Subdivision schemes in geometric modelling

Abstract

Subdivision schemes are efficient computational methods for the design and representation of 3D surfaces of arbitrary topology. They are also a tool for the generation of refinable functions, which are instrumental in the construction of wavelets. This paper presents various flavours of subdivision, seasoned by the personal viewpoint of the authors, which is mainly concerned with geometric modelling. Our starting point is the general setting of scalar multivariate nonstationary schemes on regular grids. We also briefly review other classes of schemes, such as schemes on general nets, matrix schemes, nonuniform schemes and nonlinear schemes. Different representations of subdivision schemes, and several tools for the analysis of convergence, smoothness and approximation order are discussed, followed by explanatory examples.

Normal Multi-scale Transforms for Curves

“…We start with recalling known facts for linear subdivision schemes [4,8]. Let the dyadic S be given by a locally supported mask a = (a k ) k∈Z of scalar coefficients, i.e., the action of S on a scalar sequence x = (x i ) i∈Z is given by…”

Optimality of multilevel preconditioning for nonconforming P1 finite elements

“…The remainder of this subsection is devoted to the verification of (20). There is a standard way of dealing with such recursive inequalities which has been developed for studying regularity of subdivision schemes and vector refinement equations (see, e.g., [5,11,12,20]). The basic ingredients are as follows.…”

“…The main new ingredient is a H 1+ Bernstein inequality for the IPOs which replaces the H 1 Bernstein estimates used previously, and allows to get an optimal upper spectral bound for the preconditioners under consideration. Verifying such Bernstein inequalities requires subtle techniques from the regularity theory of subdivision methods and refinable functions (for general expositions on these research areas, see [5,11,12,20]). …”

From approximating to interpolatory non-stationary subdivision schemes with the same generation properties

“…Two important areas where interpolatory subdivision schemes play a crucial role are Computer Aided Geometric Design (CAGD) and wavelets construction (see [11] and [19], respectively). In these fields a fundamental issue that recently emerged is concerned with the study of numerical algorithms for converting known approximating schemes into new interpolatory ones.…”