Graphics Reference
In-Depth Information
Path-Straightening Approach
: This is an approach for finding geodesic paths between
points on a manifold by minimizing the energy functional:
•
1
d
d
t
d
t
,
d
d
t
α
=
.
E
[
]
0
Klassen & Srivastava (2006) derived a particularly convenient form of the gradient of
E
in
cases where
M
is a submanifold of a larger vector space
V
and the Riemannian metric on
M
is a restriction of the metric on
V
. This is the case for all of the manifolds studied in
this topic.
•
Quotient Space of a Manifold
: Let a group
G
act on the manifold
M
as defined earlier.
Then, the set of all orbits of all elements of
M
is called the quotient of
M
under
G
.ITis
denoted by
M
/
G
.Wehave:
M
/
G
={
[
p
]
|
p
∈
M
}
,
where [
p
]
={
(
g
·
p
)
|
g
∈
G
}
.
Reparametrization Action on Curves
: Let
B
be an appropriate space of curves of the type
•
n
β
:[0
,
1]
→ R
and
be the reparametrization group. Then,
acts on
B
according to the
2
metric on
action (
,
then this action is not by isometries. However, if one assumes an elastic metric, as defined by
Mio et al. [2007], then this action is by isometries and one can define the inherited distance
on the quotient space
γ,β
)
=
β
◦
γ
. It is interesting to note that if we assume the standard
L
B
B/
.
Reparametrization Group
: Let
be the set of all corner-preserving diffeomorphisms from
•
[0
,
1] to itself. That is,
is the set of all
γ
such that
γ
is a diffeomorphisms and
γ
(0)
=
0
and
γ
(1)
=
1.
is a group under concatenation: for any
γ
1
,γ
2
∈
the group operation is
γ
1
◦
γ
2
. The identity element of the group is given by
γ
id
(
t
)
=
t
.
, the curve
˜
n
is a parametrized curve in
n
, then for a
If
β
:[0
,
1]
→ R
R
γ
∈
β
(
t
)
=
β
(
γ
(
t
))
is called a reparametrization of
β
; therefore,
is also called the reparametrization group.
•
Riemannian Metric
: A Riemannian metric is map that smoothly associates to each point
p
M
a form for computing inner product between elements of
T
p
(
M
). A manifold with a
Riemannian metric on it is called a Riemannian manifold.
∈
∈
•
Tangent Space
: For a point
p
M
,
T
p
(
M
) is the set of all vectors that are tangent to
M
at
point
p
. A tangent vector can be defined by constructing a differentiable curve on
M
passing
through
p
and evaluating the velocity to the curve at that point.
T
p
(
M
) is a vector space of
the same dimension as the manifold
M
.
References
Antini G, Berretti S, del Bimbo A, Pala P. 3D mesh partitioning for retrieval by parts applications. In: Proceeding of
the IEEE International Conference on Multimedia and Expo; 2005 Jul 6-9 Amsterdam, The Netherlands. New
York: IEEE. 2005. p. 1210-1213.
Berretti S, Bimbo AD, Pala P. Distinguishing facial features for ethnicity-based 3d face recognition. ACM Transactions
on Intelligent Systems and Technology 2012;3(3):45.
Berretti S, del Bimbo A, Pala P. Description and retrieval of 3D face models using iso-geodesic stripes. In: Proceeding
of the 8th ACM International Workshop on Multimedia Information Retrieval; 2006 Oct 26-27, Santa Barbara,
CA. New York: ACM Press. 2006. p. 13-22.