3-D shape interrogation by medial axis transform

The Medial Axis Transform is a representation of an object which has been shown to be useful in design, interrogation, animation, finite element mesh generation, performance analysis, manufacturing simulation, path planning, and tolerance specification. In this thesis, the theory of the Medial Axis Transform for 3-D objects and an algorithm to compute such transforms are considered. For objects with piecewise $C\sp2$ boundaries, relationships between the curvature of the boundary and the position of the Medial Axis are developed. For n-dimensional submanifolds of $\Re\sp{n}$ with boundaries which are piecewise $C\sp2$ and completely $G\sp1$, a deformation retract is set up between each object and its Medial Axis, which demonstrates that if the object is path connected, then so is its Medial Axes. In addition, it is proven that path connected polyhedral solids without cavities have path connected Medial Axis. An algorithm for determining the Medial Axis Transform is developed first for convex 3-D polyhedral solids and then extended to general 3-D polyhedral solids of arbitrary genus without cavities, with non-convex vertices and edges. The algorithms are based on a classification scheme which relates different pieces of the Medial Axis to one another even in the presence of degenerate Medial Axis points. Vertices of the Medial Axis are connected to one another by tracing along adjacent edges, and finally the faces of the Axis are found by traversing closed loops of vertices and edges. Systems of governing equations for different types of Medial Axis points are established using the classification scheme, and the computation of solutions of these systems is discussed. Representation of the Medial Axis and associated radius function is addressed, and pseudocode for the algorithms is given along with recommended optimizations. A connectivity theorem is proven to show the completeness of the algorithm. A further extension of the algorithm to objects with curved boundaries is also outlined. Complexity estimates and stability analysis for the polyhedral algorithms are presented. Finally, examples illustrate the computational properties of the algorithm for convex and non-convex 3-D polyhedral solids with polyhedral holes. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)