Non-linear shape approximation via the entropy scale space

©1993 SPIE--The International Society for Optical Engineering. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. The electronic version of this article is the complete one and can be found online at: http://dx.doi.org/10.1117/12.146627DOI: 10.1117/12.146627There are two classical approaches to approximating the shape of objects. The first is based on diffusion and often leads to the (Gaussian) smoothing of contour information. The resulting scale-space may often be viewed as generated by parabolic operators which progressively and globally smooth shapes. The second approach is based on morphological morphology operations which represent the interior of shapes as sets, e.g., a collection of disks. The resulting morphological space can be viewed as being defined via a hyperbolic operator whose weak or viscosity solutions progressively smooth shapes in a local manner. 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. For example, imagine an image of don Quixote holding a lance. Within the entropy scale space the lance can be removed without disturbing the shape of his body. In contrast, if Gaussian smoothing techniques were used, the extent of the lance would effect the amount of distortion introduced to the rest of his body. As such, the entropy scale space is a combination of smoothing due to shocks as "black holes" of information and the subsequent rarefaction wave reconstruction, and the anisotropic diffusion process spreading of contour information. Our technique is numerically stable, and several examples will be shown