On the intrinsic reconstruction of shape from its symmetries

We address the issue of the use of symmetry-based representations, such as the medial axis and an augmented form of it, the shock structure, to regenerate shapes. First, we address pointwise reconstruction of the boundary from points of the medial axis. As classified into three generic types (A²/1 mid-branch, A₃ end point of a branch, and A₁³ junction). Second, we examine the intrinsic reconstruction of shape when differential properties of the axis are also available. We show the surprising result that the tangent and curvature of the medial axis, coupled with the speed and acceleration of the shock flowing along the´s axis, i.e., first and second order properties, are sufficient to determine the boundary tangents and curvatures at corresponding points of the boundary. This implies that for a rather coarse sampling of the symmetry axis, the location together with its tangent, curvature: speed, and acceleration is sufficient to accurately regenerate a local neighborhood of shape at this point. Together with reconstruction properties at junction (A³₁) and end points (A₃), these results lead to the full intrinsic regeneration of a shape from a representation of it as a directed planar graph (where the links represent curvature and acceleration functions, and where the nodes contain tangent and speed information): a representation ideally suited for the design and manipulation of free-form shape