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of expression recognition when emotions were manifested by subtle changes of the face.
In another solution presented by Sandbach et al. (2011, 2012), the authors used an existing
nonrigid registration algorithm (FFD) (Rueckert et al., 1999) on the basis of the B-splines
interpolation between a lattice of control points. Here, dense matching was a preprocessing
step used to estimate a motion vector field between 3D frames
t
and
t
-1. The problem of
quantifying subtle deformations along the sequence still remains a challenging task, and the
results presented in Sandbach et al. (2011) were limited to just three facial expressions:
happiness, anger, and surprise.
In the following paragraphs we discuss an approach that uses a collection of radial curves
to represent a 3D face. A Riemannian shape analysis is applied to effectively quantify defor-
mations induced by facial expressions in subsequent 3D frames. This is obtained by a dense
scalar field defined in Chapter 3 (Section 3.8.5), which denotes the shooting directions of the
geodesics paths constructed between pairs of corresponding radial curves of two expressive
faces. This approach has been originally proposed in Drira et al. (2012). The first step to
capture the deformation between two given 3D faces
F
1
and
F
2
is to extract the radial curves
originating from the nose tip. Let
1
α
2
α
β
and
β
denote the radial curves that make an angle
α
˙
with a reference radial curve on faces
F
1
and
F
2
, respectively. The tangent vector field
ψ
α
that
1
α
2
α
represents the energy
E
needed to deform
.We
consider the magnitude of this vector field at each point
k
of the curve to construct the DSFs
of the facial surface. In this way, the DSF quantifies the local deformation between points of
radial curves
β
to
β
is then calculated for each index
α
β
2
, respectively, of the faces
F
1
and
F
2
. In the practice, we represent each
face with 100 radial curves and 50 points on each curve so that the DSFs between two 3D
faces is expressed by a 5000-dimensional vector.
β
1
and
Mean Shape Deformation with Random Forest Classifier
In this section, we propose a pattern-based approach for expression classification in 3D video.
The idea is to capture a mean deformation of the face in the sliding window on the 3D
expression sequence, and consider it as a pattern for classification. To get this pattern, the first
frame of each subsequence is considered as the reference one, and the dense deformation is
computed from this frame to each of the remaining frames of the sub-sequence. Let
F
ref
denote
the reference frame of a sub-sequence and
F
i
the
i
th successive frame in the subsequence; the
successive frame, for example, is denoted by
F
1
. The DSF is calculated between
F
ref
and
F
i
,
for different values of
i
(
i
=
1
...
n
−
1), and the mean deformation is then given by
n
−
1
1
¯
DSF
=
DSF(
F
ref
,
F
i
)
.
(5.10)
−
n
1
i
=
1
6 frames. Each expres-
sion is illustrated in two rows, the upper row gives the reference frame of the subsequence and
the
n
Figure 5.19 illustrates one subsequence for each expression with
n
=
1 successive frames of the subsequences. Later, the corresponding dense scalar fields
computed for each frame are shown. The mean deformation map is reported at the right and
represents the feature vector for each subsequence. The feature vector for this subsequence is
built on the basis of the mean deformation of the
n
−
1 calculated deformations. Thus, each
subsequence is represented by a feature vector of size equal to the number of points on the
−
