Toward a computational theory of shape
Benjamin B. Kimia · eScholarship@McGill (McGill) · 1990
Although the shape of objects is a key to their recognition, viable theories for representing and describing shape have been elusive. We propose a framework that unifies competing approaches to shape. The basis for our approach is an analysis of deformations of shape designed to induce a topology over shapes suitable to support object recognition. We show that deformations classify into constant motion and curvature motion, which intriguingly lead to conservation laws for shape. These conservation laws are nonlinear and lead to singularities. A notion of entropy for shape is developed which limits the singularities of shape to shocks. The formation of shocks and their classification under arbitrary deformations is the basis of our representation for shape. The space of deformations leads to a reaction-diffusion space for shape in which the formation of shocks is studied. This leads us to propose parts, protrusions, and bends as the computational elements fo shape. A notion of scale on these elements then naturally emerges, which is captured by the entropy scale-space. Any particular shape is finally described as the interaction between processes for computing parts, protrusions, and bends, the perceptual reality of which is illustrated via qualitative experiments.