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analyze the case of perfect symmetry and show it results in a degenerate eigende-
composition. We then resolve this issue and explain why perfect symmetry rarely
appears in real data such as images. The proposed scheme requires no apriori knowl-
edge of the type of symmetry (reflection/rotation) and holds for both. Our scheme,
denoted as Spectral Symmetry Analysis (SSA), can detect partial symmetry and
is robust to outliers. In our last contribution we apply the SSA to the analysis of
symmetry in images, for which we utilize local image features [Low03, SM97] to
represent images as sets of points. For that we utilize image descriptors to reduce
the computational complexity.
The chapter is organized as follows: we start by presenting the geometrical prop-
erties of symmetries in Section 2 and then survey previous results on symmetry
detection, local features and spectral alignment in Section 3. Our approach to sym-
metry analysis is presented in Section 4 and experimentally verified in Section 5.
Concluding remarks are given in Section 6.
2
Symmetries and Their Properties
In this work we study the symmetry properties of sets of points S
= {
x i }
,such
n . The common types of symmetries are the rotational and reflectional
symmetries. Sets having only the first type are described by the cyclic group C K ,
while others have both rotational and reflectional symmetry, and are described by
the dihedral group D K ,where K is an order of the respective symmetry.
In this section we define the rotational (cyclic) and reflectional (dihedral) sym-
metries, denoted C K and D K , respectively. By considering the subsets of points
that x i R
S I
S I
S that are invariant under the corresponding symmetry transforms T C K
and T D K , we are able to recover the rotation centers and reflection axes. These in-
variant sets are shown to be related to the spectral properties of T C K and T D K . Finally,
we derive an analytical relationship between T C K and T D K in a two-dimensional case,
which allows us to infer the rotational symmetry transform T C K , given two reflec-
tional transforms T D K .
,
2.1
Rotational Symmetry
n is rotationally symmetric with a
Definition 1 (Rotational symmetry). AsetS
R
rotational symmetry transform T C K ,
of order K if
x i
S
, ∃
x j
S, s.t.
x j =
T C K x i .
(1)
2 ,T C K
Fo r S
R
is given by
cos
β k
sin
β k 0
x
y
1
,
T C K (
x
,
y
)=
sin
β k
cos
β k
0
(2)
0
0
1
 
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