Graphics Reference
In-Depth Information
Neutral
Happy
Angry
Surprise
Sad
Fear
Disgust
Low
deformations
High
deformations
Figure 3.15 Deformation maps computed between the neutral face of a sample subject and the apex
frame of the six prototypical expressions sequences of the same subject
In Figure 3.15, an example of the deformation field computed on the 3D frames of a sample
subject is reported. In particular, a neutral mesh is reported on the left, and the vector field
is computed between the 3D neutral face and the 3D apex frames of each expression of the
same subject. The values of the vector field needed to be applied on the neutral face to convey
the six different universal expressions reported using a color scale. In particular, colors from
green to black represent the highest deformations, whereas blue represents the lower values of
the vector field. It can be observed, as the regions with high deformation lie in different parts
of the face for different expressions. For example, as intuitively expected, the corners of the
mouth and the cheeks are strongly deformed for happiness expression, whereas the eyebrows
are strongly deformed for the angry expression.
3.7 Statistical Shape Analysis
As mentioned earlier, an important advantage of our Riemannian approach over many past
papers is its ability to compute summary statistics of a set of faces.
3.7.1 Statistics on Manifolds: Karcher Mean
What are the challenges in applying classical statistics if the underlying domain is nonlinear?
Take the case of the simplest statistic, the sample mean, for a sample set
{
x 1 ,
x 2 ,...,
x k }
on
R
n :
k
1
k
n
x k =
x i ,
x i ∈ R
(3.17)
i = 1
n but nonlinear manifold? In this situation, the
summation in Equation 3.17 is not valid operation, and the equation is not useful. So, how do
we define the sample mean in this case?
For example, one can use the notion of Karcher mean (Karcher, 1977) to define an average
face that can serve as a representative face of a group of faces.
Now what if underlying space is not
R
 
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