Thus, the 4 dimensional symmetry data
is reduced to 3
dimensions in
*S*_{rhorizontal}(*p*). The projected symmetry (or axial
symmetry) maps for *r*=1 to *r*=6 are displayed in Figure .
These maps are derived from the rotated intensity image in
Figure (c).

The 6 scales (*r*=1 to *r*=6) form our axial symmetry scale-space. The scale
or *r* represents the vertical thickness of the horizontal symmetries detected
in the image. A thin, closed mouth usually would generate a line of symmetry
points at *r*=1. An open mouth, on the other hand, will generate a cloud of
points at a larger *r* within its center. An open mouth's extremities taper
off (since it is closed on both ends) regardless of its size. Thus, the
mouth's extremities will appear as clouds at small *r*.

Note that the symmetry points in Figure are also blurred horizontally with a Gaussian window to improve connectivity and reduce small gaps. This allows us to use a linking procedure that connects the discrete points in the maps to form connected structures.