A PhD thesis @ NALAG

TW 2004_01

Evelyne Vanraes
Powell-Sabin splines and multiresolution techniques

Advisor: Adhemar Bultheel

Abstract

Powell-Sabin splines are piecewise quadratic polynomials with global C¹ continuity. In contrast to the widely used tensor product B-splines and NURBS, they are defined on triangular patches which has certain advantages. In this dissertation we place Powell-Sabin splines in a multiresolution context. Multiresolution techniques are concerned with the generation, representation and manipulation of geometric objects at different levels of detail. The main ingredient here is a subdivision scheme to compute a representation of a surface on a refined triangulation. Such a refined triangulation has more vertices and more and smaller triangles than the original triangulation. The new basis functions after subdivision have smaller support and give the designer more local control. It makes it possible to represent a surface on different levels of detail. A standard dyadic scheme cannot be used for non uniform triangulations where the vertices are not regularly spaced. Instead we propose a sqrt(3) scheme. Applying this scheme twice results in a triadic scheme. When going back from a fine resolution level to a coarser resolution level, fine detail is lost. We develop a wavelet decomposition algorithm that transforms a fine scale surface into a coarse scale approximation and a detail part that lives in a complement space. We use the lifting paradigm with the triadic subdivision scheme as prediction step. Because the domain triangles need not to be uniform, we pay special attention to stability in the design of the update step. The multiresolution techniques for splines can only be applied in the functional case or when the object is defined as a parametric surface. We propose an extension of a dyadic subdivision algorithm voor uniform Powell-Sabin splines to arbitrary topologies. In this setting a surface is defined procedurally as the limit of a refinement process.