A Continuous Shape Descriptor by Orientation Diffusion

Abstract

We propose a continuous description for 2-D shapes that calculates convexity, symmetry and is able to account for size. Convexity and size are known to be critical in deciding figure/ground (F/G) separation, with the study initiated by the Gestalt school [9][11]. However, few quantitative discussions were made before. Thus, we emphasize the convexity/size measurement for the purpose of F/G prediction. A Kullback-Leibler measure is introduced. In addition, the symmetry information is studied through the same platform. All these shape properties are collected for shape representations. Overall, our representations are given in a continuous manner. For convexity measurement, unlike the 1/0 mathematical definition where shapes are categorized as convex or concave, we give a measure describing shapes as “more” or “less” convex than others. In symmetry information (skeleton) retrieval, a 2-D intensity map is provided with the intensity value specifying “strength” of the skeleton. The proposed representations are robust in the sense that small fine-scale perturbations on shape boundaries will cause minor effects on the final representations. All these shape properties are intergrated into one description. To apply to the F/G separation, the shape measure can be flexibly chosen between a size-invariant convexity measure or a convexity measure with the small size preference. The model is established on an orientation diffusion framework, where the local features, served as inputs, are intensity edge locations and their orientations. The approach is a variational one, rooted in a Markov random field (MRF) formulation. A quadratic form is used to assure simplicity and the existence of solution.