Graphics Reference
In-Depth Information
application - face analysis under facial expression variations. This is because of the following
reasons: (1) Such analysis uses the square-root velocity function representation, which allows
comparison local facial shapes in the presence of elastic deformations. (2) This method uses
a square-root representation under which the elastic metric reduces to the standard
2
metric
and thus simplifies the analysis (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 follows: For an interval
I
=
[0
,
1],
3
3
let
β
:
I
−→ R
be a curve and define
q
:
I
−→ R
to be its square-root velocity function
(SRVF), given by
˙
β
(
t
)
=
q
(
t
)
.
(3.5)
˙
|
β
(
t
)
|
∈
|·|
R
3
. We note that
q
(
t
)isa
Here
t
is a parameter
I
and
is the Euclidean norm in
special 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
=
I
2
-metric. Also, the squared
2
-norm of
q
, given by
2
L
L
q
q
(
t
)
,
q
(
t
)
d
t
=
˙
1, implying all curves are rescaled to unit length,
then translation and scaling variability have been removed by this mathematical representation
of curves.
β
(
t
)
d
t
,
is the length of
β
.Ifweset
q
=
β
1
−
β
2
=
β
1
◦
γ
−
β
2
◦
γ
.
(3.6)
Consider the two curves in Figure 3.6
a
. Let us fix the parametrization of the top curve to
be arc-length, that is, we are going to traverse that curve with speed equal to one. To have
the best matching of the curves, we should know at what rate we must move along the two
1
2
1
2
11
12
12
11
(b)
(a)
Figure 3.6
Illustration of elastic metric. In order to compare the two curves in (a), some combination
of stretching and bending are needed. The elastic metric measures the amounts of these deformations.
The optimal matching between the two curves is illustrated in (b). Copyright
C
2012, IEEE
