Graphics Reference
In-Depth Information
3
. Although there are several ways to analyze
shapes of closed curves, an elastic analysis of the parametrized curves is particularly appro-
priate in 3D curves analysis. This is because (1) such analysis uses a square-root velocity
function representation which allows us to compare local facial shapes in presence of elastic
deformations, (2) this method uses a square-root representation under which the elastic metric
reduces to the standard
We start by considering a closed curve
β
in
R
2
metric and thus simplifies the analysis, and (3) under this metric
the Riemannian distance between curves is invariant to the reparametrization. To analyze the
shape of
L
β
, we shall represent it mathematically using a square-root representation of
β
as
3
be a curve and define
q
:
I
3
to be
follows ; for an interval
I
=
[0
,
1], let
β
:
I
−→ R
−→ R
its square-root velocity function (SRVF), given by
˙
β
(
t
)
=
q
(
t
)
.
(5.6)
˙
β
(
t
)
3
. We note that
q
(
t
) is a special
Here
t
is a parameter
∈
I
and
.
is the Euclidean norm in
R
function that captures the shape of
and is particularly convenient for shape analysis, as
we describe next. The classical elastic metric for comparing shapes of curves becomes the
L
β
2
-metric under the SRVF representation (Srivastava et al., 2011). This point is very important
as it simplifies the calculus of elastic metric to the well-known calculus of functional analysis
under the
=
S
1
<
2
-metric. Also, the squared
2
-norm of
q
, given by:
2
L
L
q
q
(
t
)
,
q
(
t
)
>
=
S
1
˙
d
t
β
(
t
)
d
t
,
which is the length of
β
. To restrict our shape analysis to closed curves,
|
S
1
q
(
t
)
1
3
2
(
1
3
). Notice that the
we define the set:
C
={
q
:
S
−→ R
q
(
t
)
d
t
=
0
}⊂L
S
,
R
elements of
are allowed to have different lengths. Because of a nonlinear (closure) constraint
on its elements,
C
is a non-linear manifold. We can make it a Riemannian manifold by using
the metric: For any
u
C
,
v
∈
T
q
(
C
), we define
,
=
1
,
.
u
v
u
(
t
)
v
(
t
)
d
t
(5.7)
S
So far we have described a set of closed curves and have endowed it with a Riemannian
structure. Next we consider the issue of representing the
shapes
of these curves. It is easy
to see that several elements of
C
can represent curves with the same shape. For example, if
3
, we get a different SRVF but its shape remains unchanged. Another
similar situation arises when a curve is reparametrized; a reparameterization changes the SRVF
of curve but not its shape. To handle this variability, we define orbits of the rotation group
SO
(3), and the reparameterization group
R
we rotate a curve in
as the equivalence classes in
C
. Here,
is the
1
set of all orientation-preserving diffeomorphisms of
S
(to itself) and the elements of
are
1
3
and a function
viewed as reparameterization functions. For example, for a curve
β
:
S
→ R
1
1
,
γ
:
S
→ S
γ
∈
, the curve
β
◦
γ
is a reparameterization of
β
. The corresponding SRVF
→
√
˙
changes according to
q
(
t
)
γ
(
t
)
q
(
γ
(
t
)). We set the elements of the orbit
˙
[
q
]
={
γ
(
t
)
Oq
(
γ
(
t
))
|
O
∈
SO
(3)
,γ
∈
}
(5.8)
to be equivalent from the perspective of shape analysis. The set of such equivalence classes,
denoted by
=
C/
3
.
S
(
SO
(3)
×
) is called the
shape space
of closed curves in
R
S
inherits a
Riemannian metric from the larger space
C
because of the quotient structure.
