Graphics Reference
In-Depth Information
-1
(
q
i
)
v
= exp
μ
T
(
S
)
μ
μ
q
i
Figure 3.19
Illustration of mapping shapes onto the tangent space of
μ
,
T
μ
(
S
)
We summarize the calculation of eigencurves in Algorithm 4.
Algorithm 4
Eigencurves computation
Input
: Training faces (without missing data)
G
={
y
i
}
1
≤
i
≤
N
G
Output
:
B
: eigenvectors
K : number of curves in each face;
for
k
={
v
k
j
}
←
1
to
K
do
μ
k
=
intrinsic mean of
SRVF
(
y
k
i
) (Karcher mean).
for
i
←
1
to
N
G
do
exp
−
1
β
k
i
=
μ
k
(
SRVF
(
y
k
i
))
end
S
k
=
N
G
T
k
i
i
=
1
β
k
i
β
v
k
j
= eigenvectors of
S
k
end
3.8 Applications of Statistical Shape Analysis
3.8.1 3D Face Restoration
Now returning to the problem of completing a partially occluded curve, let us assume that it
is observed for parameter value
t
in [0
,τ
⊂
,
]
[0
1]. In other words, the SRVF of this curve
q
(
t
) is known for
t
∈
[0
,τ
] and unknown for
t
>τ
. Then, we can estimate the coefficients of
u
j
,α
≈
0
q
under the chosen basis according to
c
j
,α
=
q
,
q
(
t
)
,
u
j
,α
(
t
)
d
t
, and estimate the
SRVF of the full curve according to
J
q
α
(
t
)
=
c
j
,α
u
j
,α
(
t
)
,
t
∈
[0
,
1]
.
j
=
1
