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 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.
 
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