A Graph Matching Approach to Symmetry Detection and Analysis - Computational Intelligence Paradigms in Advanced Pattern Classification - page 117

2.3 Interrelations between Rotational and Reflectional

Symmetries

Theorem 1. If a set S has rotational symmetry of order K, then it either has reflec-
tional symmetry of order K or has no reflectional symmetry at all [Cox69, Wey52].
If S has both rotational and reflectional symmetry then the axes of reflectional sym-
metry are given by

1
2 β k ,

α n = α 0 + k = 0 ,..., K − 1 , (9)

where α 0 is the angle of one of the reflection axes, and β k are the angles of rotational

symmetry.

Theorem 2. Given two distinct reflectional transforms T D 1 and T D 2 , one can recover

the corresponding symmetrical rotational transforms T C K

T C K = T D 1 · T D 2 (10)

The proof is given In Appendix A.

2.4 Discussion

The geometrical properties presented above, pave the way for a computational
scheme for the analysis of a given set of prospective symmetry transforms

.
By imposing Eqs. 3 and 7, erroneous transforms can be discarded, and the spec-
tral analysis, given in Eqs. 4 and 8 can be used to recover the center of rotational
symmetry and the axis of the reflectional one. Moreover, based on Theorem 2, one
can start with a pair of reflection transforms

{ T i }

and recover the rotational
transform T C K . Note that in practice, the prospective transforms

{ T D 1 , T D 2 }

, are computed
by the spectral matching algorithm discussed in Section 4. That procedure provides
an alternative geometrical approach to recovering the symmetry centers and axes.
Using both methods, we are able to cross-validate the results.
The norm tests of the symmetrical transforms (Eqs. 3 and 7), can be applied to
higher dimensional data sets, as well as the spectral analysis in Eqs. 4 and 8. The
equivalent in three-dimensional data is given by Euler's theorem [TV98] .

{ T i }

3 Previous Work

This section overviews previous work related to our scheme. Section 3.1 provides
a survey of recent results in symmetry analysis, while Section 3.2 presents the no-
tion of Local Features that is used to represent an image as a set of salient points.
A combinatorial formulation of the alignment of sets of points in R n is discussed
in Section 3.3, and a computationally efficient solution, via spectral relaxation is