Graphics Reference
In-Depth Information
between
q
1
and
q
2
,
˙
where
ψ
denotes a path on the manifold
C
ψ
∈
T
ψ
(
C
) is the tangent vector
2
field on the curve
ψ
∈
C
, and
<.>
denotes the
L
inner product on the tangent space.
2
Because elements of
C
have an unit
L
norm,
C
is a hypersphere in the Hilbert space
2
(
I
3
). As a consequence, the geodesic path between any two points
q
1
,
L
,
R
q
2
∈
C
is simply
ψ
∗
given by the minor arc of the great circle connecting them on this hypersphere,
:[0
,
1]
→
C
.
This is given by the following expression:
1
sin(
ψ
∗
(
τ
)
=
)
(sin((1
−
τ
)
θ
)
q
1
+
sin(
θτ
)
q
2
)
,
(3.13)
θ
cos
−
1
(
where
θ
=
d
C
(
q
1
,
q
2
)
=
q
1
,
q
2
). We point out that sin(
θ
)
=
0, if the distance between
ψ
∗
(
the two curves is zero, in other words
q
1
=
q
2
. In this case, for each
τ
,
τ
)
=
q
1
=
q
2
. The
d
ψ
∗
d
τ
tangent vector field on this geodesic is then written as
:[0
,
1]
→
T
ψ
(
C
), and is obtained
by the following equation:
ψ
∗
d
d
θ
sin(
=−
)
(cos((1
−
τ
)
θ
)
q
1
−
cos(
θτ
)
q
2
)
.
(3.14)
τ
θ
Knowing that on geodesic path, the covariant derivative of its tangent vector field is equal
to 0,
ψ
∗
d
ψ
∗
d
d
d
ψ
∗
, and it can be represented with
is parallel along the geodesic
τ
|
τ
=
0
without any
τ
loss of information. Accordingly, Equation 3.14 becomes
ψ
∗
d
d
θ
sin(
|
τ
=
0
=
)
(
q
2
−
θ
θ
=
.
cos(
)
q
1
) (
0)
(3.15)
τ
θ
A graphical interpretation of this mathematical representation is shown in Figure 3.14. In
Figure 3.14
a
, we show a sample face with the happy expression and all the extracted radial
curves. In Figures 3.14
b
and 3.14
c
two corresponding radial curves (i.e., radial curves at the
same angle
), respectively, on neutral and happy faces of the same person are highlighted.
These curves are reported together in Figure 3.14
d
, where the amount of the deformation
between them can be appreciated, although the two curves lie at the same angle
α
α
and belong
to the same person. The amount of deformation between the two curves is calculated using
Equation 3.15, and the plot of the magnitude of this vector at each point of the curve is reported
in Figure 3.14
e
(i.e., 50 points are used to sample each of the two radial curves and reported
in
x
axis, the magnitude of DSF is reported in
y
axis).
Finally, Figure 3.14
f
illustrates the idea to map the two radial curves on the hypersphere
C
in the Hilbert space through their SRVFs
q
1
and
q
2
and shows the geodesic path connecting
these two points on the hypersphere. The tangent vectors of this geodesic path represent a
vector field whose covariant derivative is zero. According to this,
d
ψ
∗
d
τ
|
τ
=
0
becomes sufficient
to represent this vector field, with the remaining vectors generatable by parallel transport of
d
ψ
∗
d
ψ
∗
.
On the basis of the preceding representation, we define a DSF capable of capturing defor-
mations between two corresponding radial curves
τ
|
τ
=
0
along the geodesic
1
α
2
α
β
and
β
of two faces approximated by a
collection of radial curves.
