The notion of similarity can be naturally formalized if the distal shape space and the internal representation space are both endowed with metric structure, in which case (dis)similarity in each space can be derived from the appropriate distance function. Now, to demonstrate the existence of a metric for a given shape family (i.e., a subspace of the shape space), it suffices to exhibit at least one parameterization scheme common to all the shapes in the family. A metric on shapes can then be defined simply as the Euclidean distance in the underlying parameter space [IMAGE ]. Some examples of objects produced using a common parameterization for a family of animal-like shapes are shown in Figure 7.
[IMAGE ]
Figure: Four shapes jointly parameterized by a set of 57
parameters [Manolache and Edelman, 1993]. The possibility of
such parameterization for a given shape family is a prerequisite for
its veridical representation, as argued in
section 4.3.