Shapes, Shocks, and Deformations I: The Components of Two-Dimensional Shape and the Reaction-Diffusion Space

We undertake to develop a general theory of two-dimensional shape by elucidating several principles which any such theory should meet. The principles are organized around two basic intuitions: first, if a boundary were changed only slightly, then, in general, its shape would change only slightly. This leads us to propose an operational theory of shape based on incremental contour deformations. The second intuition is that not all contours are shapes, but rather only those that can enclose "physical" material. A theory of contour deformation is derived from these principles, based on abstract conservation principles and Hamilton-Jacobi theory. These principles are based on the work of Sethian [82, 86], the Osher-Sethian level set formulation [65], the classical shock theory of Lax [53, 54], as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of Gage-Hamilton-Grayson [32, 37]. The result is a characterization of th..