Furthermore, the center of cocircularity has an orientation associated with
it. This orientation is determined by the phase of the cocircular edges
contributing to this center of cocircularity. The phase value of an edge is
the orientation of the normal of the edge. This normal points along the
direction of the change of intensity from dark to bright. As displayed in
Figure , the edges have phases
and which are computed from the Sobel edge detection. The symmetry orientation at
the center of cocircularity, ,
is the line that bisects the normals of
the two edges. Equation returns the value of .

The lines of symmetry are formed by linking all centres of circularity found in the image. In other words, circles are constructed from all pairs of cocircular edges and their centers (the centers of cocircularity) are used to trace out lines of symmetry. For each point

Furthermore, the centers of cocircularity can be assigned different strengths
depending upon the orientation of the edges and the intensity of the edges
that contribute to forming them. For each point in the image *p*, at each
scale or radius *r* and for each symmetry orientation
we find the set
of cocircular pairs of edges
.
Sela defines the
magnitude of symmetry in the (*p*, *r*, )
space as follows:

where and are the edge intensities of the two circular edges and is the angle separating their normals:

(2.3) |

Cocircular edges with a larger value of , increasingly face and oppose each other and a stronger sense of symmetry is perceived at the point of cocircularity,

Thus, the magnitude of the symmetry
at each point *p*, at
each radius *r* and at each orientation
is obtained and represents the
desired ``lines of symmetry''. It is possible to combine the lines of symmetry
from multiple radii so that an overall, *r*-independent value of
is found as follows:

Note that these lines of symmetry are not really lines. Rather, we compute a
symmetry magnitude at each combination of *p*, *r* and
so the result is
a set of points with an orientation value. If true connected lines are
required, these discrete points must be linked into curves using their
orientation and scale value (see Chapter 3).