Graphics Reference
In-Depth Information
w(g 1 ,g 1 )
w(g 1 ,g 2 )
w(g 1 ,g 3 )
w(p 1 ,p 1 )
w(p 1 ,p 2 )
w(p 1 ,p 3 )
D (w(p 1 ,p 1 ),w(g 1 ,g 1 ))
D (w(p 1 ,p 2 ),w(g 1 ,g 2 ))
D (w(p 1 ,p 3 ),w(g 1 ,g 3 ))
D (w(p 1 ,p 4 ),w(g 1 ,g 4 ))
p 1
g 1
p 2
g 2
p 3
g 3
.
.
.
p 9
g 9
Figure 3.35 The first nine iso-geodesic stripes for two sample face scans. The graphs constructed on a
subset of the stripes and their matching are also shown
where the first summation in Equation 3.27 accounts for the intrastripe 3DWWs similarity
measure, and the second summation evaluates the interstripe 3DWWs similarity measure. The
α
parameter permits to weight differently the two distance components and its value has been
set to 0
3 (Berretti et al., 2010). This value has been tuned in a pilot set of experiments carried
out on the Face Recognition Grand Challenge version 1.0 (FRGC v1.0) (Phillips et al., 2005)
database and shows that to support face recognition, interstripe spatial relationships are more
discriminant than intrastripe spatial relationships.
Implicitly, Equation 3.27 assumes that the number of nodes N P in the graph P , is not greater
than the number of nodes N G of the graph G . This can be assumed with no loss of generality,
in that if N P >
.
N G , graphs P and G can be exchanged.
Following the considerations discussed earlier, distances between faces of two individuals
are measured by computing the 3DWW for each pair of iso-geodesic stripes, separately in the
three face parts, and then comparing the 3DWWs of homologous pairs of the two faces. The
final dissimilarity measure is obtained by averaging distances in the three parts.
According to Equation 3.27, the overall runtime complexity can be estimated as O ( N P T D ),
being T D the complexity in computing
, that is estimated to be a constant value (Berretti
et al., 2010). This permits efficient implementation for face identification in large datasets,
also with the use of appropriate index structures, with great savings in performance. More
details on the index structure and its performance are discussed in Berretti et al. (2001).
D
Exercises
1. Let S 1 , S 2 , and S 3 be regular surfaces, Prove that
A. If
φ 1
φ
: S 1
S 2 is an isometry, then
: S 2
S 1 is also an isometry.
B. If
φ
: S 1
S 2 is an isometry,
φ
: S 1
S 3 are isometries, then,
ψ φ
: S 1
S 3 is
an isometry.
C. Prove that the isometry of regular surface S constitutes a group. This group is called
the group of isometries .
2. Compute square-root velocity function of the circle
x
=
cos 2 t
,
y
=
sin 2 t
,
t
[0
,
2
π
]
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