S
we can map these training shapes in the tangent space using the inverse exponential map,
that is, obtain
v
i
,α
=
exp
−
1
μ
α
(
q
i
α
), where
θ
sin(
exp
−
1
q
1
)
(
q
2
−
cos
−
1
(
q
2
(
q
2
)
=
cos(
θ
)
q
1
)
,θ
=
q
1
,
)
,
θ
and where
q
2
is the optimal rotation and reparametrization of
q
2
to be aligned with
q
1
,as
discussed earlier. A PCA of the tangent vectors
{
v
i
}
leads to the principal basis vectors
u
1
,α
,
u
2
,α
,...,
u
J
,α
, where
J
represents the number of significant basis elements.
Figure 3.19 illustrates a mapping of elements of the shape space onto the tangent space on
the mean shape
μ
.
