Graphics Reference
In-Depth Information
(a)
(b)
(c)
(d)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
q
2
d
ψ
*
d
τ
q
1
τ
=0
0
5 0 2 0 5
(e)
30
35
40
45
50
(f)
Figure 3.14
The figure illustrates: (a) the extracted radial curves; (b)-(c) A radial curve on a neutral
face, and the correspondent radial curve on the same face with happy expression, respectively; (d) the
two radial curves are plotted together; (e) the values of the magnitude of
ψ
∗
d
τ
|
τ
=
0
(
k
) computed between
the curves in (d) are reported for each point
k
of the curves; (f) the parallel vector field across the geodesic
between
q
1
and
q
2
in the space of curves
C
d
a
reference
radial curve. In particular, the reference radial curve (i.e., the curve with
0) is
chosen as oriented along the vertical axis, whereas the other radial curves are separated each
other by a fixed angle and are ordered in a clockwise manner. As an example, Figure 3.14
a
shows the radial curves extracted for a sample face with happy expression. To extract the
radial curves, the nose tip is accurately detected, and each face scan is rotated to the upright
position so as to establish a direct correspondence between radial curves that have the same
index in different facial scans (the preprocessing steps, including nose tip detection and pose
normalization are discussed in more detail in Sect. 5.5.1). In Figure 3.14
b
,
c
, two radial curves
at
α
=
90
◦
in the neutral and happy scans of a same subject are shown. As shown in the plot
Figure 3.14
d
, facial expressions can induce consistent variations in the shape of corresponding
curves. These variations are not the same in strength from expression to expression and for
different parts of the face. To effectively capture these variations, a dense scalar field is
proposed, which relies on a Riemannian analysis of facial shapes.
Each radial curve is represented on the manifold
α
=
by its SRVF. According to this, given
the SRVFs
q
1
and
q
2
of two radial curves, the
geodesic path
C
ψ
∗
on the manifold
C
between
q
1
and
q
2
is a critical point of the following energy function:
1
2
˙
˙
E
(
ψ
)
=
<
ψ
(
τ
)
,
ψ
(
τ
)
>
d
τ,
(3.12)
