A Graph Matching Approach to Symmetry Detection and Analysis - Computational Intelligence Paradigms in Advanced Pattern Classification - page 120

Three-dimensional symmetry was analyzed in [KCD + 02, MSHS06]. The scheme
computes a reflectional symmetry descriptor that measures the amount of reflec-
tional symmetry of 3D volumes, for all planes through the center of mass. The de-
scriptor maps any 3D volume to a sphere, where each point on the sphere represents
the amount of symmetry in the object with respect to the plane perpendicular to the
direction of the point. As each point on the sphere also represents the integration
over the entire volume, the descriptor is resilient to noise and to small variations be-
tween objects. We show that our approach is directly applicable to three-dimensional
meshes.
SIFT local image features [Low03] were applied to symmetry analysis by Loy
and Eklundh in [LE06]. In their scheme, a set of feature points is detected over
the image, and the corresponding SIFT descriptors are computed. Feature points
are then matched in pairs by the similarity of their SIFT descriptors. These local
pairwise symmetries are then agglomerated by a Hough voting space of symmetry
axes. The vote of each pair in the Hough domain is given by a weight function that
measures the discrepancy in the dominant angles and scales [Low03] of the feature
points. As the SIFT descriptors are not reflection invariant, reflections are handled
by mirroring the SIFT descriptors.
In contrast, our scheme is based on a spectral relaxation of the self alignment
problem. It recovers the self assignment directly. Thus, we avoid the quantization the
Hough space, and our scheme can be applied, as is, to analyzing higher dimensional
data and will not suffer the curse dimensionality manifested by a density (voting)
estimation scheme, such as the Hough transform. Also, our scheme does not require
a local symmetry measure, such as the dominant angle, and is purely geometric. It
can be applied with any local feature, such as correlators and texture descriptors
[OPM02].
The work of Hays et al. in [HLEL06] is of particular interest to us, as it combines
the use of local image descriptors and spectral high-order assignment for transla-
tional symmetry analysis. Translational symmetry is a problem in texture analysis,
where one aims to identify periodic or near-regular repeating textures, commonly
known as lattices. Hays et al. propose to detect translational symmetry, by detecting
feature points and computing a single, high order, spectral self-alignment. The as-
signments are then locally pruned and regularized using thin-plate spline warping.
The corresponding motion field is elastic and nearly translational, hence the term
translational symmetry.
In contrast, our scheme deals with rotational and reflectional symmetries where
the estimated self-alignments relate to rotational motion. The core of our work is the
analysis of multiple self assignments and their manifestation via multiple eigenvec-
tors and eigenvalues. Moreover, based on the spectral properties of geometric trans-
form operators, we introduce a global assignment pruning measure, able to detect
erroneous self-assignments. This comes out to be essential in analyzing symmetries
in real images, which are often embedded in clutter.