Nonlinear shape approximation via the entropy scale space

There are two classical approaches to approximating the shape of objects. We are developing a general theory of shape which unifies these two different approaches in the entropy scale space. The theory is 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 novel theory of contour deformation is derived from these intuitions, based on abstract conservation principles and the Hamilton-Jacobi theory. The result is a characterization of the computational elements of shape: protrusions, parts, bends, and seeds (which show where to place the components of a shape); and leads to a space of shapes (the reaction-diffusion space) which places shapes within a neighborhood of `similar' ones. Previously, these elements of shape have been used for description. We now show how they can be used to generate another space for shapes, the entropy scale space, which is obtained from the reaction-diffusion space by running the `reaction' portion of the equations `backwards' in time. As a result distinct components of a shape can be removed by introducing a minimal disturbance to the remainder of the shape. Our technique is numerically stable, and several examples are shown.