How relevant is local approximate preservation of distance ranks, offered by a quasiconformal distal-to-proximal mapping M, to the representation of real-world shapes? A part of the answer to this question emerges from a consideration of the hierarchical structure of perceived categories. Numerous studies in cognitive science reveal that in the hierarchical structure of object categories there exists a certain level, called basic level, which is the most salient according to a variety of criteria [Rosch et al., 1976]. Taking as an example the hierarchy quadruped, dog, terrier, the basic level is that of dog. Objects whose recognition implies less detailed distinctions than those required for basic-level categorization are said to belong to a superordinate level. A reasonable assumption is that faithful representation of similarities is required within superordinate categories (e.g., within quadruped: giraffe to camel, leopard, horse), and, of course, within basic categories, but not between superordinate categories. In other words, the informativeness (and, indeed, the well-posedness) of the statement that a giraffe is more similar to a banana than to a beetle is questionable.
The above considerations suggest that the locality constraint per se does not preclude basing representation upon the principle of conformality of the distal-to-proximal mapping M. The issue at hand is, therefore, practical: for a randomly chosen point in the distal shape space, how large is the domain over which the distortion coefficient q of the mapping M is likely to be close enough to 1? Theoretical analysis and computer simulations [Duvdevani-Bar and Edelman, 1995], as well as psychophysical data [Edelman, 1995b,Cutzu and Edelman, 1995], suggest that the answer to that question is, probably, ``large enough.''
[IMAGE ]
Figure: The new Pandemonium. The computational properties of the
proposed framework for object categorization are discussed in
section 5.1). Note that, unlike in the
original Pandemonium, here the outcome depends on the distribution
of responses of the ``cognitive demons'' (recognition modules tuned
to individual ``reference'' objects), and not merely on the identity
of the demon which responds the strongest.