Symmetrization of Feature Points in 2-D Images #

: In this work, we consider the symmetrization problem, that is the problem to obtain more accurate information about location of points based on a priori knowledge of their symmetries. Methods to solve the symmetrization problem with respect to vertical and inclined axes of reflectional symmetry are considered jointly with the more general symmetrization with respect to an indefinite reflection axis. Then the case of rotational symmetry is considered. The methods produce the minimal deformation that enhances approximate symmetries present in a given arrangement of points


Introduction
Symmetry is a central concept in many natural and man-made objects and plays a crucial role in visual perception, design and engineering. Several recent efforts in shape analysis have focused on detecting symmetries in 2-D and 3-D shapes [1]- [3]. Numerous applications have successfully utilized this type of information, e.g., for model reduction [3], scan completion [4], segmentation [5], shape matching [1], etc. In many cases lowlevel symmetry analysis is based on investigations of so-called feature points, whose exact meaning depends on the resolving problem. One of the most common problems, where methods of "refinement by symmetry" can be efficiently used, is the biometrical identification, when the correct location of feature points is crucial. In particular, the accuracy of human face detection and recognition strongly depends on the measurement precision of pupils of eyes location [6]. Then the fact of near, but imperfect, reflective symmetry of human full-faces can be used to improve the accuracy [7]. In fact, usually methods of human face detection are based on the position analysis of several dozens feature points, which are either coupled in pairs, symmetric with respect to a vertical axis, or situated in the axis. In this work, we present several methods to obtain more accurate information about location of feature points, based on a priori knowledge of their symmetry. Note that positions of points in images are always known with some drift that depends on such factors as the image quality, noise levels in a vicinity of the points, the processing algorithm, and so on. As a result, evaluated coordinates could fail the symmetry conditions even for those points, which are in fact symmetric. So it is reasonable to use the information about symmetry to specify positions of feature points. Besides, the symmetrization itself should be done with minimal deformation of points positions that enhances approximate symmetries present in a given arrangement of points. The presented methods produce such optimal arrangements of feature points under reflectional and rotational symmetries.  Fig. 1, whose left hand part demonstrates a set of feature points, the central part shows its partition into classes R P , L P , O P , and the right hand part shows the feature points after symmetrization. We associate with the ordered set P the following 2ndimensional vector:

Symmetrization with Respect to a Vertical Axis
To complete the solution, note that To symmetrize a point from O P , we need to zeroize its xcoordinate, and leave its y-coordinate without changes.

Symmetrization with Respect to an Arbitrary Axis
Now, let the reflection axis be defined by the equation y ax b  , (where 0 a  ), as it is shown in Fig. 2. Then the symmetrization problem can be solved by switching to a new coordinate system Namely, assume that the y -axis of the new system coincides with the reflection axis, the origin O is in the point, where y -axis crosses Oy , and Ox  is orthogonal with Oy  in such a way that   O x y    is a "right-handed" system. Let ( , ) xy be coordinates of an arbitrary point in the "old"   Oxy -system, and let ( , ) xy  be coordinates of the same point where the values of cos and sin follow from the condition cot 0 a     , so that 2   It converts the symmetrization with respect to an arbitrary axis into the previously solved problem. Indeed, we may symmetrize X  in the same way, as it has been done in the previous section, and then return the symmetrized vector s X  back into the  

Symmetrization with Respect to an Unknown Axis
Assume now, that the parameters a and b of a reflection symmetry axis are unknown, though the existence of such axis follows from the nature of the problem under solution. In this section we use the previously developed methods to determine the parameters in such a way that symmetrization could be achieved by the minimal deformation. For this we need the useful notations: is the resulting vector of coordinates of symmetrized points. In other words, to achieve the optimal symmetrization we need to find the parameters  , b , and the minimizing vector Y .
Let   ,, F Y b  be the expression under the minimization. Writing it in a matrix form and using the minimization criteria, we would get the following equations: A series of appropriate calculations and transformations produce the system The first (matrix) equation gives us the symmetrized vector in relation with  and b . The last two scalar equations can be used to find the parameters  and b of the reflection axis.
To write down the final solution, assume that the vector X is partitioned into the "centralized" blocks   are averages of x-and y-coordinates of the evaluated feature points. Then, after some tedious transformations of the last two equations of the system, the next result follows: Substitute the evaluated parameters  and b into the first equation of the system, we can find the symmetrization AY Z  of the original vector X . As the last step of the solution, the symmetrized vector should be transformed into the original "old" coordinate system. An example of symmetrization with an unknown axis is given in Fig. 3. Distorted feature points with respect to the original axis are shown in its upper right-hand part, while its lower left part shows the reconstructed axis and the result of symmetrization with respect to it. The two axes are compared in the lower righthand part of Fig. 3. and note that all such X fill the space 2n R . Using matrices, the rotation symmetry condition can be written as p . So the second factor in the expression for s X above may be considered as the "center of the cloud" with average coordinates. Multiplication by the block matrix produce the optimal rotational symmetrized arrangement of feature points.

Conclusion
The considered problem of symmetrization of characteristic points relative to the axial and rotational symmetry has numerous applications, since it is the simplest and most common type of symmetry. We have shown how to symmetrize feature points of an image with respect to vertical and inclined axes of reflectional symmetry and considered the more general symmetrization with respect to an indefinite reflection axis. Together with it the case of rotational symmetry is considered. It is worth to mention that all the methods produce the minimal deformation that enhances approximate symmetries present in a given arrangement of points. However, there are also other types of symmetry -dihedral, translational and others, that can be met in the tasks related to image processing. The particular interest is the symmetrization under affine distortions, which occur in most real developments. A rigorous mathematical solution of these problems may not be easy, but it will provide additional opportunities for high-quality image processing.