Graphics Reference
In-Depth Information
UL
UR
Y
X
Z
Y
L
X
(a)
(b)
Figure 3.31
(a) Sample face models, where the fourth and seventh iso-geodesic stripes are evidenced.
The 3DWW relationship descriptors are computed for the UL, UR, and L parts of the stripe pair; (b)
Projection of the pairs of iso-geodesic stripes in (a) on the XY plane, with the partitioning of the
iso-geodesic stripes into three parts
of faces with expression variations (results of the iso-geodesic stripes approach without face
partitioning were first reported in Berretti et al. (2006)).
Once facial stripes are extracted, distinctive structural features of 3D face scans are captured
by describing the pointwise 3D spatial relationships between homologous parts of pairs of
iso-geodesic stripes. To this end, the
3D weighted walkthroughs
(3DWWs) descriptor has been
used. The 3DWW was first introduced in Berretti et al. (2006), and their use and properties in
the context of 3D face recognition have been extensively discussed in Berretti et al. (2010). It
defines a set of integral measures over the points of two regions,
A
and
B
, in the 3D domain.
These measures are captured through weights
w
i
,
j
,
k
(
A
B
) that encode the number of pairs
of points belonging to
A
and
B
, whose displacement is captured by the walkthrough
,
i
,
j
,
k
(with
i
,
j
,
k
taking values in
{−
1
,
0
,
+
1
}
):
1
K
i
,
j
,
k
·
z
a
)d
b
d
w
i
,
j
,
k
(
A
,
B
)
=
C
i
(
x
b
−
x
a
)
C
j
(
y
b
−
y
a
)
C
k
(
z
b
−
a
,
(3.21)
A
B
where d
b
=
dx
b
dy
b
dz
b
and d
a
=
dx
a
dy
a
dz
a
;
K
i
,
j
,
k
acts as a normalization factor to guarantee
that
w
i
,
j
,
k
takes value in [0
,
1];
C
±
1
(
.
) are the characteristic functions of the positive and
negative real semi-axis (0
) denotes the Dirac's function
that is used to reduce the dimensionality of the integration domain to enable a finite non-null
measure. In particular,
C
±
1
(
t
) are defined in the following way:
,
+∞
) and (
−∞
,
0), respectively;
C
0
(
·
1 f
t
1 f
t
>
0
<
0
C
+
1
(
t
)
=
C
−
1
(
t
)
=
(3.22)
0
otherwise
0
otherwise
,
being the Dirac's function, by definition:
1 f
t
==
0
,
C
0
(
t
)
=
(3.23)
0
otherwise
.
