Stationary Subdivision and Multiresolution Surface Representations

Abstract

Stationary subdivision is an important tool for generating smooth free-form surfaces used in CAGD and computer graphics. One of the challenges in the construction of subdivision schemes for arbitrary meshes is to guarantee that the surfaces produced by the algorithm are $C\sp1$-continuous. First results in this direction were obtained only recently. In this thesis we derive necessary and sufficient criteria for $C\sp{k}$-continuity that generalize and extend most known conditions. We present a new method for analysis of smoothness of subdivision which allows us to analyze subdivision schemes which do not generate surfaces admitting closed-form parameterization on regular meshes, such as the Butterfly scheme and schemes with modified rules for tagged edges. The theoretical basis for analysis of subdivision that we develop allows us to suggest methods for constructing new subdivision schemes with improved behavior. We present a new interpolating subdivision scheme based on the Butterfly scheme, which generates $C\sp1$-continuous surfaces from arbitrary meshes. We describe a multiresolution representation for meshes based on subdivision. Combining subdivision and the smoothing algorithms of Taubin, allows us to construct a set of algorithms for interactive multiresolution editing of complex hierarchical meshes of arbitrary topology.