Salient Shape Selection and Representation

Courant Institute of Mathematical Sciences

New York University

We propose a new framework for shape representation and scenery shape selection. Various topics including figure/ground separation, shape axis construction, junction detection and illusory figure finding will be discussed.

The model construction is inspired by the Gestalt studies. They suggest proximity, convexity, symmetry, etc, as cues for figure/ground separation and visual organization. By our distributed systems, we quantify those attributes for complete/partial shapes and use them for shape evaluations and representations. In particular, the shape convexity instead of other well-studied shape attributes such as the symmetry axis or size, will be emphasized.

Two models are proposed. The decay diffusion process is applied in predicting figure/ground phenomenon, based on a convexity measure for figure/ground sharing the same area. The orientation diffusion process, adopting orientation information on shape boundaries/edges, will discuss the figure/ground separation or shape convexity comparison for regions not owning the same size. A Kullback-Leibler convexity measure is proposed, with a flexible scenario. Through a parameter, we are allowed to choose between a size-invariant convexity measure or one with small-size preference. For convexity comparison of perfectly convex shapes, a preference of circles over triangles will be given, as well as the preference of squares over rectangles. These two models are also used in generating the symmetry information. In particular, the symmetry information suggested by the orientation process is computed by only local operations. The junction information will be derived similarly, where junctions are considered no more than ``boundary axis points''.

Our framework, based on variational formulations will produce the static-state results. The simulation is continuous, rely on no artificial binary thresholds. For convexity measurement, other than the mathematical 0-1 definition, we distinguish between ``more'' or ``less'' convex shapes. For axis construction, we provide the information which continuously describes strength of the axes for natural axis pruning. For junction detection, the transition from low-curvature or high-curvature curves to curves with a discontinuous curvature will be seen.

The decay diffusion process, with help of the convexity/entropy measure will also be applied in shape selection. Hence, our framework integrally combining many different functions is useful as a universal low- to middle-level vision simulation.