Graphics Reference
In-Depth Information
d
ψ
α
d
Definition 3.6.1
Dense Scalar Field (DSF)
Let x
α
(
t
)
, which corresponds to
the values of the magnitude computed between the curves q
1
and q
2
for each point t of the
curves. Let T be the number of sampled points per curve and
=||
τ
|
τ
=
0
(
t
)
||
|
|
be the number of curves
used per face. So, we define the function f by:
R
+
)
T
,
f
:
C
×
C
−→
(
f
q
1
,
q
2
=
x
1
.
x
k
x
T
α
α
,...,
α
,...,
{
β
α
|
α
∈
}
1
{
β
α
|
α
∈
}
2
Assuming that
and
be the collections of radial curves associated with
the two faces F
1
and F
2
and let
{
q
1
α
and
{
q
2
α
be their SRVFS, the DSF vector is defined by:
=
f
q
1
,
q
2
,...,
f
q
1
,
q
2
,...,
f
q
|
|
1
q
|
2
.
DSF(
F
1
,
,
F
2
)
(
3.16)
The dimension of the vector DSF vector is
|
|×
T.
Algorithm 2 summarizes the proposed approach.
Algorithm 2
Computation of the dense scalar field
Input
: Facial surfaces
F
1
and
F
2
;
T
: number of points on a curve;
α
0
: angle between
successive radial curves;
|
|
: number of curves per face.
Output
: DSF(
F
1
,
F
2
): the Dense Scalar Field between the two faces.
α
=
0
;
while
α<
|
|
do
for
i
1
to
2
do
Extract the curve
←
;
Compute corresponding square-root velocity function
q
i
α
β
i
α
=
˙
β
i
α
(
t
)
√
β
i
α
(
t
)
(
t
)
∈
C
;
=
,
...
t
1
2
T
.
end
Compute
the distance between
q
1
α
and
q
2
α
d
C
(
q
1
q
2
α
cos
−
1
(
q
1
α
q
2
θ
as:
θ
=
α
,
)
=
α
)
d
ψ
∗
d
Compute the deformation vector
τ
|
τ
=
0
using Equation 3.15 as:
f
q
1
,
q
2
=
T
+
(
x
α
(1)
,
x
α
(2)
,...,
x
α
(
T
))
∈
;
,
)
q
2
θ
sin(
)
q
1
α
x
α
(
t
)
=
α
−
cos(
θ
t
=
1
,
2
...
T
;
θ
end
Compute the local deformation DSF(
F
1
,
d
ψ
∗
d
τ
|
τ
=
0
(
k
);
F
2
) as the magnitude of
f
(
q
|
|
1
q
|
|
2
(
f
(
q
1
,
q
2
)
f
(
q
1
,
q
2
)
DSF(
F
1
,
F
2
)
=
,...,
,...,
,
))