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(a)
(b)
(c)
(d)
Fig. 6 Rotational and reflectional symmetries.
( a ) The eigenvalues
λ i
( b )
The symmetric
transform corresponding to
λ 2
( c ) The symmetric transform corresponding to
λ 3
( d )
The
symmetric transform corresponding to λ 4
as can be seen in Fig. 7b. We analyzed the P
15 eigenvectors corresponding to the
P leading eigenvalues, and applied the norm test to the corresponding transforms
{
=
P
1 . The norms
P
1 are depicted in Fig. 7a and reveal that only T C 1
T C i }
{
(
T C i ) }
det
and T C 2
correspond to symmetries. the other transformations are pruned. As both
det
are close to 1, both are rotations. The corresponding point
alignments and rotation centers are shown in Figs. 7c and 7d.
Figure 8 takes us further with a real life object, where despite the object being
both rotationally and reflectionally symmetric, only reflectional symmetries are de-
tected, due to the lack of appropriate local descriptors. The detected reflectional
alignments corresponding to the three largest eigenvalues are shown in Figs. 8b-8e.
Examining
(
T C 1 )
and det
(
T C 2 )
15
i
4 does not reveal the expected rotational symmetry transforms.
Examining the image it is evident that the geometrically corresponding points show
dissimilar intensities. Namely, the SSA was able to detect pseudo-symmetry. Apat-
tern of geometrical symmetry lacking the corresponding intensity symmetry. This
property is intriguing, as most human viewers would indeed consider the Pentagon
a symmetric object. By applying Theorem 2 we are able to infer the rotational sym-
metry based on the detected reflections.
Figure 9 demonstrates successful symmetry detection in the presence of heavily
cluttered background by applying the transformation pruning measure. Figure 10 is
another example (as the Pentagon in Fig. 8) of a pseudo-symmetric object whose
symmetry is far from being perfect and can be argued by some. Background clutter
and low repeatability of interest regions challenge the symmetry detection algorithm
{
T C i }
=
 
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