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being the integration along y independent from x and z. Then, by applying the usual integration rules,
it results in the following:
L
+
T
T
2
K
+
1
,
−
1
,
+
1
D
+
T
D
+
T
w
+
1
,
−
1
,
+
1
(
a
n
,
b
m
)
=
(
L
+
T
−
x
a
)d
x
a
d
z
b
d
z
a
=
D
z
a
L
(
L
+
T
)
x
a
−
2
L
+
T
D
+
T
D
+
T
T
2
K
+
1
,
−
1
,
+
1
x
a
=
d
z
b
d
z
a
=
L
D
z
a
T
2
K
+
1
,
−
1
,
+
1
D
+
T
D
+
T
T
2
2
=
d
z
b
d
z
a
=
D
z
a
D
+
T
T
2
K
+
1
,
−
1
,
+
1
T
2
2
1
K
+
1
,
−
1
,
+
1
T
6
4
.
=
(
D
+
T
−
z
a
)d
z
a
=
D
To obtain an adimensional value, the normalization factor K
+
1
,
−
1
,
+
1
is posed equal to the products of
the volumes of the two voxels, that is:
|
a
n
||
b
m
|=
T
6
. This results in a final value for the coefficient
w
+
1
,
−
1
,
+
1
(
a
n
,
b
m
)
=
1
/
4
. A similar approach can be used in the derivation of the other coefficients.
The properties of 3DWW, joined with the geodesic distance computation and the face
partitioning provide the method with robustness to expression variations. In fact, geodesic
distances between two facial points keep sufficiently stable under expression changes resulting
into the fact that the large majority of the points of each stripe still remain within the same stripe,
even under facial expression changes. In addition, because of the constrained elasticity of the
skin tissue, neighbor points can be assumed to feature very similar motion for moderate facial
expressions in most parts of the face. For all these points the mutual displacement between the
two points is mainly determined by the geometry of the neutral face. This property is preserved
by 3DWWs that provide an integral measure of displacements between pairs of points.
3.9.3 Face Representation and Matching Using Iso-geodesic Stripes
A generic face model
F
, is represented through a set of
N
F
stripes. In that 3DWWs are
computed for every pair of iso-geodesic stripes (including the pair composed by a stripe and
itself), a face is represented by a set of
N
F
·
2 relationship matrixes. According
to the proposed representation, iso-geodesic stripes and 3DWW computed between pairs of
stripes (
interstripe
3DWW) and between each stripe and itself (
intrastripe
3DWW), have been
cast to a graph representation where
intrastripe
3DWW are used to label the graph nodes and
interstripe
3DWWs to label the graph edges (see Figure 3.35).
To compare graph representations, distance measures for node labels and for edge labels
have been defined. Both of them rely on the
L
1
distance measure
(
N
F
+
1)
/
defined between 3DWWs
(Berretti et al., 2006). The similarity measure between two face models represented through
the graphs
P
and
G
with nodes
p
k
and
g
k
, is then derived as follows:
D
N
P
N
P
·
μ
(
P
,
G
)
=
1
D
(
w
(
p
k
,
p
k
)
,
w
(
g
k
,
g
k
))
(3.27)
k
=
N
P
k
−
1
)
N
P
(
N
P
−
2(1
−
α
+
1)
·
1
D
(
w
(
p
k
,
p
h
)
,
w
(
g
k
,
g
h
))
,
k
=
1
h
=
