ABSTRACT
We now turn to the study of the face as a three-dimensional object. We are interested in features of the face which are invariant under rotations, translations and scaling. This is not just an idle wish for mathematical elegance, but is the only possible way to find meaningful features since there is no distinguished coordinate system in the space around us and faces are always moving to new positions and orientations. The eigenhead approach of Atick et al. (96), for instance, depends on choosing a particular vertical axis through the head and expanding the function giving the distance of points on the face from this axis. In contrast, the principal curvatures of a surface (whose definition will be recalled below) are invariant under all Euclidean coordinate changes and lead to invariant features on every smooth surface. To motivate this excursion into differential geometry, it is helpful to see that they do express something significant about the face: Figure 6.1 shows an example of the two principal curvatures calculated for a face. It is an interesting exercise to 'explain' what these figures show by touching your own face and mentally estimating its principal curvatures at various points. These curvatures and the parabolic and ridge curves to be introduced shortly create a remarkable pattern on the human face and reveal how complex a surface the face really is.
