General Geometric Good Continuation: From Taylor to Laplace via Level Sets

Good continuation is the Gestalt observation that parts often group in particular ways to form coherent wholes. Perceptual integration of edges, for example, involves orientation good continuation, a property which has been exploited computationally very extensively. But more general local-global relationships, such as for shading or color, have been elusive. While Taylor’s Theorem suggests certain modeling and smoothness criteria, the consideration of level set geometry indicates a different approach. Using such first principles we derive, for the first time, a generalization of good continuation to all those visual structures that can be abstracted as scalar functions over the image plane. Based on second order differential constraints that reflect good continuation, our analysis leads to a unique class of harmonic models and a cooperative algorithm for structure inference. Among the different applications of good continuation, here we apply these results to the denoising of shading and intensity distributions and demonstrate how our approach eliminates spurious measurements while preserving both singularities and regular structure, a property that facilitates higher level processes which depend so critically on both of these classes of visual structures.