Horizontal Symmetry Projection

Once the symmetry points have been computed for dark symmetry at the appropriate scale, we project the symmetry points along the horizontal. There is no need to consider non-horizontal symmetry points since the mouth is mainly a horizontally oriented limb. The symmetry maps derived by Equation  are projected using Equation . Recall that we compute only 8 values of $\psi \in \{0^{\circ}, 22.5^{\circ}, 45^{\circ}, 67.5^{\circ}, 90^{\circ}, 112.5^{\circ}, 135^{\circ}, 157.5^{\circ} \}$ (where $0^{\circ}$ is aligned with the horizontal).

$$S_{r_{horizontal}}(p) = 2 \times S_{r,0^{\circ}} + S_{r,22.5^{\circ}} + S_{r,157.5^{\circ}}$$ (3.2)

Thus, the 4 dimensional symmetry data $S_{r, \psi}(p)$ 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).

  

Figure 3.17: Horizontal Projection of Symmetry Points

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 $5\times1$ 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.