Graphics Reference
In-Depth Information
The interpolation deformation model is given in terms of the warping function
F
(
u
), with
4
Au
4
×
1
K
(
u
)
L
×
W
T
=
+
,
F
(
u
)
(1.40)
4
×
L
where
A
(affine transformation) and
W
(non-affine warping) are TPS parameters and
K
(
u
)
=
)
T
is the control point influence vector.
The warping coefficients
A
and
W
are computed by the equation:
(
|
u
−
u
1
|
;
|
u
−
u
2
|
;
...
;
|
u
−
u
m
|
W
)
U
I U
T
0
(
A
|
=
(
V
|
0)
(1.41)
K
+
L
λ
In other words, any point on
close to a source landmark
v
k
will be moved to a place
close to the corresponding target landmark
u
k
in
M
. The points in between are interpolated
smoothly using Bookstein's Thin Plate Spline algorithm Bookstein (1989).
P
•
Non-rigid ICP
. Register in a non-rigid way a template
by non-rigid
ICP requires estimating both correspondence and a suitable warping function that matches
the shape difference between them. In Allen et al. (2003) and Amberg et al. (2007) similar
ideas are presented for scan-template warping applied on human body in Allen et al. (2003)
and on human faces in Amberg et al. (2007). Both of them proposed an energy-minimization
framework, as given by
M
and an input scan
P
E
data
(
T
)
E
smoothness
E
la
ndm
arks
w
i
dist
2
(
T
i
v
i
, P
T
i
T
j
2
T
i
v
i
−
u
j
2
E
=
α
)
+
β
−
F
+
γ
,
v
i
∈
M
i
,
j
|{
vi
,
vj
}∈
edges(
M
)
}
i
(1.42)
where minimizing the term
E
data
guarantee that the distance between the deformed template
M
is small. The term
E
smoothness
is used to regularize the deformation. In
other words, it penalizes large displacement differences between neighboring vertices. The
term
E
landmarks
is introduced to guide the deformation by using corresponding control points
that are simply the anthropometric markers in human body and facial landmarks in the case
of face fitting. Similar formulation are presented in Zhang et al. (2004) for template fitting.
The Figure 1.16 illustrates an example of template fitting results. A similar formulation is
used in Weise et al. (2009) for personalized template building.
and the target data
P
Template Tracking
In Zhang et al. (2004), after the template fitting step, the authors proposed a tracking procedure
which yields point correspondence across the entire sequence. They obtained time-varying
face models (of the deformed template) without using markers. Once this template sequence
is acquired, they propose to interactively manipulate it to create new expressions. To achieve
template tracking, they first compute optical flow from the sequence. The flow represents
