Three - Dimensional Shape Description Using

Blum's transform, variously known as the symmetric axis transform, medial axis transform, or skeleton, and his associated two-dimensional shape description methodology are generalized to three-dimensions. Bookstein's twodimensional algorithm for finding an approximation to the symmetric axis is also generalized. The symmetric axis (SA) of an object with a smooth boundary is the locus of points inside the object having at least two nearest neighbors on the object boundary. In three dimensions, the SA is, in general, a collection of smooth surface patches, called simplified segments, connected together in a tree-like structure. Together with the radius function, the distance from each point on the SA to a nearest boundary point, the SA forms the symmetric axis transform. The three-dimensional symmetric axis transform defines a unique, coordinatesystem-independent decomposition of an object into disjoint, two-sided pieces, each with its own simplified segment and associated object boundary patches. Four principal contributions are presented. {1) A relationship among the Gaussian and mean curvatures of a simplified segment, the Gaussian and mean curvatures of the associated object boundary patches, and radius function measures 1s derived. {2) A further decomposition is proposed wherein each two-sided piece is partitioned into primitives drawn from three separate, but not completely independent, primitive sets: width primitives, boundary primitives, and axis primitives. Width primitives are regions derived from derivatives of the radius function; hence, they capture the behavior of the boundary patches with respect to the simplified segment. Axis and boundary primitives are regions of constant signs of Gaussian and mean curvatures of the simplified segment and boundary patches respectively. The aforementioned curvature relationship is used to derive relationships among the primitive sets. (3) In the course of studying width primitives, it is proved that, under certain non-degeneracy assumptions, the regions of the graph defined by the critical points, ridge lines, and course lines of a scalar valued function over a surface have one of three types of cycle as boundary~ (4) An almost linear algorithm that takes a polyhedral approximation to a three-dimensional object and yields a polyhedral surface approximation to that object's SA is developed.