SHAPES, SHOCKS, AND DEFORMATIONS .1. THE COMPONENTS OF 2-DIMENSIONAL SHAPE AND THE REACTION-DIFFUSION SPACE

We undertake to develop a general theory of two-dimensional shape by e lucidating several principles which any such theory should meet. The p rinciples are organized around two basic intuitions: first, if a bound ary were changed only slightly, then, in general, its shape would chan ge only slightly. This leads us to propose an operational theory of sh ape based on incremental contour deformations. The second intuition is that not all contours are shapes, but rather only those that can encl ose ''physical'' material. A theory of contour deformation is derived from these principles, based on abstract conservation principles and H amilton-Jacobi theory. These principles are based on the work of Sethi an (1985a, c), the Osher-Sethian (1988), level set formulation the cla ssical shock theory of Lax (1971; 1973), as well as curve evolution th eory for a curve evolving as a function of the curvature and the relat ion to geometric smoothing of Gage-Hamilton-Grayson (1986; 1989). The result is a characterization of the computational elements of shape: d eformations, parts, bends, and seeds, which show where to place the co mponents of a shape. The theory unifies many of the diverse aspects of shapes, and leads to a space of shapes (the reaction/diffusion space) , which places shapes within a neighborhood of ''similar'' ones. Such similarity relationships underlie descriptions suitable for recognitio n.