A Graph Matching Approach to Symmetry Detection and Analysis - Computational Intelligence Paradigms in Advanced Pattern Classification - page 119

difference of the Fourier magnitude in Cartesian coordinates along rays. The his-
togram's maxima correspond to the direction of the zero crossing. In [KS06], Keller
et al. extended Lucchese's work, by applying the PseudoPolar Fourier transform to
computing algebraically-accurate line integral in the Fourier domain. The symmetry
resulted in a periodic pattern in the line integral result. This was detected by spectral
analysis (MUSIC). These algorithms are by nature global, being able to effectively
detect fully symmetric images, such as synthetic symmetric patterns. Yet, some of
them [KS06], struggle at detecting small localized symmetric objects embedded in
clutter.
The frequency domain was also utilized by Lee et al. in [LCL08], where Frieze-
expansions were applied to the input image. Thus converting planar rotational sym-
metries into periodic one-dimensional signals, whose period corresponds to the
order of the symmetry. This period is estimated by recovering the maxima of the
Fourier spectrum.
Recent work emphasizes the use of local image features. The local information
is then agglomerated to detected the global symmetry. Reisfeld et al. [RWY95] sug-
gested a low-level, operator for interest points detection where symmetry is con-
sidered a cue. This symmetry operator constructs the symmetry map of the image
by computing an edge map, where the magnitude and orientation of each edge de-
pend on the symmetry associated with each of its pixels. The proposed operator is
able to process different symmetry scales, enabling it to be used in multi-resolution
schemes.
A related approach was presented in [LW99], where both reflectional and ro-
tational symmetries can be detected, even under a weak perspective projection. A
Hough transform is used to derive the symmetry axes from edge contours. A refine-
ment algorithm discards erroneous symmetry axes by imposing geometrical con-
straints using a voting scheme.
An approach related to our work was introduced in [ZPA95], where the symmetry
is analyzed as a symmetry of a set of points. For an object, given by a sequence of
points, the symmetry distance is defined as the minimum distance in which we need
to move the points of the original object in order to obtain a symmetric object.
This also defines the symmetry transform of an object as the symmetric object that
is closest to the given one. This approach requires finding point correspondences,
which is in often difficult, and an exhaustive search over all potential symmetry axes
is performed.
Shen et al. [SIT01] used an affine invariant feature vector, computed over a set of
interest points. The symmetry was detected by analyzing the cross-similarity matrix
of this vectors. Rotational and reflectional symmetries can be analyzed by finding
the loci corresponding to its minima.
The gradient vector flow field was used in [PY04] to compute a local feature
vector. For each point, its location, orientation, and magnitude were retained. Local
features in the form of Taylor coefficients of the field were computed and a hashing
algorithm is then applied to detect pairs of points with symmetric fields, while a
voting scheme is used to robustly identify the location of symmetry axis.