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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 )
=
,...,
,...,
,
))
 
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