Graphics Reference
In-Depth Information
In our case, the exponential mapping exp maps a vector v
T μ (
S
) to a point of
S
. In other
words, to reach the point exp μ ( v ), one starts at
, and then moves for time 1 along the unique
constant speed geodesic whose velocity vector at q is v . The inverse of an exponential map
takes a point q i on the manifold
μ
S
and maps it to an element (or multiple elements) of the
tangent space T μ (
S
). A vector v is said to be the inverse exponential map of a point q i S
,at
exp 1
μ
the point
μ S
,ifexp μ ( v )
=
q i . It is denoted by v
=
( q i ).
As a result, the exponential map, exp : T μ (
S
)
S
, has also a simple expression. Let v be
a vector v
T
μ
(
S
), the exponential mapping of v gives an element of the manifold
S
as:
v
exp μ ( v )
=
cos(
v
)
μ +
sin(
v
)
,
v
The exponential map is a bijection if we restrict
v
so that
v
[0
). For a point q i S
,
exp 1
μ
such that ( q i = μ
), the inverse exponential map v
=
( q i ) projects q i on the tangent space
of
μ
as u :
θ
sin(
) ( q i
θ
μ
,
u
=
cos(
)
)
θ
cos 1 (
where
θ =
<μ,
q
>
). The result of this projection is elements on the tangent space
{
. It is possible to perform traditional operations and work as in an Euclidean
space using the projected elements. We summarize this procedure in Algorithm 5.
v 1 ,
v 2 ,...,
v N }
Algorithm 5 Projection of facial curves on eigencurves space
Input : P : face without missing data, eigenvectors B k j ,
k
=
1
...
K
Output : P restored
K : number of curves in each face;
for k
1 to K do
q k =
SRVF ( c k );
exp 1
v k =
μ k ( q k );
μ k (
N B k
q k =
exp
j = 1
c k v k j
v k j );
c k =
SRVF ( q k );
end
Figure 3.20 illustrates the restoration of three different curves. As illustrated respectively in
Figure 3.20 a - c , the restoration of curve generates the same curve. This validates our model,
which represents curves in a new basis while keeping 90% of the information. The same idea
is followed to recover 3D faces: Curves of different index are restored. The collection of the
restored curves represents the 3D face. Notice that the number of eigenvectors differs from one
level to another. Figure 3.21 illustrates the face projection. The first row illustrates the original
face (to the left) and the restored one (to the right). In the second row, we see in the middle
 
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