Browse Theses

This thesis presents a particular set of spheres as new representation of the shape of a 3D solid. The spheres considered are maximal inscribed spheres in the solid and their centres are chosen in such a way that at most one sphere centre lies in a cubic region of space.

The shape representation proposed is a discretization of the medial surface transform of a solid. Part I of this thesis presents algorithms for the computation of this representation given a boundary representation of a solid by approximating its medial surface transform. Properties of those medial spheres that are not detected by our algorithm in 3D are described and a complete characterization of those medial circles that are not detected by a 2D version of our algorithm is given.

In Part II, recent results from differential geometry are used to compute principal curvatures and principal curvature directions on the boundary of the smooth solid represented using the union of medial spheres. This computation is performed using only the medial sphere centres and a pair of points on each medial sphere that lies on the surface of the solid being modeled. It is shown how the union of medial spheres allows a part-based description of the solid, with a significance measure associated with each part.

In Part III, it is shown that our shape representation can offer a tight volumetric fit to a polyhedron, using a small number of spheres. The spheres used in our representation can be quickly updated as the solid undergoes a certain class of deformations. It is shown how our set of medial spheres allows efficient and accurate proximity queries between polyhedra.