A Symmetry-Based Generative Model for Shape

We propose a novel generative language for shape that is based on the shock graph: given a shock graph topology, we explore constraints on the geometry and dynamics of the shock graph branches at each point required to generate a valid shape, i.e., with no self-intersection, cusps, or crossovers. We model the shape boundary as a piece-wise smooth circular arc spline, which is dense in the space of piecewise smooth curves. Using this model we derive an independent set of parameters which generate a variety of shapes and satisfy the reconstruction constraints. We show simple examples of using this generative model as an active deformable shape and for morphing between two shapes. The results illustrate that it is possible to generate any generic shape with relatively few parameters, further reduced if prior knowledge of shape is available.