Three-Dimensional Shape Description Using the Symmetric Axis Transform I: Theory

Abstract

Blum's two-dimensional shape description method based on the symmetric axis transform (SAT) is generalized to three dimensions. The method uniquely decomposes an object into a collection of sub-objects each drawn from three separate, but not completely independent, primitive sets defined in the paper: width primitives, based on radius function properties; axis primitives, based on symmetric axis curvatures; and boundary primitives, based on boundary surface curvatures. Width primitives are themselves comprised of two components: slope districts and curvature districts. Visualizing the radius function as if it were the height function of some mountainous terrain, each slope district corresponds to a mountain face together with the valley below it. Curvature districts further partition each slope district into regions that are locally convex, concave, or saddle-like. Similarly, axis (boundary) primitives are regions of the symmetric surface where the symmetric surface (boundary surfaces) are locally convex, concave, or saddle-like. Relations among the primitive sets are discussed.