Fluctuating Distance Fields, Parts, Three-Partite Skeletons

Shapes are continuous objects, even when they are drawn on digital media or composed using finite number of elements. As such, they defy analytic approach; explicitization of their parts, hierarchies, skeletons, or even centroids is ill-posed. I describe a novel approach to perceptually organize shapes and explicate their features without being negligent of their continuous nature. The basic construct is an unusual phase field that can be conceived in a number of varying ways using varying mathematical machinery, so highlighting the field itself rather than how it is being computed. Connections among the field, Mumford-Shah and Tari-Shah-Pien models, and reaction-diffusion equation suggest that the field may bridge low-level and high-level visual processing. 20.1 Shapes Are Continuous Shapes are continuous objects, even when they are drawn on digital media or composed using finite number of elements. In favor of the argument, Gestalt school produced numerous illustrations, e.g., (1) a finite set of isolated dots placed at a fixed distance from a fixed location creates a circle percept; (2) a peanut shape prevailingly reveals two discrete blobs or approaches to a blob, as its neck thins or thickens; (3) two squares juxtaposed to form an eight-pointed star gives way to new parts, namely eight triangles and an octagon. S. Tari ( ) Middle East Technical University, 06531, Ankara, Turkey e-mail: [email protected] M. Breuß et al. (eds.), Innovations for Shape Analysis, Mathematics and Visualization, DOI 10.1007/978-3-642-34141-0 20, © Springer-Verlag Berlin Heidelberg 2013 439