Y. Bérubé-Lauzière, K. Siddiqi, A. Tannenbaum (University of Minnesota), S.W. Zucker The notion of scale for shape is extremely subtle. The observation that features relevant at one level of resolution, e.g., ruffles from individual feathers on a bird's wing, are best described as noise at other levels of resolution, e.g., that at which the global shape of the wing emerges, is intuitive. Yet, formalizing this notion for computer vision has proved quite difficult. A significant advance in this direction was the introduction by Kimia, Tannenbaum and Zucker of a reaction-diffusion space for shape. Here a family of shapes is generated by evolving an initial shape by a partial differential equation. The motion of each point in the normal direction is governed by a prescribed velocity, consisting of two components: a curvature (smoothing) term plus an inflationary (hyperbolic) constant speed term. The idea of such a model is that one wants to play off the diffusive term, which `smoothes' structure, against the hyperbolic term, which produces shocks that are key for representing shape. A drawback of this model is that the diffusive term dominates. In order to obtain a representation in terms of shocks one must shut it off in a rather ad hoc manner after all the structure up to a given scale has been removed. Our solution to this problem is to introduce an entirely hyperbolic flow for which the velocity in the normal direction has a smoothing effect in a specified neighborhood of the initial shape but which gradually decays to a constant outside this neighborhood, after which the motion is equivalent to pure reaction. Hence the transition from smoothing to shock formation is accomplished by a single process, and no ad hoc shutting off of the diffusive term is required. The amount of smoothing is determined by the size of the neighborhood and thus a scale-space for shape is introduced in a natural way. Shocks computed during the pure reaction regime of the flow give a coarse to fine description of the shape.

Mon Apr 7 12:54:24 EDT 1997