Graphics Reference
In-Depth Information
The main ingredient in comparing and analysing shapes of curves is the construction of
a geodesic between any two elements of
S
, under the Riemannian metric given in Equation
5.7. Given any two curves
β
2
represented by their SRVFs
q
1
and
q
2
, we want to
compute a geodesic path between the orbits [
q
1
] and [
q
2
] in the shape space
β
1
and
. This task
is accomplished using a
path-straightening approach
, which was introduced in Klassen and
Srivastava (2006). The basic idea here is to connect the two points [
q
1
] and [
q
2
]byan
arbitrary initial path
S
α
and to iteratively update this path using the negative gradient of an
2
s
1
energy function
E
[
ds
. The interesting part is that the gradient of
E
has
been derived analytically and can be used directly for updating
α
]
=
α
˙
(
s
)
,
α
˙
(
s
)
. As shown in Klassen
and Srivastava (2006), the critical points of
E
are actually geodesic paths in
α
S
. Thus, this
gradient-based update leads to a critical point of
E
, which, in turn, is a geodesic path between
the given points. In the remainder of the chapter, we will use the notation
d
S
(
β
1
,β
2
)to
denote the length of the geodesic in the
shape space
S
between orbits
q
1
and
q
2
to reduce
the notation.
3D Patches Shape Analysis
Now, we extend ideas developed in the previous section from analyzing shapes of curves to
the shapes of patches. As mentioned earlier, we are going to represent a number of
l
patches
of a facial surface
S
with an indexed collection of the level curves of the
r
l
−
.
function
c
l
where
c
l
λ
(Euclidean distance from the reference point
r
l
). That is,
P
l
↔{
λ
,λ
∈
[0
,λ
0
]
}
,
is the
level set associated with
r
l
−
.
=
λ
. Through this relation, each patch has been represented
,λ
0
]
. In our framework, the shapes of any two patches are compared
by comparing their corresponding level curves. Given any two patches
P
1
and
P
2
, and their
level curves
[0
as an element of the set
S
c
1
c
2
{
λ
,λ
∈
[0
,λ
0
]
}
and
{
λ
,λ
∈
[0
,λ
0
]
}
, respectively, our idea is to compare the
patches curves
c
1
λ
and
c
2
λ
, and to accumulate these differences over all
λ
. More formally, we
define a distance
d
S
,λ
0
]
given by
[0
L
d
S
(
c
1
c
2
λ
d
S
[0
,λ
0
]
(
P
1
,
P
2
)
=
λ
,
)
d
λ.
(5.9)
0
P
2
), which is useful in biometry and other classification
experiments, we also have a geodesic path in
In addition to the distance
d
S
[0
,λ
0
]
(
P
1
,
,λ
0
]
between the two points represented by
P
1
and
P
2
. This geodesic corresponds to the optimal elastic deformations of facial curves
and, thus, facial surfaces from one to another. Figure 5.12 shows some examples of geodesic
paths that are computed between corresponding patches associated with shape models sharing
the same expression, and termed
intraclass geodesics
. In the first column we illustrate the
source, which represents scan models of the same subject, but under different expressions.
The third column represents the targets as scan models of different subjects. As for the middle
column, it shows the geodesic paths. In each row we have both the shape and the mean
curvature mapping representations of the patches along the geodesic path from the source to
the target. The mean curvature representation is added to identify concave/convex areas on the
source and target patches and equally spaced steps of geodesics. This figure shows that certain
patches, belonging to the same class of expression, are deformed in a similar way. In contrast,
S
[0
