On the Intrinsic Reconstruction of Shape from its Symmetries

The main question we address is: What is the minimal information required to generate closed, nonintersecting planar boundaries? For this paper, we restrict “shape” to this meaning. More precisely, we examine whether the medial axis, together with dynamics, can serve as a language to design shapes and to effect shape changes, e.g., for modeling, to generate a morph sequence, etc. We represent the medial axis together with a direction of flow along the axis as the shock graph and examine the reconstruction of shape along each of the three types of medial axis points, (labeledA1,A 3 1,A3; see below for theA notation) and the associated six types of shock points. First, we show that the tangent and curvature of the medial axis and the speed and acceleration of the shock with respect to time of propagation, i.e., first and second order geometric and dynamic properties, are sufficient to determine the boundary tangent and curvature at corresponding points of the boundary. This implies that a rather coarse sampling of the symmetry axis, its tangent, curvature, speed, and acceleration is sufficient to regenerate accurately a local neighborhood of shape at regular axis points (A1). We also show how higher order differential properties of the axis can be related to the higher-order differential properties of the boundary of the same order. Second, we examine the reconstruction of shape at branch points (A1) where three regular branches are joined. We show that the three pairs of geometry (that is, curvature) and dynamics (that is, acceleration) must satisfy certain constraints. Finally, we derive similar results for the end points of shock branches (A3 points). These formulas completely specify the local reconstruction of a shape from its shock-graph or medial axis and the conditions required to form a coherent shape from the medial axis.