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The purpose of the work is to study the possibil-
ity of image comparison using information about the
symmetry. Figure 1 shows examples of axes for both
perfectly symmetrical images (a figure with a Jaccard
measure equal to 1) and asymmetric ones. In identi-
cal images, the axes of symmetry will be located on
the figure in approximately the same place. For al-
most similar images (for example, images of butter-
flies with opened wings), the axis of symmetry will
always be located along the body. Thus, on the found
axis of symmetry, two images of butterflies will be
correctly combined, and it will be necessary to cal-
culate the measure of similarity. In this work we use
the Jaccard similarity as a measure of the similarity of
two images. We will not use the comparison of parts
of one image, but two different shapes – A and B:
µ(A, B) =
|
S(A) ∩ S(B)
|
|
S(A) ∪ S(B)
|
. (2)
The quality of the obtained measure and its appli-
cation for image recognition are also investigated.
2 RELATED WORKS
The task of symmetry detection and symmetry mea-
sure evaluation for 2D shapes is well-known, and
there are many effective methods for its solution
based on: 1) Fourier series expansion of paramet-
ric contour representation (Van Otterloo, 1988), 2)
contour representation by turning function (Sheynin
et al., 1999), 3) contour representation by critical
points and computation of similarity measure for two
sub-contours via vectors of geodesic distances (Yang
et al., 2008), 4) model of Electrical Charge Distri-
bution on the Shape (ECDS) (Li et al., 2014), 5)
Boundary-Skeleton Function (BSF) (Niu et al., 2015),
6) pair-wise comparison of sub-sequences of skeleton
primitives (Kushnir et al., 2016), 7) Fourier descriptor
of the image contour (Mestetskiy and Zhuravskaya,
2019), 8) image gradient (Sun and Si, 1999).
However, there are a few works devoted to the use
of the symmetry information in other applications,
such as image comparison. In particular, (Hauagge
and Snavely, 2012) describes the use of local symme-
tries of architectural structures to compare images of
buildings.
3 COMPARISON OF IMAGES
In this paper, we propose to use information about the
symmetry of shapes to compare images. We will rely
on the procedures for finding the axis of symmetry
proposed in previous works. To achieve reliability
in the verification of the proposed procedure, an ex-
act algorithm for determining the reflection symmetry
of binary raster images (Kushnir et al., 2016) and its
parallel version (Fedotova et al., 2017) will be used,
requiring a complete search of all potential axes of
symmetry. Calculations were carried out on a super-
computer, however, in the works (Kushnir et al., 2016;
Kushnir et al., 2019) methods of significant accelera-
tion of the computational procedure were developed.
This algorithm searches for the axis of symmetry
by iterating over all possible lines passing through a
pair of points on the contour of the figure. The value
of symmetry is calculated with respect to each line.
The line with respect to which the symmetry measure
is maximal is considered to be the symmetry axis.
To estimate the value of symmetry, the Jaccard mea-
sure is used, which shows the degree of similarity of
two sets. The sets are the pixels of the binary image.
When searching for a symmetry measure, the image
is mirrored relative to the selected line and overlays
the original one. Consequently, the areas of intersec-
tion and union of two sets are formed, and the Jaccard
measure is calculated as their ratio (1). It is worth not-
ing that the axes of symmetry found with the help of
the Jaccard measure do not always coincide with the
visual assessment by a person, a discussion is given
in Section 7 of the paper (Kushnir et al., 2016).
After the symmetry axes are found for the pair of
images being compared, it is proposed to align the
axes in the images with each other and calculate the
Jaccard measure (2), which will show the value of
similarity, as shown in figure 2.
Figure 2: Example of intersection of two images (black
color in the central shape indicates the intersection zone,
blue and green-areas that do not match).
It is obvious that two different shapes can be dif-
ferent sizes, rotated through a certain angle and bi-
ased. Just using information about the location of the
axes of symmetry we will achieve invariance to the
shift, rotation and scale.
In similar images, the axes of symmetry will be
in the shape approximately the same. Knowing the
location of the axis of symmetry, we can calculate
the affine transformation, which will allow to over-
lay (match) one image on another, aligning their axes
of symmetry.
Three points are required to calculate the affine