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a given scenario than any other pattern. A person will first focus his attention on an
object with symmetry than other objects in a picture [EWC00].
Rotational and reflectional symmetries are the most common types of symme-
tries. An object is said to have rotational symmetry of order
K
if it is invariant under
rotations of
2
K
1 about a point denoted the symmetry center. Whereas
an object has reflectional symmetry if it is invariant under a reflection transformation
about a line, denoted the reflection axis. Figure 1 presents both types of symmetry.
k
,
k
=
0
...
K
−
(a)
(b)
Fig. 1
Rotational and reflectional symmetries.
(
a
)
Rotational symmetry of order eight with-
out reflectional symmetry.
(
b
)
Reflectional symmetry of order one.
The problem of symmetry detection and analysis has been studied by many
researches [CPML07, DG04, KCD
+
02, KG98, LE06, Luc04, RWY95, SIT01,
SNP05]. Most of them deal with two-dimensional symmetries, while a few ana-
lyze three-dimensional data [KCD
+
02]. A recent survey [PLC
+
08] by Chen
et al.
found that despite the significant research effort made, there is still a need for a
robust, widely applicable “symmetry detector”.
We propose an effective scheme for detection and analysis of rotational and re-
flectional symmetries in
n
-dimensions. The scheme is based on the self-alignment
of points using a spectral relaxation technique as proposed by Leordeanu in [LH05].
Our core contribution is to show that the symmetry of a sets of points
S
n
is man-
ifested by a multiplicity of the leading eigenvalues and corresponding eigenvectors.
This leads to a purely geometric symmetry detection approach that only utilizes the
coordinates
of the set
S
, and can thus be applied to abstract sets of points in
∈
R
n
.
Given the eigendecomposition, we show how to recover the intrinsic properties of
reflectional and rotational symmetries (center of rotation, point correspondences,
symmetry axes). In our second contribution, we derive a geometrical pruning mea-
sure, by representing the alignments as geometrical transforms in
R
n
and enforcing
a norm constraint. This allows us to reject erroneous matchings, and analyze real
data, where symmetries are often embedded in clutter. In our third contribution, we
R
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